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rfc:rfc8017

Internet Engineering Task Force (IETF) K. Moriarty, Ed. Request for Comments: 8017 EMC Corporation Obsoletes: 3447 B. Kaliski Category: Informational Verisign ISSN: 2070-1721 J. Jonsson

                                                             Subset AB
                                                              A. Rusch
                                                                   RSA
                                                         November 2016
        PKCS #1: RSA Cryptography Specifications Version 2.2

Abstract

 This document provides recommendations for the implementation of
 public-key cryptography based on the RSA algorithm, covering
 cryptographic primitives, encryption schemes, signature schemes with
 appendix, and ASN.1 syntax for representing keys and for identifying
 the schemes.
 This document represents a republication of PKCS #1 v2.2 from RSA
 Laboratories' Public-Key Cryptography Standards (PKCS) series.  By
 publishing this RFC, change control is transferred to the IETF.
 This document also obsoletes RFC 3447.

Status of This Memo

 This document is not an Internet Standards Track specification; it is
 published for informational purposes.
 This document is a product of the Internet Engineering Task Force
 (IETF).  It represents the consensus of the IETF community.  It has
 received public review and has been approved for publication by the
 Internet Engineering Steering Group (IESG).  Not all documents
 approved by the IESG are a candidate for any level of Internet
 Standard; see Section 2 of RFC 7841.
 Information about the current status of this document, any errata,
 and how to provide feedback on it may be obtained at
 http://www.rfc-editor.org/info/rfc8017.

Moriarty, et al. Informational [Page 1] RFC 8017 PKCS #1 v2.2 November 2016

Copyright Notice

 Copyright (c) 2016 IETF Trust and the persons identified as the
 document authors.  All rights reserved.
 This document is subject to BCP 78 and the IETF Trust's Legal
 Provisions Relating to IETF Documents
 (http://trustee.ietf.org/license-info) in effect on the date of
 publication of this document.  Please review these documents
 carefully, as they describe your rights and restrictions with respect
 to this document.  Code Components extracted from this document must
 include Simplified BSD License text as described in Section 4.e of
 the Trust Legal Provisions and are provided without warranty as
 described in the Simplified BSD License.

Moriarty, et al. Informational [Page 2] RFC 8017 PKCS #1 v2.2 November 2016

Table of Contents

 1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   4
   1.1.  Requirements Language . . . . . . . . . . . . . . . . . .   5
 2.  Notation  . . . . . . . . . . . . . . . . . . . . . . . . . .   6
 3.  Key Types . . . . . . . . . . . . . . . . . . . . . . . . . .   8
   3.1.  RSA Public Key  . . . . . . . . . . . . . . . . . . . . .   8
   3.2.  RSA Private Key . . . . . . . . . . . . . . . . . . . . .   9
 4.  Data Conversion Primitives  . . . . . . . . . . . . . . . . .  11
   4.1.  I2OSP . . . . . . . . . . . . . . . . . . . . . . . . . .  11
   4.2.  OS2IP . . . . . . . . . . . . . . . . . . . . . . . . . .  12
 5.  Cryptographic Primitives  . . . . . . . . . . . . . . . . . .  12
   5.1.  Encryption and Decryption Primitives  . . . . . . . . . .  12
     5.1.1.  RSAEP . . . . . . . . . . . . . . . . . . . . . . . .  13
     5.1.2.  RSADP . . . . . . . . . . . . . . . . . . . . . . . .  13
   5.2.  Signature and Verification Primitives . . . . . . . . . .  15
     5.2.1.  RSASP1  . . . . . . . . . . . . . . . . . . . . . . .  15
     5.2.2.  RSAVP1  . . . . . . . . . . . . . . . . . . . . . . .  16
 6.  Overview of Schemes . . . . . . . . . . . . . . . . . . . . .  17
 7.  Encryption Schemes  . . . . . . . . . . . . . . . . . . . . .  18
   7.1.  RSAES-OAEP  . . . . . . . . . . . . . . . . . . . . . . .  19
     7.1.1.  Encryption Operation  . . . . . . . . . . . . . . . .  22
     7.1.2.  Decryption Operation  . . . . . . . . . . . . . . . .  25
   7.2.  RSAES-PKCS1-v1_5  . . . . . . . . . . . . . . . . . . . .  27
     7.2.1.  Encryption Operation  . . . . . . . . . . . . . . . .  28
     7.2.2.  Decryption Operation  . . . . . . . . . . . . . . . .  29
 8.  Signature Scheme with Appendix  . . . . . . . . . . . . . . .  31
   8.1.  RSASSA-PSS  . . . . . . . . . . . . . . . . . . . . . . .  32
     8.1.1.  Signature Generation Operation  . . . . . . . . . . .  33
     8.1.2.  Signature Verification Operation  . . . . . . . . . .  34
   8.2.  RSASSA-PKCS1-v1_5 . . . . . . . . . . . . . . . . . . . .  35
     8.2.1.  Signature Generation Operation  . . . . . . . . . . .  36
     8.2.2.  Signature Verification Operation  . . . . . . . . . .  37
 9.  Encoding Methods for Signatures with Appendix . . . . . . . .  39
   9.1.  EMSA-PSS  . . . . . . . . . . . . . . . . . . . . . . . .  40
     9.1.1.  Encoding Operation  . . . . . . . . . . . . . . . . .  42
     9.1.2.  Verification Operation  . . . . . . . . . . . . . . .  44
   9.2.  EMSA-PKCS1-v1_5 . . . . . . . . . . . . . . . . . . . . .  45
 10. Security Considerations . . . . . . . . . . . . . . . . . . .  47
 11. References  . . . . . . . . . . . . . . . . . . . . . . . . .  48
   11.1.  Normative References . . . . . . . . . . . . . . . . . .  48
   11.2.  Informative References . . . . . . . . . . . . . . . . .  48

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 Appendix A.  ASN.1 Syntax . . . . . . . . . . . . . . . . . . . .  54
   A.1.  RSA Key Representation  . . . . . . . . . . . . . . . . .  54
     A.1.1.  RSA Public Key Syntax . . . . . . . . . . . . . . . .  54
     A.1.2.  RSA Private Key Syntax  . . . . . . . . . . . . . . .  55
   A.2.  Scheme Identification . . . . . . . . . . . . . . . . . .  57
     A.2.1.  RSAES-OAEP  . . . . . . . . . . . . . . . . . . . . .  57
     A.2.2.  RSAES-PKCS-v1_5 . . . . . . . . . . . . . . . . . . .  60
     A.2.3.  RSASSA-PSS  . . . . . . . . . . . . . . . . . . . . .  60
     A.2.4.  RSASSA-PKCS-v1_5  . . . . . . . . . . . . . . . . . .  62
 Appendix B.  Supporting Techniques  . . . . . . . . . . . . . . .  63
   B.1.  Hash Functions  . . . . . . . . . . . . . . . . . . . . .  63
   B.2.  Mask Generation Functions . . . . . . . . . . . . . . . .  66
     B.2.1.  MGF1  . . . . . . . . . . . . . . . . . . . . . . . .  67
 Appendix C.  ASN.1 Module . . . . . . . . . . . . . . . . . . . .  68
 Appendix D.  Revision History of PKCS #1  . . . . . . . . . . . .  76
 Appendix E.  About PKCS . . . . . . . . . . . . . . . . . . . . .  77
 Acknowledgements  . . . . . . . . . . . . . . . . . . . . . . . .  78
 Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  78

1. Introduction

 This document provides recommendations for the implementation of
 public-key cryptography based on the RSA algorithm [RSA], covering
 the following aspects:
 o  Cryptographic primitives
 o  Encryption schemes
 o  Signature schemes with appendix
 o  ASN.1 syntax for representing keys and for identifying the schemes
 The recommendations are intended for general application within
 computer and communications systems and as such include a fair amount
 of flexibility.  It is expected that application standards based on
 these specifications may include additional constraints.  The
 recommendations are intended to be compatible with the standards IEEE
 1363 [IEEE1363], IEEE 1363a [IEEE1363A], and ANSI X9.44 [ANSIX944].
 This document supersedes PKCS #1 version 2.1 [RFC3447] but includes
 compatible techniques.
 The organization of this document is as follows:
 o  Section 1 is an introduction.
 o  Section 2 defines some notation used in this document.

Moriarty, et al. Informational [Page 4] RFC 8017 PKCS #1 v2.2 November 2016

 o  Section 3 defines the RSA public and private key types.
 o  Sections 4 and 5 define several primitives, or basic mathematical
    operations.  Data conversion primitives are in Section 4, and
    cryptographic primitives (encryption-decryption and signature-
    verification) are in Section 5.
 o  Sections 6, 7, and 8 deal with the encryption and signature
    schemes in this document.  Section 6 gives an overview.  Along
    with the methods found in PKCS #1 v1.5, Section 7 defines an
    encryption scheme based on Optimal Asymmetric Encryption Padding
    (OAEP) [OAEP], and Section 8 defines a signature scheme with
    appendix based on the Probabilistic Signature Scheme (PSS)
    [RSARABIN] [PSS].
 o  Section 9 defines the encoding methods for the signature schemes
    in Section 8.
 o  Appendix A defines the ASN.1 syntax for the keys defined in
    Section 3 and the schemes in Sections 7 and 8.
 o  Appendix B defines the hash functions and the mask generation
    function (MGF) used in this document, including ASN.1 syntax for
    the techniques.
 o  Appendix C gives an ASN.1 module.
 o  Appendices D and E outline the revision history of PKCS #1 and
    provide general information about the Public-Key Cryptography
    Standards.
 This document represents a republication of PKCS #1 v2.2 [PKCS1_22]
 from RSA Laboratories' Public-Key Cryptography Standards (PKCS)
 series.

1.1. Requirements Language

 The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
 "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
 document are to be interpreted as described in [RFC2119].

Moriarty, et al. Informational [Page 5] RFC 8017 PKCS #1 v2.2 November 2016

2. Notation

 The notation in this document includes:
    c              ciphertext representative, an integer between 0 and
                   n-1
    C              ciphertext, an octet string
    d              RSA private exponent
    d_i            additional factor r_i's CRT exponent,
                   a positive integer such that
                     e * d_i == 1 (mod (r_i-1)), i = 3, ..., u
    dP             p's CRT exponent, a positive integer such that
                     e * dP == 1 (mod (p-1))
    dQ             q's CRT exponent, a positive integer such that
                     e * dQ == 1 (mod (q-1))
    e              RSA public exponent
    EM             encoded message, an octet string
    emBits         (intended) length in bits of an encoded message EM
    emLen          (intended) length in octets of an encoded message
                   EM
    GCD(. , .)     greatest common divisor of two nonnegative integers
    Hash           hash function
    hLen           output length in octets of hash function Hash
    k              length in octets of the RSA modulus n
    K              RSA private key
    L              optional RSAES-OAEP label, an octet string
    LCM(., ..., .) least common multiple of a list of nonnegative
                   integers

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    m              message representative, an integer between 0 and
                   n-1
    M              message, an octet string
    mask           MGF output, an octet string
    maskLen        (intended) length of the octet string mask
    MGF            mask generation function
    mgfSeed        seed from which mask is generated, an octet string
    mLen           length in octets of a message M
    n              RSA modulus, n = r_1 * r_2 * ... * r_u , u >= 2
    (n, e)         RSA public key
    p, q           first two prime factors of the RSA modulus n
    qInv           CRT coefficient, a positive integer less than
                   p such that q * qInv == 1 (mod p)
    r_i            prime factors of the RSA modulus n, including
                   r_1 = p, r_2 = q, and additional factors if any
    s              signature representative, an integer between 0 and
                   n-1
    S              signature, an octet string
    sLen           length in octets of the EMSA-PSS salt
    t_i            additional prime factor r_i's CRT coefficient, a
                   positive integer less than r_i such that
                     r_1 * r_2 * ... * r_(i-1) * t_i == 1 (mod r_i) ,
                   i = 3, ... , u
    u              number of prime factors of the RSA modulus, u >= 2
    x              a nonnegative integer
    X              an octet string corresponding to x
    xLen           (intended) length of the octet string X

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    0x             indicator of hexadecimal representation of an octet
                   or an octet string: "0x48" denotes the octet with
                   hexadecimal value 48; "(0x)48 09 0e" denotes the
                   string of three consecutive octets with hexadecimal
                   values 48, 09, and 0e, respectively
    \lambda(n)     LCM(r_1-1, r_2-1, ... , r_u-1)
    \xor           bit-wise exclusive-or of two octet strings
    \ceil(.)       ceiling function; \ceil(x) is the smallest integer
                   larger than or equal to the real number x
    ||             concatenation operator
    ==             congruence symbol; a == b (mod n) means that the
                   integer n divides the integer a - b
 Note: The Chinese Remainder Theorem (CRT) can be applied in a non-
 recursive as well as a recursive way.  In this document, a recursive
 approach following Garner's algorithm [GARNER] is used.  See also
 Note 1 in Section 3.2.

3. Key Types

 Two key types are employed in the primitives and schemes defined in
 this document: RSA public key and RSA private key.  Together, an RSA
 public key and an RSA private key form an RSA key pair.
 This specification supports so-called "multi-prime" RSA where the
 modulus may have more than two prime factors.  The benefit of multi-
 prime RSA is lower computational cost for the decryption and
 signature primitives, provided that the CRT is used.  Better
 performance can be achieved on single processor platforms, but to a
 greater extent on multiprocessor platforms, where the modular
 exponentiations involved can be done in parallel.
 For a discussion on how multi-prime affects the security of the RSA
 cryptosystem, the reader is referred to [SILVERMAN].

3.1. RSA Public Key

 For the purposes of this document, an RSA public key consists of two
 components:
       n        the RSA modulus, a positive integer
       e        the RSA public exponent, a positive integer

Moriarty, et al. Informational [Page 8] RFC 8017 PKCS #1 v2.2 November 2016

 In a valid RSA public key, the RSA modulus n is a product of u
 distinct odd primes r_i, i = 1, 2, ..., u, where u >= 2, and the RSA
 public exponent e is an integer between 3 and n - 1 satisfying
 GCD(e,\lambda(n)) = 1, where \lambda(n) = LCM(r_1 - 1, ..., r_u - 1).
 By convention, the first two primes r_1 and r_2 may also be denoted p
 and q, respectively.
 A recommended syntax for interchanging RSA public keys between
 implementations is given in Appendix A.1.1; an implementation's
 internal representation may differ.

3.2. RSA Private Key

 For the purposes of this document, an RSA private key may have either
 of two representations.
 1.  The first representation consists of the pair (n, d), where the
     components have the following meanings:
          n       the RSA modulus, a positive integer
          d       the RSA private exponent, a positive integer
 2.  The second representation consists of a quintuple (p, q, dP, dQ,
     qInv) and a (possibly empty) sequence of triplets (r_i, d_i,
     t_i), i = 3, ..., u, one for each prime not in the quintuple,
     where the components have the following meanings:
          p      the first factor, a positive integer
          q      the second factor, a positive integer
          dP     the first factor's CRT exponent, a positive integer
          dQ     the second factor's CRT exponent, a positive integer
          qInv   the (first) CRT coefficient, a positive integer
          r_i    the i-th factor, a positive integer
          d_i    the i-th factor's CRT exponent, a positive integer
          t_i    the i-th factor's CRT coefficient, a positive integer
 In a valid RSA private key with the first representation, the RSA
 modulus n is the same as in the corresponding RSA public key and is
 the product of u distinct odd primes r_i, i = 1, 2, ..., u, where u
 >= 2.  The RSA private exponent d is a positive integer less than n
 satisfying
    e * d == 1 (mod \lambda(n)),
 where e is the corresponding RSA public exponent and \lambda(n) is
 defined as in Section 3.1.

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 In a valid RSA private key with the second representation, the two
 factors p and q are the first two prime factors of the RSA modulus n
 (i.e., r_1 and r_2); the CRT exponents dP and dQ are positive
 integers less than p and q, respectively, satisfying
    e * dP == 1 (mod (p-1))
    e * dQ == 1 (mod (q-1)) ,
 and the CRT coefficient qInv is a positive integer less than p
 satisfying
    q * qInv == 1 (mod p).
 If u > 2, the representation will include one or more triplets (r_i,
 d_i, t_i), i = 3, ..., u.  The factors r_i are the additional prime
 factors of the RSA modulus n.  Each CRT exponent d_i (i = 3, ..., u)
 satisfies
    e * d_i == 1 (mod (r_i - 1)).
 Each CRT coefficient t_i (i = 3, ..., u) is a positive integer less
 than r_i satisfying
    R_i * t_i == 1 (mod r_i) ,
 where R_i = r_1 * r_2 * ... * r_(i-1).
 A recommended syntax for interchanging RSA private keys between
 implementations, which includes components from both representations,
 is given in Appendix A.1.2; an implementation's internal
 representation may differ.
 Notes:
 1.  The definition of the CRT coefficients here and the formulas that
     use them in the primitives in Section 5 generally follow Garner's
     algorithm [GARNER] (see also Algorithm 14.71 in [HANDBOOK]).
     However, for compatibility with the representations of RSA
     private keys in PKCS #1 v2.0 and previous versions, the roles of
     p and q are reversed compared to the rest of the primes.  Thus,
     the first CRT coefficient, qInv, is defined as the inverse of q
     mod p, rather than as the inverse of R_1 mod r_2, i.e., of
     p mod q.
 2.  Quisquater and Couvreur [FASTDEC] observed the benefit of
     applying the CRT to RSA operations.

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4. Data Conversion Primitives

 Two data conversion primitives are employed in the schemes defined in
 this document:
 o  I2OSP - Integer-to-Octet-String primitive
 o  OS2IP - Octet-String-to-Integer primitive
 For the purposes of this document, and consistent with ASN.1 syntax,
 an octet string is an ordered sequence of octets (eight-bit bytes).
 The sequence is indexed from first (conventionally, leftmost) to last
 (rightmost).  For purposes of conversion to and from integers, the
 first octet is considered the most significant in the following
 conversion primitives.

4.1. I2OSP

 I2OSP converts a nonnegative integer to an octet string of a
 specified length.
 I2OSP (x, xLen)
 Input:
    x        nonnegative integer to be converted
    xLen     intended length of the resulting octet string
 Output:
       X corresponding octet string of length xLen
 Error:  "integer too large"
 Steps:
    1.  If x >= 256^xLen, output "integer too large" and stop.
    2.  Write the integer x in its unique xLen-digit representation in
        base 256:
           x = x_(xLen-1) 256^(xLen-1) + x_(xLen-2) 256^(xLen-2) + ...
           + x_1 256 + x_0,
        where 0 <= x_i < 256 (note that one or more leading digits
        will be zero if x is less than 256^(xLen-1)).

Moriarty, et al. Informational [Page 11] RFC 8017 PKCS #1 v2.2 November 2016

    3.  Let the octet X_i have the integer value x_(xLen-i) for 1 <= i
        <= xLen.  Output the octet string
           X = X_1 X_2 ... X_xLen.

4.2. OS2IP

 OS2IP converts an octet string to a nonnegative integer.
 OS2IP (X)
 Input:  X octet string to be converted
 Output:  x corresponding nonnegative integer
 Steps:
    1.  Let X_1 X_2 ... X_xLen be the octets of X from first to last,
        and let x_(xLen-i) be the integer value of the octet X_i for 1
        <= i <= xLen.
    2.  Let x = x_(xLen-1) 256^(xLen-1) + x_(xLen-2) 256^(xLen-2) +
        ...  + x_1 256 + x_0.
    3.  Output x.

5. Cryptographic Primitives

 Cryptographic primitives are basic mathematical operations on which
 cryptographic schemes can be built.  They are intended for
 implementation in hardware or as software modules and are not
 intended to provide security apart from a scheme.
 Four types of primitive are specified in this document, organized in
 pairs: encryption and decryption; and signature and verification.
 The specifications of the primitives assume that certain conditions
 are met by the inputs, in particular that RSA public and private keys
 are valid.

5.1. Encryption and Decryption Primitives

 An encryption primitive produces a ciphertext representative from a
 message representative under the control of a public key, and a
 decryption primitive recovers the message representative from the
 ciphertext representative under the control of the corresponding
 private key.

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 One pair of encryption and decryption primitives is employed in the
 encryption schemes defined in this document and is specified here:
 RSA Encryption Primitive (RSAEP) / RSA Decryption Primitive (RSADP).
 RSAEP and RSADP involve the same mathematical operation, with
 different keys as input.  The primitives defined here are the same as
 Integer Factorization Encryption Primitive using RSA (IFEP-RSA) /
 Integer Factorization Decryption Primitive using RSA (IFDP-RSA) in
 IEEE 1363 [IEEE1363] (except that support for multi-prime RSA has
 been added) and are compatible with PKCS #1 v1.5.
 The main mathematical operation in each primitive is exponentiation.

5.1.1. RSAEP

 RSAEP ((n, e), m)
 Input:
       (n, e) RSA public key
       m message representative, an integer between 0 and n - 1
 Output:  c ciphertext representative, an integer between 0 and n - 1
 Error:  "message representative out of range"
 Assumption:  RSA public key (n, e) is valid
 Steps:
    1.  If the message representative m is not between 0 and n - 1,
        output "message representative out of range" and stop.
    2.  Let c = m^e mod n.
    3.  Output c.

5.1.2. RSADP

 RSADP (K, c)
 Input:
       K RSA private key, where K has one of the following forms:
       +  a pair (n, d)

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       +  a quintuple (p, q, dP, dQ, qInv) and a possibly empty
          sequence of triplets (r_i, d_i, t_i), i = 3, ..., u
       c ciphertext representative, an integer between 0 and n - 1
 Output:  m message representative, an integer between 0 and n - 1
 Error:  "ciphertext representative out of range"
 Assumption:  RSA private key K is valid
 Steps:
    1.  If the ciphertext representative c is not between 0 and n - 1,
        output "ciphertext representative out of range" and stop.
    2.  The message representative m is computed as follows.
        a.  If the first form (n, d) of K is used, let m = c^d mod n.
        b.  If the second form (p, q, dP, dQ, qInv) and (r_i, d_i,
            t_i) of K is used, proceed as follows:
            i.   Let m_1 = c^dP mod p and m_2 = c^dQ mod q.
            ii.  If u > 2, let m_i = c^(d_i) mod r_i, i = 3, ..., u.
            iii. Let h = (m_1 - m_2) * qInv mod p.
            iv.  Let m = m_2 + q * h.
            v.   If u > 2, let R = r_1 and for i = 3 to u do
                 1.  Let R = R * r_(i-1).
                 2.  Let h = (m_i - m) * t_i mod r_i.
                 3.  Let m = m + R * h.
    3.  Output m.
 Note: Step 2.b can be rewritten as a single loop, provided that one
 reverses the order of p and q.  For consistency with PKCS #1 v2.0,
 however, the first two primes p and q are treated separately from the
 additional primes.

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5.2. Signature and Verification Primitives

 A signature primitive produces a signature representative from a
 message representative under the control of a private key, and a
 verification primitive recovers the message representative from the
 signature representative under the control of the corresponding
 public key.  One pair of signature and verification primitives is
 employed in the signature schemes defined in this document and is
 specified here: RSA Signature Primitive, version 1 (RSASP1) / RSA
 Verification Primitive, version 1 (RSAVP1).
 The primitives defined here are the same as Integer Factorization
 Signature Primitive using RSA, version 1 (IFSP-RSA1) / Integer
 Factorization Verification Primitive using RSA, version 1 (IFVP-RSA1)
 in IEEE 1363 [IEEE1363] (except that support for multi-prime RSA has
 been added) and are compatible with PKCS #1 v1.5.
 The main mathematical operation in each primitive is exponentiation,
 as in the encryption and decryption primitives of Section 5.1.
 RSASP1 and RSAVP1 are the same as RSADP and RSAEP except for the
 names of their input and output arguments; they are distinguished as
 they are intended for different purposes.

5.2.1. RSASP1

 RSASP1 (K, m)
 Input:
    K        RSA private key, where K has one of the following forms:
             - a pair (n, d)
             - a quintuple (p, q, dP, dQ, qInv) and a (possibly empty)
               sequence of triplets (r_i, d_i, t_i), i = 3, ..., u
    m        message representative, an integer between 0 and n - 1
 Output:
    s        signature representative, an integer between 0 and n - 1
 Error:  "message representative out of range"
 Assumption:  RSA private key K is valid

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 Steps:
    1.  If the message representative m is not between 0 and n - 1,
        output "message representative out of range" and stop.
    2.  The signature representative s is computed as follows.
        a.  If the first form (n, d) of K is used, let s = m^d mod n.
        b.  If the second form (p, q, dP, dQ, qInv) and (r_i, d_i,
            t_i) of K is used, proceed as follows:
            1.  Let s_1 = m^dP mod p and s_2 = m^dQ mod q.
            2.  If u > 2, let s_i = m^(d_i) mod r_i, i = 3, ..., u.
            3.  Let h = (s_1 - s_2) * qInv mod p.
            4.  Let s = s_2 + q * h.
            5.  If u > 2, let R = r_1 and for i = 3 to u do
                a.  Let R = R * r_(i-1).
                b.  Let h = (s_i - s) * t_i mod r_i.
                c.  Let s = s + R * h.
    3.  Output s.
 Note: Step 2.b can be rewritten as a single loop, provided that one
 reverses the order of p and q.  For consistency with PKCS #1 v2.0,
 however, the first two primes p and q are treated separately from the
 additional primes.

5.2.2. RSAVP1

 RSAVP1 ((n, e), s)
 Input:
       (n, e) RSA public key
       s signature representative, an integer between 0 and n - 1

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 Output:
       m message representative, an integer between 0 and n - 1
 Error:  "signature representative out of range"
 Assumption:  RSA public key (n, e) is valid
 Steps:
    1.  If the signature representative s is not between 0 and n - 1,
        output "signature representative out of range" and stop.
    2.  Let m = s^e mod n.
    3.  Output m.

6. Overview of Schemes

 A scheme combines cryptographic primitives and other techniques to
 achieve a particular security goal.  Two types of scheme are
 specified in this document: encryption schemes and signature schemes
 with appendix.
 The schemes specified in this document are limited in scope in that
 their operations consist only of steps to process data with an RSA
 public or private key, and they do not include steps for obtaining or
 validating the key.  Thus, in addition to the scheme operations, an
 application will typically include key management operations by which
 parties may select RSA public and private keys for a scheme
 operation.  The specific additional operations and other details are
 outside the scope of this document.
 As was the case for the cryptographic primitives (Section 5), the
 specifications of scheme operations assume that certain conditions
 are met by the inputs, in particular that RSA public and private keys
 are valid.  The behavior of an implementation is thus unspecified
 when a key is invalid.  The impact of such unspecified behavior
 depends on the application.  Possible means of addressing key
 validation include explicit key validation by the application; key
 validation within the public-key infrastructure; and assignment of
 liability for operations performed with an invalid key to the party
 who generated the key.
 A generally good cryptographic practice is to employ a given RSA key
 pair in only one scheme.  This avoids the risk that vulnerability in
 one scheme may compromise the security of the other and may be
 essential to maintain provable security.  While RSAES-PKCS1-v1_5

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 (Section 7.2) and RSASSA-PKCS1-v1_5 (Section 8.2) have traditionally
 been employed together without any known bad interactions (indeed,
 this is the model introduced by PKCS #1 v1.5), such a combined use of
 an RSA key pair is NOT RECOMMENDED for new applications.
 To illustrate the risks related to the employment of an RSA key pair
 in more than one scheme, suppose an RSA key pair is employed in both
 RSAES-OAEP (Section 7.1) and RSAES-PKCS1-v1_5.  Although RSAES-OAEP
 by itself would resist attack, an opponent might be able to exploit a
 weakness in the implementation of RSAES-PKCS1-v1_5 to recover
 messages encrypted with either scheme.  As another example, suppose
 an RSA key pair is employed in both RSASSA-PSS (Section 8.1) and
 RSASSA-PKCS1-v1_5.  Then the security proof for RSASSA-PSS would no
 longer be sufficient since the proof does not account for the
 possibility that signatures might be generated with a second scheme.
 Similar considerations may apply if an RSA key pair is employed in
 one of the schemes defined here and in a variant defined elsewhere.

7. Encryption Schemes

 For the purposes of this document, an encryption scheme consists of
 an encryption operation and a decryption operation, where the
 encryption operation produces a ciphertext from a message with a
 recipient's RSA public key, and the decryption operation recovers the
 message from the ciphertext with the recipient's corresponding RSA
 private key.
 An encryption scheme can be employed in a variety of applications.  A
 typical application is a key establishment protocol, where the
 message contains key material to be delivered confidentially from one
 party to another.  For instance, PKCS #7 [RFC2315] employs such a
 protocol to deliver a content-encryption key from a sender to a
 recipient; the encryption schemes defined here would be suitable key-
 encryption algorithms in that context.
 Two encryption schemes are specified in this document: RSAES-OAEP and
 RSAES-PKCS1-v1_5.  RSAES-OAEP is REQUIRED to be supported for new
 applications; RSAES-PKCS1-v1_5 is included only for compatibility
 with existing applications.
 The encryption schemes given here follow a general model similar to
 that employed in IEEE 1363 [IEEE1363], combining encryption and
 decryption primitives with an encoding method for encryption.  The
 encryption operations apply a message encoding operation to a message
 to produce an encoded message, which is then converted to an integer
 message representative.  An encryption primitive is applied to the
 message representative to produce the ciphertext.  Reversing this,
 the decryption operations apply a decryption primitive to the

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 ciphertext to recover a message representative, which is then
 converted to an octet-string-encoded message.  A message decoding
 operation is applied to the encoded message to recover the message
 and verify the correctness of the decryption.
 To avoid implementation weaknesses related to the way errors are
 handled within the decoding operation (see [BLEICHENBACHER] and
 [MANGER]), the encoding and decoding operations for RSAES-OAEP and
 RSAES-PKCS1-v1_5 are embedded in the specifications of the respective
 encryption schemes rather than defined in separate specifications.
 Both encryption schemes are compatible with the corresponding schemes
 in PKCS #1 v2.1.

7.1. RSAES-OAEP

 RSAES-OAEP combines the RSAEP and RSADP primitives (Sections 5.1.1
 and 5.1.2) with the EME-OAEP encoding method (Step 2 in
 Section 7.1.1, and Step 3 in Section 7.1.2).  EME-OAEP is based on
 Bellare and Rogaway's Optimal Asymmetric Encryption scheme [OAEP].
 It is compatible with the Integer Factorization Encryption Scheme
 (IFES) defined in IEEE 1363 [IEEE1363], where the encryption and
 decryption primitives are IFEP-RSA and IFDP-RSA and the message
 encoding method is EME-OAEP.  RSAES-OAEP can operate on messages of
 length up to k - 2hLen -2 octets, where hLen is the length of the
 output from the underlying hash function and k is the length in
 octets of the recipient's RSA modulus.
 Assuming that computing e-th roots modulo n is infeasible and the
 mask generation function in RSAES-OAEP has appropriate properties,
 RSAES-OAEP is semantically secure against adaptive chosen-ciphertext
 attacks.  This assurance is provable in the sense that the difficulty
 of breaking RSAES-OAEP can be directly related to the difficulty of
 inverting the RSA function, provided that the mask generation
 function is viewed as a black box or random oracle; see [FOPS] and
 the note below for further discussion.
 Both the encryption and the decryption operations of RSAES-OAEP take
 the value of a label L as input.  In this version of PKCS #1, L is
 the empty string; other uses of the label are outside the scope of
 this document.  See Appendix A.2.1 for the relevant ASN.1 syntax.
 RSAES-OAEP is parameterized by the choice of hash function and mask
 generation function.  This choice should be fixed for a given RSA
 key.  Suggested hash and mask generation functions are given in
 Appendix B.

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 Note: Past results have helpfully clarified the security properties
 of the OAEP encoding method [OAEP]  (roughly the procedure described
 in Step 2 in Section 7.1.1).  The background is as follows.  In 1994,
 Bellare and Rogaway [OAEP] introduced a security concept that they
 denoted plaintext awareness (PA94).  They proved that if a
 deterministic public-key encryption primitive (e.g., RSAEP) is hard
 to invert without the private key, then the corresponding OAEP-based
 encryption scheme is plaintext aware (in the random oracle model),
 meaning roughly that an adversary cannot produce a valid ciphertext
 without actually "knowing" the underlying plaintext.  Plaintext
 awareness of an encryption scheme is closely related to the
 resistance of the scheme against chosen-ciphertext attacks.  In such
 attacks, an adversary is given the opportunity to send queries to an
 oracle simulating the decryption primitive.  Using the results of
 these queries, the adversary attempts to decrypt a challenge
 ciphertext.
 However, there are two flavors of chosen-ciphertext attacks, and PA94
 implies security against only one of them.  The difference relies on
 what the adversary is allowed to do after she is given the challenge
 ciphertext.  The indifferent attack scenario (denoted CCA1) does not
 admit any queries to the decryption oracle after the adversary is
 given the challenge ciphertext, whereas the adaptive scenario
 (denoted CCA2) does (except that the decryption oracle refuses to
 decrypt the challenge ciphertext once it is published).  In 1998,
 Bellare and Rogaway, together with Desai and Pointcheval [PA98], came
 up with a new, stronger notion of plaintext awareness (PA98) that
 does imply security against CCA2.
 To summarize, there have been two potential sources for
 misconception: that PA94 and PA98 are equivalent concepts, or that
 CCA1 and CCA2 are equivalent concepts.  Either assumption leads to
 the conclusion that the Bellare-Rogaway paper implies security of
 OAEP against CCA2, which it does not.
 (Footnote: It might be fair to mention that PKCS #1 v2.0 cites [OAEP]
 and claims that "a chosen ciphertext attack is ineffective against a
 plaintext-aware encryption scheme such as RSAES-OAEP" without
 specifying the kind of plaintext awareness or chosen ciphertext
 attack considered.)
 OAEP has never been proven secure against CCA2; in fact, Victor Shoup
 [SHOUP] has demonstrated that such a proof does not exist in the
 general case.  Put briefly, Shoup showed that an adversary in the
 CCA2 scenario who knows how to partially invert the encryption
 primitive but does not know how to invert it completely may well be
 able to break the scheme.  For example, one may imagine an attacker
 who is able to break RSAES-OAEP if she knows how to recover all but

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 the first 20 bytes of a random integer encrypted with RSAEP.  Such an
 attacker does not need to be able to fully invert RSAEP, because she
 does not use the first 20 octets in her attack.
 Still, RSAES-OAEP is secure against CCA2, which was proved by
 Fujisaki, Okamoto, Pointcheval, and Stern [FOPS] shortly after the
 announcement of Shoup's result.  Using clever lattice reduction
 techniques, they managed to show how to invert RSAEP completely given
 a sufficiently large part of the pre-image.  This observation,
 combined with a proof that OAEP is secure against CCA2 if the
 underlying encryption primitive is hard to partially invert, fills
 the gap between what Bellare and Rogaway proved about RSAES-OAEP and
 what some may have believed that they proved.  Somewhat
 paradoxically, we are hence saved by an ostensible weakness in RSAEP
 (i.e., the whole inverse can be deduced from parts of it).
 Unfortunately, however, the security reduction is not efficient for
 concrete parameters.  While the proof successfully relates an
 adversary A against the CCA2 security of RSAES-OAEP to an algorithm I
 inverting RSA, the probability of success for I is only approximately
 \epsilon^2 / 2^18, where \epsilon is the probability of success for
 A.
 (Footnote: In [FOPS], the probability of success for the inverter was
 \epsilon^2 / 4.  The additional factor 1 / 2^16 is due to the eight
 fixed zero bits at the beginning of the encoded message EM, which are
 not present in the variant of OAEP considered in [FOPS].  (A must be
 applied twice to invert RSA, and each application corresponds to a
 factor 1 / 2^8.))
 In addition, the running time for I is approximately t^2, where t is
 the running time of the adversary.  The consequence is that we cannot
 exclude the possibility that attacking RSAES-OAEP is considerably
 easier than inverting RSA for concrete parameters.  Still, the
 existence of a security proof provides some assurance that the
 RSAES-OAEP construction is sounder than ad hoc constructions such as
 RSAES-PKCS1-v1_5.
 Hybrid encryption schemes based on the RSA Key Encapsulation
 Mechanism (RSA-KEM) paradigm offer tight proofs of security directly
 applicable to concrete parameters; see [ISO18033] for discussion.
 Future versions of PKCS #1 may specify schemes based on this
 paradigm.

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7.1.1. Encryption Operation

 RSAES-OAEP-ENCRYPT ((n, e), M, L)
 Options:
    Hash     hash function (hLen denotes the length in octets of
             the hash function output)
    MGF      mask generation function
 Input:
    (n, e)   recipient's RSA public key (k denotes the length in
             octets of the RSA modulus n)
    M        message to be encrypted, an octet string of length mLen,
             where mLen <= k - 2hLen - 2
    L        optional label to be associated with the message; the
             default value for L, if L is not provided, is the empty
             string
 Output:
    C        ciphertext, an octet string of length k
 Errors:  "message too long"; "label too long"
 Assumption:  RSA public key (n, e) is valid
 Steps:
    1.  Length checking:
        a.  If the length of L is greater than the input limitation
            for the hash function (2^61 - 1 octets for SHA-1), output
            "label too long" and stop.
        b.  If mLen > k - 2hLen - 2, output "message too long" and
            stop.
    2.  EME-OAEP encoding (see Figure 1 below):
        a.  If the label L is not provided, let L be the empty string.
            Let lHash = Hash(L), an octet string of length hLen (see
            the note below).
        b.  Generate a padding string PS consisting of k - mLen -
            2hLen - 2 zero octets.  The length of PS may be zero.

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        c.  Concatenate lHash, PS, a single octet with hexadecimal
            value 0x01, and the message M to form a data block DB of
            length k - hLen - 1 octets as
               DB = lHash || PS || 0x01 || M.
        d.  Generate a random octet string seed of length hLen.
        e.  Let dbMask = MGF(seed, k - hLen - 1).
        f.  Let maskedDB = DB \xor dbMask.
        g.  Let seedMask = MGF(maskedDB, hLen).
        h.  Let maskedSeed = seed \xor seedMask.
        i.  Concatenate a single octet with hexadecimal value 0x00,
            maskedSeed, and maskedDB to form an encoded message EM of
            length k octets as
               EM = 0x00 || maskedSeed || maskedDB.
    3.  RSA encryption:
        a.  Convert the encoded message EM to an integer message
            representative m (see Section 4.2):
               m = OS2IP (EM).
        b.  Apply the RSAEP encryption primitive (Section 5.1.1) to
            the RSA public key (n, e) and the message representative m
            to produce an integer ciphertext representative c:
               c = RSAEP ((n, e), m).
        c.  Convert the ciphertext representative c to a ciphertext C
            of length k octets (see Section 4.1):
               C = I2OSP (c, k).

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    4.  Output the ciphertext C.
    _________________________________________________________________
                              +----------+------+--+-------+
                         DB = |  lHash   |  PS  |01|   M   |
                              +----------+------+--+-------+
                                             |
                   +----------+              |
                   |   seed   |              |
                   +----------+              |
                         |                   |
                         |-------> MGF ---> xor
                         |                   |
                +--+     V                   |
                |00|    xor <----- MGF <-----|
                +--+     |                   |
                  |      |                   |
                  V      V                   V
                +--+----------+----------------------------+
          EM =  |00|maskedSeed|          maskedDB          |
                +--+----------+----------------------------+
    _________________________________________________________________
                 Figure 1: EME-OAEP Encoding Operation
 Notes:
  1. lHash is the hash of the optional label L.
  1. The decoding operation follows reverse steps to recover M and

verify lHash and PS.

  1. If L is the empty string, the corresponding hash value lHash has

the following hexadecimal representation for different choices of

    Hash:
    SHA-1:   (0x)da39a3ee 5e6b4b0d 3255bfef 95601890 afd80709
    SHA-256: (0x)e3b0c442 98fc1c14 9afbf4c8 996fb924 27ae41e4 649b934c
                 a495991b 7852b855
    SHA-384: (0x)38b060a7 51ac9638 4cd9327e b1b1e36a 21fdb711 14be0743
                 4c0cc7bf 63f6e1da 274edebf e76f65fb d51ad2f1 4898b95b
    SHA-512: (0x)cf83e135 7eefb8bd f1542850 d66d8007 d620e405 0b5715dc
                 83f4a921 d36ce9ce 47d0d13c 5d85f2b0 ff8318d2 877eec2f
                 63b931bd 47417a81 a538327a f927da3e

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7.1.2. Decryption Operation

 RSAES-OAEP-DECRYPT (K, C, L)
 Options:
    Hash     hash function (hLen denotes the length in octets of
             the hash function output)
    MGF      mask generation function
 Input:
    K        recipient's RSA private key (k denotes the length in
             octets of the RSA modulus n), where k >= 2hLen + 2
    C        ciphertext to be decrypted, an octet string of length k
    L        optional label whose association with the message is to
             be verified; the default value for L, if L is not
             provided, is the empty string
 Output:
    M        message, an octet string of length mLen, where
             mLen <= k - 2hLen - 2
 Error:  "decryption error"
 Steps:
    1.  Length checking:
        a.  If the length of L is greater than the input limitation
            for the hash function (2^61 - 1 octets for SHA-1), output
            "decryption error" and stop.
        b.  If the length of the ciphertext C is not k octets, output
            "decryption error" and stop.
        c.  If k < 2hLen + 2, output "decryption error" and stop.
    2.  RSA decryption:
        a.  Convert the ciphertext C to an integer ciphertext
            representative c (see Section 4.2):
               c = OS2IP (C).

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        b.  Apply the RSADP decryption primitive (Section 5.1.2) to
            the RSA private key K and the ciphertext representative c
            to produce an integer message representative m:
               m = RSADP (K, c).
            If RSADP outputs "ciphertext representative out of range"
            (meaning that c >= n), output "decryption error" and stop.
        c.  Convert the message representative m to an encoded message
            EM of length k octets (see Section 4.1):
               EM = I2OSP (m, k).
    3.  EME-OAEP decoding:
        a.  If the label L is not provided, let L be the empty string.
            Let lHash = Hash(L), an octet string of length hLen (see
            the note in Section 7.1.1).
        b.  Separate the encoded message EM into a single octet Y, an
            octet string maskedSeed of length hLen, and an octet
            string maskedDB of length k - hLen - 1 as
               EM = Y || maskedSeed || maskedDB.
        c.  Let seedMask = MGF(maskedDB, hLen).
        d.  Let seed = maskedSeed \xor seedMask.
        e.  Let dbMask = MGF(seed, k - hLen - 1).
        f.  Let DB = maskedDB \xor dbMask.
        g.  Separate DB into an octet string lHash' of length hLen, a
            (possibly empty) padding string PS consisting of octets
            with hexadecimal value 0x00, and a message M as
               DB = lHash' || PS || 0x01 || M.
            If there is no octet with hexadecimal value 0x01 to
            separate PS from M, if lHash does not equal lHash', or if
            Y is nonzero, output "decryption error" and stop.  (See
            the note below.)

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    4.  Output the message M.
    Note: Care must be taken to ensure that an opponent cannot
    distinguish the different error conditions in Step 3.g, whether by
    error message or timing, and, more generally, that an opponent
    cannot learn partial information about the encoded message EM.
    Otherwise, an opponent may be able to obtain useful information
    about the decryption of the ciphertext C, leading to a chosen-
    ciphertext attack such as the one observed by Manger [MANGER].

7.2. RSAES-PKCS1-v1_5

 RSAES-PKCS1-v1_5 combines the RSAEP and RSADP primitives (Sections
 5.1.1 and 5.1.2) with the EME-PKCS1-v1_5 encoding method (Step 2 in
 Section 7.2.1, and Step 3 in Section 7.2.2).  It is mathematically
 equivalent to the encryption scheme in PKCS #1 v1.5.
 RSAES-PKCS1-v1_5 can operate on messages of length up to k - 11
 octets (k is the octet length of the RSA modulus), although care
 should be taken to avoid certain attacks on low-exponent RSA due to
 Coppersmith, Franklin, Patarin, and Reiter when long messages are
 encrypted (see the third bullet in the notes below and [LOWEXP];
 [NEWATTACK] contains an improved attack).  As a general rule, the use
 of this scheme for encrypting an arbitrary message, as opposed to a
 randomly generated key, is NOT RECOMMENDED.
 It is possible to generate valid RSAES-PKCS1-v1_5 ciphertexts without
 knowing the corresponding plaintexts, with a reasonable probability
 of success.  This ability can be exploited in a chosen-ciphertext
 attack as shown in [BLEICHENBACHER].  Therefore, if RSAES-PKCS1-v1_5
 is to be used, certain easily implemented countermeasures should be
 taken to thwart the attack found in [BLEICHENBACHER].  Typical
 examples include the addition of structure to the data to be encoded,
 rigorous checking of PKCS #1 v1.5 conformance (and other redundancy)
 in decrypted messages, and the consolidation of error messages in a
 client-server protocol based on PKCS #1 v1.5.  These can all be
 effective countermeasures and do not involve changes to a protocol
 based on PKCS #1 v1.5.  See [BKS] for a further discussion of these
 and other countermeasures.  It has recently been shown that the
 security of the SSL/TLS handshake protocol [RFC5246], which uses
 RSAES-PKCS1-v1_5 and certain countermeasures, can be related to a
 variant of the RSA problem; see [RSATLS] for discussion.
 Note: The following passages describe some security recommendations
 pertaining to the use of RSAES-PKCS1-v1_5.  Recommendations from PKCS
 #1 v1.5 are included as well as new recommendations motivated by
 cryptanalytic advances made in the intervening years.

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 o  It is RECOMMENDED that the pseudorandom octets in Step 2 in
    Section 7.2.1 be generated independently for each encryption
    process, especially if the same data is input to more than one
    encryption process.  Haastad's results [HAASTAD] are one
    motivation for this recommendation.
 o  The padding string PS in Step 2 in Section 7.2.1 is at least eight
    octets long, which is a security condition for public-key
    operations that makes it difficult for an attacker to recover data
    by trying all possible encryption blocks.
 o  The pseudorandom octets can also help thwart an attack due to
    Coppersmith et al.  [LOWEXP] (see [NEWATTACK] for an improvement
    of the attack) when the size of the message to be encrypted is
    kept small.  The attack works on low-exponent RSA when similar
    messages are encrypted with the same RSA public key.  More
    specifically, in one flavor of the attack, when two inputs to
    RSAEP agree on a large fraction of bits (8/9) and low-exponent RSA
    (e = 3) is used to encrypt both of them, it may be possible to
    recover both inputs with the attack.  Another flavor of the attack
    is successful in decrypting a single ciphertext when a large
    fraction (2/3) of the input to RSAEP is already known.  For
    typical applications, the message to be encrypted is short (e.g.,
    a 128-bit symmetric key), so not enough information will be known
    or common between two messages to enable the attack.  However, if
    a long message is encrypted, or if part of a message is known,
    then the attack may be a concern.  In any case, the RSAES-OAEP
    scheme overcomes the attack.

7.2.1. Encryption Operation

 RSAES-PKCS1-V1_5-ENCRYPT ((n, e), M)
 Input:
    (n, e)   recipient's RSA public key (k denotes the length in
             octets of the modulus n)
    M        message to be encrypted, an octet string of length
             mLen, where mLen <= k - 11
 Output:
    C        ciphertext, an octet string of length k
 Error:  "message too long"

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 Steps:
    1.  Length checking: If mLen > k - 11, output "message too long"
        and stop.
    2.  EME-PKCS1-v1_5 encoding:
        a.  Generate an octet string PS of length k - mLen - 3
            consisting of pseudo-randomly generated nonzero octets.
            The length of PS will be at least eight octets.
        b.  Concatenate PS, the message M, and other padding to form
            an encoded message EM of length k octets as
               EM = 0x00 || 0x02 || PS || 0x00 || M.
    3.  RSA encryption:
        a.  Convert the encoded message EM to an integer message
            representative m (see Section 4.2):
               m = OS2IP (EM).
        b.  Apply the RSAEP encryption primitive (Section 5.1.1) to
            the RSA public key (n, e) and the message representative m
            to produce an integer ciphertext representative c:
               c = RSAEP ((n, e), m).
        c.  Convert the ciphertext representative c to a ciphertext C
            of length k octets (see Section 4.1):
               C = I2OSP (c, k).
    4.  Output the ciphertext C.

7.2.2. Decryption Operation

 RSAES-PKCS1-V1_5-DECRYPT (K, C)
 Input:
    K        recipient's RSA private key
    C        ciphertext to be decrypted, an octet string of length k,
             where k is the length in octets of the RSA modulus n

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 Output:
    M        message, an octet string of length at most k - 11
 Error:  "decryption error"
 Steps:
    1.  Length checking: If the length of the ciphertext C is not k
        octets (or if k < 11), output "decryption error" and stop.
    2.  RSA decryption:
        a.  Convert the ciphertext C to an integer ciphertext
            representative c (see Section 4.2):
               c = OS2IP (C).
        b.  Apply the RSADP decryption primitive (Section 5.1.2) to
            the RSA private key (n, d) and the ciphertext
            representative c to produce an integer message
            representative m:
               m = RSADP ((n, d), c).
            If RSADP outputs "ciphertext representative out of range"
            (meaning that c >= n), output "decryption error" and stop.
        c.  Convert the message representative m to an encoded message
            EM of length k octets (see Section 4.1):
               EM = I2OSP (m, k).
    3.  EME-PKCS1-v1_5 decoding: Separate the encoded message EM into
        an octet string PS consisting of nonzero octets and a message
        M as
           EM = 0x00 || 0x02 || PS || 0x00 || M.
        If the first octet of EM does not have hexadecimal value 0x00,
        if the second octet of EM does not have hexadecimal value
        0x02, if there is no octet with hexadecimal value 0x00 to
        separate PS from M, or if the length of PS is less than 8
        octets, output "decryption error" and stop.  (See the note
        below.)
    4.  Output M.

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    Note: Care shall be taken to ensure that an opponent cannot
    distinguish the different error conditions in Step 3, whether by
    error message or timing.  Otherwise, an opponent may be able to
    obtain useful information about the decryption of the ciphertext
    C, leading to a strengthened version of Bleichenbacher's attack
    [BLEICHENBACHER]; compare to Manger's attack [MANGER].

8. Signature Scheme with Appendix

 For the purposes of this document, a signature scheme with appendix
 consists of a signature generation operation and a signature
 verification operation, where the signature generation operation
 produces a signature from a message with a signer's RSA private key,
 and the signature verification operation verifies the signature on
 the message with the signer's corresponding RSA public key.  To
 verify a signature constructed with this type of scheme, it is
 necessary to have the message itself.  In this way, signature schemes
 with appendix are distinguished from signature schemes with message
 recovery, which are not supported in this document.
 A signature scheme with appendix can be employed in a variety of
 applications.  For instance, the signature schemes with appendix
 defined here would be suitable signature algorithms for X.509
 certificates [ISO9594].  Related signature schemes could be employed
 in PKCS #7 [RFC2315], although for technical reasons the current
 version of PKCS #7 separates a hash function from a signature scheme,
 which is different than what is done here; see the note in
 Appendix A.2.3 for more discussion.
 Two signature schemes with appendix are specified in this document:
 RSASSA-PSS and RSASSA-PKCS1-v1_5.  Although no attacks are known
 against RSASSA-PKCS1-v1_5, in the interest of increased robustness,
 RSASSA-PSS is REQUIRED in new applications.  RSASSA-PKCS1-v1_5 is
 included only for compatibility with existing applications.
 The signature schemes with appendix given here follow a general model
 similar to that employed in IEEE 1363 [IEEE1363], combining signature
 and verification primitives with an encoding method for signatures.
 The signature generation operations apply a message encoding
 operation to a message to produce an encoded message, which is then
 converted to an integer message representative.  A signature
 primitive is applied to the message representative to produce the
 signature.  Reversing this, the signature verification operations
 apply a signature verification primitive to the signature to recover
 a message representative, which is then converted to an octet-string-
 encoded message.  A verification operation is applied to the message
 and the encoded message to determine whether they are consistent.

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 If the encoding method is deterministic (e.g., EMSA-PKCS1-v1_5), the
 verification operation may apply the message encoding operation to
 the message and compare the resulting encoded message to the
 previously derived encoded message.  If there is a match, the
 signature is considered valid.  If the method is randomized (e.g.,
 EMSA-PSS), the verification operation is typically more complicated.
 For example, the verification operation in EMSA-PSS extracts the
 random salt and a hash output from the encoded message and checks
 whether the hash output, the salt, and the message are consistent;
 the hash output is a deterministic function in terms of the message
 and the salt.  For both signature schemes with appendix defined in
 this document, the signature generation and signature verification
 operations are readily implemented as "single-pass" operations if the
 signature is placed after the message.  See PKCS #7 [RFC2315] for an
 example format in the case of RSASSA-PKCS1-v1_5.

8.1. RSASSA-PSS

 RSASSA-PSS combines the RSASP1 and RSAVP1 primitives with the
 EMSA-PSS encoding method.  It is compatible with the Integer
 Factorization Signature Scheme with Appendix (IFSSA) as amended in
 IEEE 1363a [IEEE1363A], where the signature and verification
 primitives are IFSP-RSA1 and IFVP-RSA1 as defined in IEEE 1363
 [IEEE1363], and the message encoding method is EMSA4.  EMSA4 is
 slightly more general than EMSA-PSS as it acts on bit strings rather
 than on octet strings.  EMSA-PSS is equivalent to EMSA4 restricted to
 the case that the operands as well as the hash and salt values are
 octet strings.
 The length of messages on which RSASSA-PSS can operate is either
 unrestricted or constrained by a very large number, depending on the
 hash function underlying the EMSA-PSS encoding method.
 Assuming that computing e-th roots modulo n is infeasible and the
 hash and mask generation functions in EMSA-PSS have appropriate
 properties, RSASSA-PSS provides secure signatures.  This assurance is
 provable in the sense that the difficulty of forging signatures can
 be directly related to the difficulty of inverting the RSA function,
 provided that the hash and mask generation functions are viewed as
 black boxes or random oracles.  The bounds in the security proof are
 essentially "tight", meaning that the success probability and running
 time for the best forger against RSASSA-PSS are very close to the
 corresponding parameters for the best RSA inversion algorithm; see
 [RSARABIN] [PSSPROOF] [JONSSON] for further discussion.
 In contrast to the RSASSA-PKCS1-v1_5 signature scheme, a hash
 function identifier is not embedded in the EMSA-PSS encoded message,
 so in theory it is possible for an adversary to substitute a

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 different (and potentially weaker) hash function than the one
 selected by the signer.  Therefore, it is RECOMMENDED that the
 EMSA-PSS mask generation function be based on the same hash function.
 In this manner, the entire encoded message will be dependent on the
 hash function, and it will be difficult for an opponent to substitute
 a different hash function than the one intended by the signer.  This
 matching of hash functions is only for the purpose of preventing hash
 function substitution and is not necessary if hash function
 substitution is addressed by other means (e.g., the verifier accepts
 only a designated hash function).  See [HASHID] for further
 discussion of these points.  The provable security of RSASSA-PSS does
 not rely on the hash function in the mask generation function being
 the same as the hash function applied to the message.
 RSASSA-PSS is different from other RSA-based signature schemes in
 that it is probabilistic rather than deterministic, incorporating a
 randomly generated salt value.  The salt value enhances the security
 of the scheme by affording a "tighter" security proof than
 deterministic alternatives such as Full Domain Hashing (FDH); see
 [RSARABIN] for discussion.  However, the randomness is not critical
 to security.  In situations where random generation is not possible,
 a fixed value or a sequence number could be employed instead, with
 the resulting provable security similar to that of FDH [FDH].

8.1.1. Signature Generation Operation

 RSASSA-PSS-SIGN (K, M)
 Input:
    K        signer's RSA private key
    M        message to be signed, an octet string
 Output:
    S        signature, an octet string of length k, where k is the
             length in octets of the RSA modulus n
 Errors:  "message too long;" "encoding error"
 Steps:
    1.  EMSA-PSS encoding: Apply the EMSA-PSS encoding operation
        (Section 9.1.1) to the message M to produce an encoded message
        EM of length \ceil ((modBits - 1)/8) octets such that the bit
        length of the integer OS2IP (EM) (see Section 4.2) is at most
        modBits - 1, where modBits is the length in bits of the RSA
        modulus n:

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           EM = EMSA-PSS-ENCODE (M, modBits - 1).
        Note that the octet length of EM will be one less than k if
        modBits - 1 is divisible by 8 and equal to k otherwise.  If
        the encoding operation outputs "message too long", output
        "message too long" and stop.  If the encoding operation
        outputs "encoding error", output "encoding error" and stop.
    2.  RSA signature:
        a.  Convert the encoded message EM to an integer message
            representative m (see Section 4.2):
               m = OS2IP (EM).
        b.  Apply the RSASP1 signature primitive (Section 5.2.1) to
            the RSA private key K and the message representative m to
            produce an integer signature representative s:
               s = RSASP1 (K, m).
        c.  Convert the signature representative s to a signature S of
            length k octets (see Section 4.1):
               S = I2OSP (s, k).
    3.  Output the signature S.

8.1.2. Signature Verification Operation

 RSASSA-PSS-VERIFY ((n, e), M, S)
 Input:
    (n, e)  signer's RSA public key
    M       message whose signature is to be verified, an octet string
    S       signature to be verified, an octet string of length k,
            where k is the length in octets of the RSA modulus n
 Output:  "valid signature" or "invalid signature"
 Steps:
    1.  Length checking: If the length of the signature S is not k
        octets, output "invalid signature" and stop.

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    2.  RSA verification:
        a.  Convert the signature S to an integer signature
            representative s (see Section 4.2):
               s = OS2IP (S).
        b.  Apply the RSAVP1 verification primitive (Section 5.2.2) to
            the RSA public key (n, e) and the signature representative
            s to produce an integer message representative m:
               m = RSAVP1 ((n, e), s).
            If RSAVP1 output "signature representative out of range",
            output "invalid signature" and stop.
        c.  Convert the message representative m to an encoded message
            EM of length emLen = \ceil ((modBits - 1)/8) octets, where
            modBits is the length in bits of the RSA modulus n (see
            Section 4.1):
               EM = I2OSP (m, emLen).
            Note that emLen will be one less than k if modBits - 1 is
            divisible by 8 and equal to k otherwise.  If I2OSP outputs
            "integer too large", output "invalid signature" and stop.
    3.  EMSA-PSS verification: Apply the EMSA-PSS verification
        operation (Section 9.1.2) to the message M and the encoded
        message EM to determine whether they are consistent:
           Result = EMSA-PSS-VERIFY (M, EM, modBits - 1).
    4.  If Result = "consistent", output "valid signature".
        Otherwise, output "invalid signature".

8.2. RSASSA-PKCS1-v1_5

 RSASSA-PKCS1-v1_5 combines the RSASP1 and RSAVP1 primitives with the
 EMSA-PKCS1-v1_5 encoding method.  It is compatible with the IFSSA
 scheme defined in IEEE 1363 [IEEE1363], where the signature and
 verification primitives are IFSP-RSA1 and IFVP-RSA1, and the message
 encoding method is EMSA-PKCS1-v1_5 (which is not defined in IEEE 1363
 but is in IEEE 1363a [IEEE1363A]).
 The length of messages on which RSASSA-PKCS1-v1_5 can operate is
 either unrestricted or constrained by a very large number, depending
 on the hash function underlying the EMSA-PKCS1-v1_5 method.

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 Assuming that computing e-th roots modulo n is infeasible and the
 hash function in EMSA-PKCS1-v1_5 has appropriate properties,
 RSASSA-PKCS1-v1_5 is conjectured to provide secure signatures.  More
 precisely, forging signatures without knowing the RSA private key is
 conjectured to be computationally infeasible.  Also, in the encoding
 method EMSA-PKCS1-v1_5, a hash function identifier is embedded in the
 encoding.  Because of this feature, an adversary trying to find a
 message with the same signature as a previously signed message must
 find collisions of the particular hash function being used; attacking
 a different hash function than the one selected by the signer is not
 useful to the adversary.  See [HASHID] for further discussion.
 Note: As noted in PKCS #1 v1.5, the EMSA-PKCS1-v1_5 encoding method
 has the property that the encoded message, converted to an integer
 message representative, is guaranteed to be large and at least
 somewhat "random".  This prevents attacks of the kind proposed by
 Desmedt and Odlyzko [CHOSEN] where multiplicative relationships
 between message representatives are developed by factoring the
 message representatives into a set of small values (e.g., a set of
 small primes).  Coron, Naccache, and Stern [PADDING] showed that a
 stronger form of this type of attack could be quite effective against
 some instances of the ISO/IEC 9796-2 signature scheme.  They also
 analyzed the complexity of this type of attack against the
 EMSA-PKCS1-v1_5 encoding method and concluded that an attack would be
 impractical, requiring more operations than a collision search on the
 underlying hash function (i.e., more than 2^80 operations).
 Coppersmith, Halevi, and Jutla [FORGERY] subsequently extended Coron
 et al.'s attack to break the ISO/IEC 9796-1 signature scheme with
 message recovery.  The various attacks illustrate the importance of
 carefully constructing the input to the RSA signature primitive,
 particularly in a signature scheme with message recovery.
 Accordingly, the EMSA-PKCS-v1_5 encoding method explicitly includes a
 hash operation and is not intended for signature schemes with message
 recovery.  Moreover, while no attack is known against the
 EMSA-PKCS-v1_5 encoding method, a gradual transition to EMSA-PSS is
 recommended as a precaution against future developments.

8.2.1. Signature Generation Operation

 RSASSA-PKCS1-V1_5-SIGN (K, M)
 Input:
    K        signer's RSA private key
    M        message to be signed, an octet string

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 Output:
    S        signature, an octet string of length k, where k is the
             length in octets of the RSA modulus n
 Errors:  "message too long"; "RSA modulus too short"
 Steps:
    1.  EMSA-PKCS1-v1_5 encoding: Apply the EMSA-PKCS1-v1_5 encoding
        operation (Section 9.2) to the message M to produce an encoded
        message EM of length k octets:
           EM = EMSA-PKCS1-V1_5-ENCODE (M, k).
        If the encoding operation outputs "message too long", output
        "message too long" and stop.  If the encoding operation
        outputs "intended encoded message length too short", output
        "RSA modulus too short" and stop.
    2.  RSA signature:
        a.  Convert the encoded message EM to an integer message
            representative m (see Section 4.2):
               m = OS2IP (EM).
        b.  Apply the RSASP1 signature primitive (Section 5.2.1) to
            the RSA private key K and the message representative m to
            produce an integer signature representative s:
               s = RSASP1 (K, m).
        c.  Convert the signature representative s to a signature S of
            length k octets (see Section 4.1):
               S = I2OSP (s, k).
    3.  Output the signature S.

8.2.2. Signature Verification Operation

 RSASSA-PKCS1-V1_5-VERIFY ((n, e), M, S)
 Input:
    (n, e)  signer's RSA public key
    M       message whose signature is to be verified, an octet string

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    S       signature to be verified, an octet string of length k,
            where k is the length in octets of the RSA modulus n
 Output  "valid signature" or "invalid signature"
 Errors:  "message too long"; "RSA modulus too short"
 Steps:
    1.  Length checking: If the length of the signature S is not k
        octets, output "invalid signature" and stop.
    2.  RSA verification:
        a.  Convert the signature S to an integer signature
            representative s (see Section 4.2):
               s = OS2IP (S).
        b.  Apply the RSAVP1 verification primitive (Section 5.2.2) to
            the RSA public key (n, e) and the signature representative
            s to produce an integer message representative m:
               m = RSAVP1 ((n, e), s).
            If RSAVP1 outputs "signature representative out of range",
            output "invalid signature" and stop.
        c.  Convert the message representative m to an encoded message
            EM of length k octets (see Section 4.1):
               EM = I2OSP (m, k).
            If I2OSP outputs "integer too large", output "invalid
            signature" and stop.
    3.  EMSA-PKCS1-v1_5 encoding: Apply the EMSA-PKCS1-v1_5 encoding
        operation (Section 9.2) to the message M to produce a second
        encoded message EM' of length k octets:
           EM' = EMSA-PKCS1-V1_5-ENCODE (M, k).
        If the encoding operation outputs "message too long", output
        "message too long" and stop.  If the encoding operation
        outputs "intended encoded message length too short", output
        "RSA modulus too short" and stop.

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    4.  Compare the encoded message EM and the second encoded message
        EM'.  If they are the same, output "valid signature";
        otherwise, output "invalid signature".
    Note: Another way to implement the signature verification
    operation is to apply a "decoding" operation (not specified in
    this document) to the encoded message to recover the underlying
    hash value, and then compare it to a newly computed hash value.
    This has the advantage that it requires less intermediate storage
    (two hash values rather than two encoded messages), but the
    disadvantage that it requires additional code.

9. Encoding Methods for Signatures with Appendix

 Encoding methods consist of operations that map between octet string
 messages and octet-string-encoded messages, which are converted to
 and from integer message representatives in the schemes.  The integer
 message representatives are processed via the primitives.  The
 encoding methods thus provide the connection between the schemes,
 which process messages, and the primitives.
 An encoding method for signatures with appendix, for the purposes of
 this document, consists of an encoding operation and optionally a
 verification operation.  An encoding operation maps a message M to an
 encoded message EM of a specified length.  A verification operation
 determines whether a message M and an encoded message EM are
 consistent, i.e., whether the encoded message EM is a valid encoding
 of the message M.
 The encoding operation may introduce some randomness, so that
 different applications of the encoding operation to the same message
 will produce different encoded messages, which has benefits for
 provable security.  For such an encoding method, both an encoding and
 a verification operation are needed unless the verifier can reproduce
 the randomness (e.g., by obtaining the salt value from the signer).
 For a deterministic encoding method, only an encoding operation is
 needed.
 Two encoding methods for signatures with appendix are employed in the
 signature schemes and are specified here: EMSA-PSS and
 EMSA-PKCS1-v1_5.

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9.1. EMSA-PSS

 This encoding method is parameterized by the choice of hash function,
 mask generation function, and salt length.  These options should be
 fixed for a given RSA key, except that the salt length can be
 variable (see [JONSSON] for discussion).  Suggested hash and mask
 generation functions are given in Appendix B.  The encoding method is
 based on Bellare and Rogaway's Probabilistic Signature Scheme (PSS)
 [RSARABIN][PSS].  It is randomized and has an encoding operation and
 a verification operation.
 Figure 2 illustrates the encoding operation.
    __________________________________________________________________
                                   +-----------+
                                   |     M     |
                                   +-----------+
                                         |
                                         V
                                       Hash
                                         |
                                         V
                           +--------+----------+----------+
                      M' = |Padding1|  mHash   |   salt   |
                           +--------+----------+----------+
                                          |
                +--------+----------+     V
          DB =  |Padding2|   salt   |   Hash
                +--------+----------+     |
                          |               |
                          V               |
                         xor <--- MGF <---|
                          |               |
                          |               |
                          V               V
                +-------------------+----------+--+
          EM =  |    maskedDB       |     H    |bc|
                +-------------------+----------+--+
    __________________________________________________________________
 Figure 2: EMSA-PSS Encoding Operation
 Note that the verification operation follows reverse steps to recover
 salt and then forward steps to recompute and compare H.

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 Notes:
 1.  The encoding method defined here differs from the one in Bellare
     and Rogaway's submission to IEEE 1363a [PSS] in three respects:
  • It applies a hash function rather than a mask generation

function to the message. Even though the mask generation

        function is based on a hash function, it seems more natural to
        apply a hash function directly.
  • The value that is hashed together with the salt value is the

string (0x)00 00 00 00 00 00 00 00 || mHash rather than the

        message M itself.  Here, mHash is the hash of M.  Note that
        the hash function is the same in both steps.  See Note 3 below
        for further discussion.  (Also, the name "salt" is used
        instead of "seed", as it is more reflective of the value's
        role.)
  • The encoded message in EMSA-PSS has nine fixed bits; the first

bit is 0 and the last eight bits form a "trailer field", the

        octet 0xbc.  In the original scheme, only the first bit is
        fixed.  The rationale for the trailer field is for
        compatibility with the Integer Factorization Signature
        Primitive using Rabin-Williams (IFSP-RW) in IEEE 1363
        [IEEE1363] and the corresponding primitive in ISO/IEC
        9796-2:2010 [ISO9796].
 2.  Assuming that the mask generation function is based on a hash
     function, it is RECOMMENDED that the hash function be the same as
     the one that is applied to the message; see Section 8.1 for
     further discussion.
 3.  Without compromising the security proof for RSASSA-PSS, one may
     perform Steps 1 and 2 of EMSA-PSS-ENCODE and EMSA-PSS-VERIFY (the
     application of the hash function to the message) outside the
     module that computes the rest of the signature operation, so that
     mHash rather than the message M itself is input to the module.
     In other words, the security proof for RSASSA-PSS still holds
     even if an opponent can control the value of mHash.  This is
     convenient if the module has limited I/O bandwidth, e.g., a smart
     card.  Note that previous versions of PSS [RSARABIN][PSS] did not
     have this property.  Of course, it may be desirable for other
     security reasons to have the module process the full message.
     For instance, the module may need to "see" what it is signing if
     it does not trust the component that computes the hash value.

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 4.  Typical salt lengths in octets are hLen (the length of the output
     of the hash function Hash) and 0.  In both cases, the security of
     RSASSA-PSS can be closely related to the hardness of inverting
     RSAVP1.  Bellare and Rogaway [RSARABIN] give a tight lower bound
     for the security of the original RSA-PSS scheme, which
     corresponds roughly to the former case, while Coron [FDH] gives a
     lower bound for the related Full Domain Hashing scheme, which
     corresponds roughly to the latter case.  In [PSSPROOF], Coron
     provides a general treatment with various salt lengths ranging
     from 0 to hLen; see [IEEE1363A] for discussion.  See also
     [JONSSON], which adapts the security proofs in [RSARABIN]
     [PSSPROOF] to address the differences between the original and
     the present version of RSA-PSS as listed in Note 1 above.
 5.  As noted in IEEE 1363a [IEEE1363A], the use of randomization in
     signature schemes -- such as the salt value in EMSA-PSS -- may
     provide a "covert channel" for transmitting information other
     than the message being signed.  For more on covert channels, see
     [SIMMONS].

9.1.1. Encoding Operation

 EMSA-PSS-ENCODE (M, emBits)
 Options:
    Hash     hash function (hLen denotes the length in octets of
             the hash function output)
    MGF      mask generation function
    sLen     intended length in octets of the salt
 Input:
    M        message to be encoded, an octet string
    emBits   maximal bit length of the integer OS2IP (EM) (see Section
             4.2), at least 8hLen + 8sLen + 9
 Output:
    EM       encoded message, an octet string of length emLen = \ceil
             (emBits/8)
 Errors:  "Encoding error"; "message too long"

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 Steps:
    1.   If the length of M is greater than the input limitation for
         the hash function (2^61 - 1 octets for SHA-1), output
         "message too long" and stop.
    2.   Let mHash = Hash(M), an octet string of length hLen.
    3.   If emLen < hLen + sLen + 2, output "encoding error" and stop.
    4.   Generate a random octet string salt of length sLen; if sLen =
         0, then salt is the empty string.
    5.   Let
            M' = (0x)00 00 00 00 00 00 00 00 || mHash || salt;
         M' is an octet string of length 8 + hLen + sLen with eight
         initial zero octets.
    6.   Let H = Hash(M'), an octet string of length hLen.
    7.   Generate an octet string PS consisting of emLen - sLen - hLen
         - 2 zero octets.  The length of PS may be 0.
    8.   Let DB = PS || 0x01 || salt; DB is an octet string of length
         emLen - hLen - 1.
    9.   Let dbMask = MGF(H, emLen - hLen - 1).
    10.  Let maskedDB = DB \xor dbMask.
    11.  Set the leftmost 8emLen - emBits bits of the leftmost octet
         in maskedDB to zero.
    12.  Let EM = maskedDB || H || 0xbc.
    13.  Output EM.

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9.1.2. Verification Operation

 EMSA-PSS-VERIFY (M, EM, emBits)
 Options:
    Hash     hash function (hLen denotes the length in octets of
             the hash function output)
    MGF      mask generation function
    sLen     intended length in octets of the salt
 Input:
    M        message to be verified, an octet string
    EM       encoded message, an octet string of length emLen = \ceil
             (emBits/8)
    emBits   maximal bit length of the integer OS2IP (EM) (see Section
             4.2), at least 8hLen + 8sLen + 9
 Output:  "consistent" or "inconsistent"
 Steps:
    1.   If the length of M is greater than the input limitation for
         the hash function (2^61 - 1 octets for SHA-1), output
         "inconsistent" and stop.
    2.   Let mHash = Hash(M), an octet string of length hLen.
    3.   If emLen < hLen + sLen + 2, output "inconsistent" and stop.
    4.   If the rightmost octet of EM does not have hexadecimal value
         0xbc, output "inconsistent" and stop.
    5.   Let maskedDB be the leftmost emLen - hLen - 1 octets of EM,
         and let H be the next hLen octets.
    6.   If the leftmost 8emLen - emBits bits of the leftmost octet in
         maskedDB are not all equal to zero, output "inconsistent" and
         stop.
    7.   Let dbMask = MGF(H, emLen - hLen - 1).
    8.   Let DB = maskedDB \xor dbMask.
    9.   Set the leftmost 8emLen - emBits bits of the leftmost octet
         in DB to zero.

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    10.  If the emLen - hLen - sLen - 2 leftmost octets of DB are not
         zero or if the octet at position emLen - hLen - sLen - 1 (the
         leftmost position is "position 1") does not have hexadecimal
         value 0x01, output "inconsistent" and stop.
    11.  Let salt be the last sLen octets of DB.
    12.  Let
            M' = (0x)00 00 00 00 00 00 00 00 || mHash || salt ;
         M' is an octet string of length 8 + hLen + sLen with eight
         initial zero octets.
    13.  Let H' = Hash(M'), an octet string of length hLen.
    14.  If H = H', output "consistent".  Otherwise, output
         "inconsistent".

9.2. EMSA-PKCS1-v1_5

 This encoding method is deterministic and only has an encoding
 operation.
 EMSA-PKCS1-v1_5-ENCODE (M, emLen)
 Option:
    Hash     hash function (hLen denotes the length in octets of
             the hash function output)
 Input:
    M        message to be encoded
    emLen    intended length in octets of the encoded message, at
             least tLen + 11, where tLen is the octet length of the
             Distinguished Encoding Rules (DER) encoding T of
             a certain value computed during the encoding operation
 Output:
    EM       encoded message, an octet string of length emLen
 Errors:  "message too long"; "intended encoded message length too
    short"

Moriarty, et al. Informational [Page 45] RFC 8017 PKCS #1 v2.2 November 2016

 Steps:
    1.  Apply the hash function to the message M to produce a hash
        value H:
           H = Hash(M).
        If the hash function outputs "message too long", output
        "message too long" and stop.
    2.  Encode the algorithm ID for the hash function and the hash
        value into an ASN.1 value of type DigestInfo (see
        Appendix A.2.4) with the DER, where the type DigestInfo has
        the syntax
             DigestInfo ::= SEQUENCE {
                 digestAlgorithm AlgorithmIdentifier,
                 digest OCTET STRING
             }
        The first field identifies the hash function and the second
        contains the hash value.  Let T be the DER encoding of the
        DigestInfo value (see the notes below), and let tLen be the
        length in octets of T.
    3.  If emLen < tLen + 11, output "intended encoded message length
        too short" and stop.
    4.  Generate an octet string PS consisting of emLen - tLen - 3
        octets with hexadecimal value 0xff.  The length of PS will be
        at least 8 octets.
    5.  Concatenate PS, the DER encoding T, and other padding to form
        the encoded message EM as
           EM = 0x00 || 0x01 || PS || 0x00 || T.
    6.  Output EM.

Moriarty, et al. Informational [Page 46] RFC 8017 PKCS #1 v2.2 November 2016

 Notes:
 1.  For the nine hash functions mentioned in Appendix B.1, the DER
     encoding T of the DigestInfo value is equal to the following:
       MD2:     (0x)30 20 30 0c 06 08 2a 86 48 86 f7 0d 02 02 05 00 04
                    10 || H.
       MD5:     (0x)30 20 30 0c 06 08 2a 86 48 86 f7 0d 02 05 05 00 04
                    10 || H.
       SHA-1:   (0x)30 21 30 09 06 05 2b 0e 03 02 1a 05 00 04 14 || H.
       SHA-224:  (0x)30 2d 30 0d 06 09 60 86 48 01 65 03 04 02 04
                    05 00 04 1c || H.
       SHA-256: (0x)30 31 30 0d 06 09 60 86 48 01 65 03 04 02 01 05 00
                    04 20 || H.
       SHA-384: (0x)30 41 30 0d 06 09 60 86 48 01 65 03 04 02 02 05 00
                    04 30 || H.
       SHA-512: (0x)30 51 30 0d 06 09 60 86 48 01 65 03 04 02 03 05 00
                    04 40 || H.
       SHA-512/224:  (0x)30 2d 30 0d 06 09 60 86 48 01 65 03 04 02 05
                         05 00 04 1c || H.
       SHA-512/256:  (0x)30 31 30 0d 06 09 60 86 48 01 65 03 04 02 06
                         05 00 04 20 || H.
 2.  In version 1.5 of this document, T was defined as the BER
     encoding, rather than the DER encoding, of the DigestInfo value.
     In particular, it is possible -- at least in theory -- that the
     verification operation defined in this document (as well as in
     version 2.0) rejects a signature that is valid with respect to
     the specification given in PKCS #1 v1.5.  This occurs if other
     rules than DER are applied to DigestInfo (e.g., an indefinite
     length encoding of the underlying SEQUENCE type).  While this is
     unlikely to be a concern in practice, a cautious implementor may
     choose to employ a verification operation based on a BER decoding
     operation as specified in PKCS #1 v1.5.  In this manner,
     compatibility with any valid implementation based on PKCS #1 v1.5
     is obtained.  Such a verification operation should indicate
     whether the underlying BER encoding is a DER encoding and hence
     whether the signature is valid with respect to the specification
     given in this document.

10. Security Considerations

 Security considerations are discussed throughout this memo.

Moriarty, et al. Informational [Page 47] RFC 8017 PKCS #1 v2.2 November 2016

11. References

11.1. Normative References

 [GARNER]   Garner, H., "The Residue Number System", IRE Transactions
            on Electronic Computers, Volume EC-8, Issue 2, pp.
            140-147, DOI 10.1109/TEC.1959.5219515, June 1959.
 [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
            Requirement Levels", BCP 14, RFC 2119,
            DOI 10.17487/RFC2119, March 1997,
            <http://www.rfc-editor.org/info/rfc2119>.
 [RSA]      Rivest, R., Shamir, A., and L. Adleman, "A Method for
            Obtaining Digital Signatures and Public-Key
            Cryptosystems", Communications of the ACM, Volume 21,
            Issue 2, pp. 120-126, DOI 10.1145/359340.359342, February
            1978.

11.2. Informative References

 [ANSIX944] ANSI, "Key Establishment Using Integer Factorization
            Cryptography", ANSI X9.44-2007, August 2007.
 [BKS]      Bleichenbacher, D., Kaliski, B., and J. Staddon, "Recent
            Results on PKCS #1: RSA Encryption Standard", RSA
            Laboratories, Bulletin No. 7, June 1998.
 [BLEICHENBACHER]
            Bleichenbacher, D., "Chosen Ciphertext Attacks Against
            Protocols Based on the RSA Encryption Standard PKCS #1",
            Lecture Notes in Computer Science, Volume 1462, pp. 1-12,
            1998.
 [CHOSEN]   Desmedt, Y. and A. Odlyzko, "A Chosen Text Attack on the
            RSA Cryptosystem and Some Discrete Logarithm Schemes",
            Lecture Notes in Computer Science, Volume 218, pp.
            516-522, 1985.
 [COCHRAN]  Cochran, M., "Notes on the Wang et al. 2^63 SHA-1
            Differential Path", Cryptology ePrint Archive: Report
            2007/474, August 2008, <http://eprint.iacr.org/2007/474>.
 [FASTDEC]  Quisquater, J. and C. Couvreur, "Fast Decipherment
            Algorithm for RSA Public-Key Cryptosystem", Electronic
            Letters, Volume 18, Issue 21, pp. 905-907,
            DOI 10.1049/el:19820617, October 1982.

Moriarty, et al. Informational [Page 48] RFC 8017 PKCS #1 v2.2 November 2016

 [FDH]      Coron, J., "On the Exact Security of Full Domain Hash",
            Lecture Notes in Computer Science, Volume 1880, pp.
            229-235, 2000.
 [FOPS]     Fujisaki, E., Okamoto, T., Pointcheval, D., and J. Stern,
            "RSA-OAEP is Secure under the RSA Assumption", Lecture
            Notes in Computer Science, Volume 2139, pp. 260-274,
            August 2001.
 [FORGERY]  Coppersmith, D., Halevi, S., and C. Jutla, "ISO 9796-1 and
            the new forgery strategy", rump session of Crypto, August
            1999.
 [HAASTAD]  Haastad, J., "Solving Simultaneous Modular Equations of
            Low Degree", SIAM Journal on Computing, Volume 17,
            Issue 2, pp. 336-341, DOI 10.1137/0217019, April 1988.
 [HANDBOOK] Menezes, A., van Oorschot, P., and S. Vanstone, "Handbook
            of Applied Cryptography", CRC Press, ISBN: 0849385237,
            1996.
 [HASHID]   Kaliski, B., "On Hash Function Firewalls in Signature
            Schemes", Lecture Notes in Computer Science, Volume 2271,
            pp. 1-16, DOI 10.1007/3-540-45760-7_1, February 2002.
 [IEEE1363] IEEE, "Standard Specifications for Public Key
            Cryptography", IEEE Std 1363-2000,
            DOI 10.1109/IEEESTD.2000.92292, August 2000,
            <http://ieeexplore.ieee.org/document/891000/>.
 [IEEE1363A]
            IEEE, "Standard Specifications for Public Key Cryptography
            - Amendment 1: Additional Techniques", IEEE Std 1363a-
            2004, DOI 10.1109/IEEESTD.2004.94612, September 2004,
            <http://ieeexplore.ieee.org/document/1335427/>.
 [ISO18033] International Organization for Standardization,
            "Information technology -- Security techniques --
            Encryption algorithms - Part 2: Asymmetric ciphers", ISO/
            IEC 18033-2:2006, May 2006.
 [ISO9594]  International Organization for Standardization,
            "Information technology - Open Systems Interconnection -
            The Directory: Public-key and attribute certificate
            frameworks", ISO/IEC 9594-8:2008, December 2008.

Moriarty, et al. Informational [Page 49] RFC 8017 PKCS #1 v2.2 November 2016

 [ISO9796]  International Organization for Standardization,
            "Information technology - Security techniques - Digital
            signature schemes giving message recovery - Part 2:
            Integer factorization based mechanisms",
            ISO/IEC 9796-2:2010, December 2010.
 [JONSSON]  Jonsson, J., "Security Proofs for the RSA-PSS Signature
            Scheme and Its Variants", Cryptology ePrint
            Archive: Report 2001/053, March 2002,
            <http://eprint.iacr.org/2001/053>.
 [LOWEXP]   Coppersmith, D., Franklin, M., Patarin, J., and M. Reiter,
            "Low-Exponent RSA with Related Messages", Lecture Notes in
            Computer Science, Volume 1070, pp. 1-9, 1996.
 [MANGER]   Manger, J., "A Chosen Ciphertext Attack on RSA Optimal
            Asymmetric Encryption Padding (OAEP) as Standardized in
            PKCS #1 v2.0", Lecture Notes in Computer Science, Volume
            2139, pp. 230-238, DOI 10.1007/3-540-44647-8_14, 2001.
 [MD4]      Dobbertin, H., "Cryptanalysis of MD4", Lecture Notes in
            Computer Science, Volume 1039, pp. 53-69,
            DOI 10.1007/3-540-60865-6_43, 1996.
 [MD4FIRST] Dobbertin, H., "The First Two Rounds of MD4 are Not One-
            Way", Lecture Notes in Computer Science, Volume 1372, pp.
            284-292, DOI 10.1007/3-540-69710-1_19, March 1998.
 [MD4LAST]  den Boer, B. and A. Bosselaers, "An Attack on the Last Two
            Rounds of MD4", Lecture Notes in Computer Science, Volume
            576, pp. 194-203, DOI 10.1007/3-540-46766-1_14, 1992.
 [NEWATTACK]
            Coron, J., Joye, M., Naccache, D., and P. Paillier, "New
            Attacks on PKCS #1 v1.5 Encryption", Lecture Notes in
            Computer Science, Volume 1807, pp. 369-381,
            DOI 10.1007/3-540-45539-6_25, May 2000.
 [OAEP]     Bellare, M. and P. Rogaway, "Optimal Asymmetric Encryption
            - How to Encrypt with RSA", Lecture Notes in Computer
            Science, Volume 950, pp. 92-111, November 1995.
 [PA98]     Bellare, M., Desai, A., Pointcheval, D., and P. Rogaway,
            "Relations Among Notions of Security for Public-Key
            Encryption Schemes", Lecture Notes in Computer
            Science, Volume 1462, pp. 26-45, DOI 10.1007/BFb0055718,
            1998.

Moriarty, et al. Informational [Page 50] RFC 8017 PKCS #1 v2.2 November 2016

 [PADDING]  Coron, J., Naccache, D., and J. Stern, "On the Security of
            RSA Padding", Lecture Notes in Computer Science, Volume
            1666, pp. 1-18, DOI 10.1007/3-540-48405-1_1, December
            1999.
 [PKCS1_22] RSA Laboratories, "PKCS #1: RSA Cryptography Standard
            Version 2.2", October 2012.
 [PREFIX]   Stevens, M., Lenstra, A., and B. de Weger, "Chosen-prefix
            collisions for MD5 and applications", International
            Journal of Applied Cryptography, Volume 2, No. 4, pp.
            322-359, July 2012.
 [PSS]      Bellare, M. and P. Rogaway, "PSS: Provably Secure Encoding
            Method for Digital Signatures", Submission to IEEE P1363a,
            August 1998, <http://grouper.ieee.org/groups/1363/
            P1363a/contributions/pss-submission.pdf>.
 [PSSPROOF] Coron, J., "Optimal Security Proofs for PSS and Other
            Signature Schemes", Lecture Notes in Computer
            Science, Volume 2332, pp. 272-287,
            DOI 10.1007/3-540-46035-7_18, 2002.
 [RFC1319]  Kaliski, B., "The MD2 Message-Digest Algorithm", RFC 1319,
            DOI 10.17487/RFC1319, April 1992,
            <http://www.rfc-editor.org/info/rfc1319>.
 [RFC1321]  Rivest, R., "The MD5 Message-Digest Algorithm", RFC 1321,
            DOI 10.17487/RFC1321, April 1992,
            <http://www.rfc-editor.org/info/rfc1321>.
 [RFC2313]  Kaliski, B., "PKCS #1: RSA Encryption Version 1.5",
            RFC 2313, DOI 10.17487/RFC2313, March 1998,
            <http://www.rfc-editor.org/info/rfc2313>.
 [RFC2315]  Kaliski, B., "PKCS #7: Cryptographic Message Syntax
            Version 1.5", RFC 2315, DOI 10.17487/RFC2315, March 1998,
            <http://www.rfc-editor.org/info/rfc2315>.
 [RFC2437]  Kaliski, B. and J. Staddon, "PKCS #1: RSA Cryptography
            Specifications Version 2.0", RFC 2437,
            DOI 10.17487/RFC2437, October 1998,
            <http://www.rfc-editor.org/info/rfc2437>.
 [RFC3447]  Jonsson, J. and B. Kaliski, "Public-Key Cryptography
            Standards (PKCS) #1: RSA Cryptography Specifications
            Version 2.1", RFC 3447, DOI 10.17487/RFC3447, February
            2003, <http://www.rfc-editor.org/info/rfc3447>.

Moriarty, et al. Informational [Page 51] RFC 8017 PKCS #1 v2.2 November 2016

 [RFC5246]  Dierks, T. and E. Rescorla, "The Transport Layer Security
            (TLS) Protocol Version 1.2", RFC 5246,
            DOI 10.17487/RFC5246, August 2008,
            <http://www.rfc-editor.org/info/rfc5246>.
 [RFC5652]  Housley, R., "Cryptographic Message Syntax (CMS)", STD 70,
            RFC 5652, DOI 10.17487/RFC5652, September 2009,
            <http://www.rfc-editor.org/info/rfc5652>.
 [RFC5958]  Turner, S., "Asymmetric Key Packages", RFC 5958,
            DOI 10.17487/RFC5958, August 2010,
            <http://www.rfc-editor.org/info/rfc5958>.
 [RFC6149]  Turner, S. and L. Chen, "MD2 to Historic Status",
            RFC 6149, DOI 10.17487/RFC6149, March 2011,
            <http://www.rfc-editor.org/info/rfc6149>.
 [RFC7292]  Moriarty, K., Ed., Nystrom, M., Parkinson, S., Rusch, A.,
            and M. Scott, "PKCS #12: Personal Information Exchange
            Syntax v1.1", RFC 7292, DOI 10.17487/RFC7292, July 2014,
            <http://www.rfc-editor.org/info/rfc7292>.
 [RSARABIN] Bellare, M. and P. Rogaway, "The Exact Security of Digital
            Signatures - How to Sign with RSA and Rabin", Lecture
            Notes in Computer Science, Volume 1070, pp. 399-416,
            DOI 10.1007/3-540-68339-9_34, 1996.
 [RSATLS]   Jonsson, J. and B. Kaliski, "On the Security of RSA
            Encryption in TLS", Lecture Notes in Computer
            Science, Volume 2442, pp. 127-142,
            DOI 10.1007/3-540-45708-9_9, 2002.
 [SHA1CRYPT]
            Wang, X., Yao, A., and F. Yao, "Cryptanalysis on SHA-1",
            Lecture Notes in Computer Science, Volume 2442, pp.
            127-142, February 2005,
            <http://csrc.nist.gov/groups/ST/hash/documents/
            Wang_SHA1-New-Result.pdf>.
 [SHOUP]    Shoup, V., "OAEP Reconsidered (Extended Abstract)",
            Lecture Notes in Computer Science, Volume 2139, pp.
            239-259, DOI 10.1007/3-540-44647-8_15, 2001.
 [SHS]      National Institute of Standards and Technology, "Secure
            Hash Standard (SHS)", FIPS PUB 180-4, August 2015,
            <http://dx.doi.org/10.6028/NIST.FIPS.180-4>.

Moriarty, et al. Informational [Page 52] RFC 8017 PKCS #1 v2.2 November 2016

 [SILVERMAN]
            Silverman, R., "A Cost-Based Security Analysis of
            Symmetric and Asymmetric Key Lengths", RSA
            Laboratories, Bulletin No. 13, 2000.
 [SIMMONS]  Simmons, G., "Subliminal Communication is Easy Using the
            DSA", Lecture Notes in Computer Science, Volume 765, pp.
            218-232, DOI 10.1007/3-540-48285-7_18, 1994.

Moriarty, et al. Informational [Page 53] RFC 8017 PKCS #1 v2.2 November 2016

Appendix A. ASN.1 Syntax

A.1. RSA Key Representation

 This section defines ASN.1 object identifiers for RSA public and
 private keys and defines the types RSAPublicKey and RSAPrivateKey.
 The intended application of these definitions includes X.509
 certificates, PKCS #8 [RFC5958], and PKCS #12 [RFC7292].
 The object identifier rsaEncryption identifies RSA public and private
 keys as defined in Appendices A.1.1 and A.1.2.  The parameters field
 has associated with this OID in a value of type AlgorithmIdentifier
 SHALL have a value of type NULL.
    rsaEncryption    OBJECT IDENTIFIER ::= { pkcs-1 1 }
 The definitions in this section have been extended to support multi-
 prime RSA, but they are backward compatible with previous versions.

A.1.1. RSA Public Key Syntax

 An RSA public key should be represented with the ASN.1 type
 RSAPublicKey:
       RSAPublicKey ::= SEQUENCE {
           modulus           INTEGER,  -- n
           publicExponent    INTEGER   -- e
       }
 The fields of type RSAPublicKey have the following meanings:
 o  modulus is the RSA modulus n.
 o  publicExponent is the RSA public exponent e.

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A.1.2. RSA Private Key Syntax

 An RSA private key should be represented with the ASN.1 type
 RSAPrivateKey:
       RSAPrivateKey ::= SEQUENCE {
           version           Version,
           modulus           INTEGER,  -- n
           publicExponent    INTEGER,  -- e
           privateExponent   INTEGER,  -- d
           prime1            INTEGER,  -- p
           prime2            INTEGER,  -- q
           exponent1         INTEGER,  -- d mod (p-1)
           exponent2         INTEGER,  -- d mod (q-1)
           coefficient       INTEGER,  -- (inverse of q) mod p
           otherPrimeInfos   OtherPrimeInfos OPTIONAL
       }
 The fields of type RSAPrivateKey have the following meanings:
 o  version is the version number, for compatibility with future
    revisions of this document.  It SHALL be 0 for this version of the
    document, unless multi-prime is used; in which case, it SHALL be
    1.
          Version ::= INTEGER { two-prime(0), multi(1) }
             (CONSTRAINED BY
             {-- version must be multi if otherPrimeInfos present --})
 o  modulus is the RSA modulus n.
 o  publicExponent is the RSA public exponent e.
 o  privateExponent is the RSA private exponent d.
 o  prime1 is the prime factor p of n.
 o  prime2 is the prime factor q of n.
 o  exponent1 is d mod (p - 1).
 o  exponent2 is d mod (q - 1).
 o  coefficient is the CRT coefficient q^(-1) mod p.

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 o  otherPrimeInfos contains the information for the additional primes
    r_3, ..., r_u, in order.  It SHALL be omitted if version is 0 and
    SHALL contain at least one instance of OtherPrimeInfo if version
    is 1.
          OtherPrimeInfos ::= SEQUENCE SIZE(1..MAX) OF OtherPrimeInfo
          OtherPrimeInfo ::= SEQUENCE {
              prime             INTEGER,  -- ri
              exponent          INTEGER,  -- di
              coefficient       INTEGER   -- ti
          }
 The fields of type OtherPrimeInfo have the following meanings:
 o  prime is a prime factor r_i of n, where i >= 3.
 o  exponent is d_i = d mod (r_i - 1).
 o  coefficient is the CRT coefficient t_i = (r_1 * r_2 * ... *
    r_(i-1))^(-1) mod r_i.
 Note: It is important to protect the RSA private key against both
 disclosure and modification.  Techniques for such protection are
 outside the scope of this document.  Methods for storing and
 distributing private keys and other cryptographic data are described
 in PKCS #12 and #15.

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A.2. Scheme Identification

 This section defines object identifiers for the encryption and
 signature schemes.  The schemes compatible with PKCS #1 v1.5 have the
 same definitions as in PKCS #1 v1.5.  The intended application of
 these definitions includes X.509 certificates and PKCS #7.
 Here are type identifier definitions for the PKCS #1 OIDs:
 PKCS1Algorithms    ALGORITHM-IDENTIFIER ::= {
     { OID rsaEncryption                PARAMETERS NULL } |
     { OID md2WithRSAEncryption         PARAMETERS NULL } |
     { OID md5WithRSAEncryption         PARAMETERS NULL } |
     { OID sha1WithRSAEncryption        PARAMETERS NULL } |
     { OID sha224WithRSAEncryption      PARAMETERS NULL } |
     { OID sha256WithRSAEncryption      PARAMETERS NULL } |
     { OID sha384WithRSAEncryption      PARAMETERS NULL } |
     { OID sha512WithRSAEncryption      PARAMETERS NULL } |
     { OID sha512-224WithRSAEncryption  PARAMETERS NULL } |
     { OID sha512-256WithRSAEncryption  PARAMETERS NULL } |
     { OID id-RSAES-OAEP   PARAMETERS RSAES-OAEP-params } |
     PKCS1PSourceAlgorithms                               |
     { OID id-RSASSA-PSS   PARAMETERS RSASSA-PSS-params },
     ...  -- Allows for future expansion --
 }

A.2.1. RSAES-OAEP

 The object identifier id-RSAES-OAEP identifies the RSAES-OAEP
 encryption scheme.
     id-RSAES-OAEP    OBJECT IDENTIFIER ::= { pkcs-1 7 }
 The parameters field associated with this OID in a value of type
 AlgorithmIdentifier SHALL have a value of type RSAES-OAEP-params:
 RSAES-OAEP-params ::= SEQUENCE {
     hashAlgorithm      [0] HashAlgorithm     DEFAULT sha1,
     maskGenAlgorithm   [1] MaskGenAlgorithm  DEFAULT mgf1SHA1,
     pSourceAlgorithm   [2] PSourceAlgorithm  DEFAULT pSpecifiedEmpty
 }
 The fields of type RSAES-OAEP-params have the following meanings:
 o  hashAlgorithm identifies the hash function.  It SHALL be an
    algorithm ID with an OID in the set OAEP-PSSDigestAlgorithms.  For
    a discussion of supported hash functions, see Appendix B.1.

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     HashAlgorithm ::= AlgorithmIdentifier {
        {OAEP-PSSDigestAlgorithms}
     }
     OAEP-PSSDigestAlgorithms    ALGORITHM-IDENTIFIER ::= {
         { OID id-sha1       PARAMETERS NULL }|
         { OID id-sha224     PARAMETERS NULL }|
         { OID id-sha256     PARAMETERS NULL }|
         { OID id-sha384     PARAMETERS NULL }|
         { OID id-sha512     PARAMETERS NULL }|
         { OID id-sha512-224 PARAMETERS NULL }|
         { OID id-sha512-256 PARAMETERS NULL },
         ...  -- Allows for future expansion --
     }
 The default hash function is SHA-1:
     sha1    HashAlgorithm ::= {
         algorithm   id-sha1,
         parameters  SHA1Parameters : NULL
     }
     SHA1Parameters ::= NULL
 o  maskGenAlgorithm identifies the mask generation function.  It
    SHALL be an algorithm ID with an OID in the set
    PKCS1MGFAlgorithms, which for this version SHALL consist of
    id-mgf1, identifying the MGF1 mask generation function (see
    Appendix B.2.1).  The parameters field associated with id-mgf1
    SHALL be an algorithm ID with an OID in the set
    OAEP-PSSDigestAlgorithms, identifying the hash function on which
    MGF1 is based.
     MaskGenAlgorithm ::= AlgorithmIdentifier { {PKCS1MGFAlgorithms} }
     PKCS1MGFAlgorithms    ALGORITHM-IDENTIFIER ::= {
         { OID id-mgf1 PARAMETERS HashAlgorithm },
         ...  -- Allows for future expansion --
     }
 o  The default mask generation function is MGF1 with SHA-1:
     mgf1SHA1    MaskGenAlgorithm ::= {
         algorithm   id-mgf1,
         parameters  HashAlgorithm : sha1
     }

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 o  pSourceAlgorithm identifies the source (and possibly the value) of
    the label L.  It SHALL be an algorithm ID with an OID in the set
    PKCS1PSourceAlgorithms, which for this version SHALL consist of
    id-pSpecified, indicating that the label is specified explicitly.
    The parameters field associated with id-pSpecified SHALL have a
    value of type OCTET STRING, containing the label.  In previous
    versions of this specification, the term "encoding parameters" was
    used rather than "label", hence the name of the type below.
     PSourceAlgorithm ::= AlgorithmIdentifier {
        {PKCS1PSourceAlgorithms}
     }
     PKCS1PSourceAlgorithms    ALGORITHM-IDENTIFIER ::= {
         { OID id-pSpecified PARAMETERS EncodingParameters },
         ...  -- Allows for future expansion --
     }
     id-pSpecified    OBJECT IDENTIFIER ::= { pkcs-1 9 }
     EncodingParameters ::= OCTET STRING(SIZE(0..MAX))
 o  The default label is an empty string (so that lHash will contain
    the hash of the empty string):
     pSpecifiedEmpty    PSourceAlgorithm ::= {
         algorithm   id-pSpecified,
         parameters  EncodingParameters : emptyString
     }
     emptyString    EncodingParameters ::= ''H
 If all of the default values of the fields in RSAES-OAEP-params are
 used, then the algorithm identifier will have the following value:
     rSAES-OAEP-Default-Identifier    RSAES-AlgorithmIdentifier ::= {
         algorithm   id-RSAES-OAEP,
         parameters  RSAES-OAEP-params : {
             hashAlgorithm       sha1,
             maskGenAlgorithm    mgf1SHA1,
             pSourceAlgorithm    pSpecifiedEmpty
         }
     }
     RSAES-AlgorithmIdentifier ::= AlgorithmIdentifier  {
         {PKCS1Algorithms}
     }

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A.2.2. RSAES-PKCS-v1_5

 The object identifier rsaEncryption (see Appendix A.1) identifies the
 RSAES-PKCS1-v1_5 encryption scheme.  The parameters field associated
 with this OID in a value of type AlgorithmIdentifier SHALL have a
 value of type NULL.  This is the same as in PKCS #1 v1.5.
     rsaEncryption    OBJECT IDENTIFIER ::= { pkcs-1 1 }

A.2.3. RSASSA-PSS

 The object identifier id-RSASSA-PSS identifies the RSASSA-PSS
 encryption scheme.
     id-RSASSA-PSS    OBJECT IDENTIFIER ::= { pkcs-1 10 }
 The parameters field associated with this OID in a value of type
 AlgorithmIdentifier SHALL have a value of type RSASSA-PSS-params:
 RSASSA-PSS-params ::= SEQUENCE {
     hashAlgorithm      [0] HashAlgorithm      DEFAULT sha1,
     maskGenAlgorithm   [1] MaskGenAlgorithm   DEFAULT mgf1SHA1,
     saltLength         [2] INTEGER            DEFAULT 20,
     trailerField       [3] TrailerField       DEFAULT trailerFieldBC
 }
 The fields of type RSASSA-PSS-params have the following meanings:
 o  hashAlgorithm identifies the hash function.  It SHALL be an
    algorithm ID with an OID in the set OAEP-PSSDigestAlgorithms (see
    Appendix A.2.1).  The default hash function is SHA-1.
 o  maskGenAlgorithm identifies the mask generation function.  It
    SHALL be an algorithm ID with an OID in the set PKCS1MGFAlgorithms
    (see Appendix A.2.1).  The default mask generation function is
    MGF1 with SHA-1.  For MGF1 (and more generally, for other mask
    generation functions based on a hash function), it is RECOMMENDED
    that the underlying hash function be the same as the one
    identified by hashAlgorithm; see Note 2 in Section 9.1 for further
    comments.
 o  saltLength is the octet length of the salt.  It SHALL be an
    integer.  For a given hashAlgorithm, the default value of
    saltLength is the octet length of the hash value.  Unlike the
    other fields of type RSASSA-PSS-params, saltLength does not need
    to be fixed for a given RSA key pair.

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 o  trailerField is the trailer field number, for compatibility with
    IEEE 1363a [IEEE1363A].  It SHALL be 1 for this version of the
    document, which represents the trailer field with hexadecimal
    value 0xbc.  Other trailer fields (including the trailer field
    HashID || 0xcc in IEEE 1363a) are not supported in this document.
     TrailerField ::= INTEGER { trailerFieldBC(1) }
 If the default values of the hashAlgorithm, maskGenAlgorithm, and
 trailerField fields of RSASSA-PSS-params are used, then the algorithm
 identifier will have the following value:
     rSASSA-PSS-Default-Identifier    RSASSA-AlgorithmIdentifier ::= {
         algorithm   id-RSASSA-PSS,
         parameters  RSASSA-PSS-params : {
             hashAlgorithm       sha1,
             maskGenAlgorithm    mgf1SHA1,
             saltLength          20,
             trailerField        trailerFieldBC
         }
     }
     RSASSA-AlgorithmIdentifier ::= AlgorithmIdentifier {
         {PKCS1Algorithms}
     }
 Note: In some applications, the hash function underlying a signature
 scheme is identified separately from the rest of the operations in
 the signature scheme.  For instance, in PKCS #7 [RFC2315], a hash
 function identifier is placed before the message and a "digest
 encryption" algorithm identifier (indicating the rest of the
 operations) is carried with the signature.  In order for PKCS #7 to
 support the RSASSA-PSS signature scheme, an object identifier would
 need to be defined for the operations in RSASSA-PSS after the hash
 function (analogous to the RSAEncryption OID for the
 RSASSA-PKCS1-v1_5 scheme).  S/MIME Cryptographic Message Syntax (CMS)
 [RFC5652] takes a different approach.  Although a hash function
 identifier is placed before the message, an algorithm identifier for
 the full signature scheme may be carried with a CMS signature (this
 is done for DSA signatures).  Following this convention, the
 id-RSASSA-PSS OID can be used to identify RSASSA-PSS signatures in
 CMS.  Since CMS is considered the successor to PKCS #7 and new
 developments such as the addition of support for RSASSA-PSS will be
 pursued with respect to CMS rather than PKCS #7, an OID for the "rest
 of" RSASSA-PSS is not defined in this version of PKCS #1.

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A.2.4. RSASSA-PKCS-v1_5

 The object identifier for RSASSA-PKCS1-v1_5 SHALL be one of the
 following.  The choice of OID depends on the choice of hash
 algorithm: MD2, MD5, SHA-1, SHA-224, SHA-256, SHA-384, SHA-512,
 SHA-512/224, or SHA-512/256.  Note that if either MD2 or MD5 is used,
 then the OID is just as in PKCS #1 v1.5.  For each OID, the
 parameters field associated with this OID in a value of type
 AlgorithmIdentifier SHALL have a value of type NULL.  The OID should
 be chosen in accordance with the following table:
       Hash algorithm   OID
       ------------------------------------------------------------
       MD2              md2WithRSAEncryption        ::= {pkcs-1 2}
       MD5              md5WithRSAEncryption        ::= {pkcs-1 4}
       SHA-1            sha1WithRSAEncryption       ::= {pkcs-1 5}
       SHA-256          sha224WithRSAEncryption     ::= {pkcs-1 14}
       SHA-256          sha256WithRSAEncryption     ::= {pkcs-1 11}
       SHA-384          sha384WithRSAEncryption     ::= {pkcs-1 12}
       SHA-512          sha512WithRSAEncryption     ::= {pkcs-1 13}
       SHA-512/224      sha512-224WithRSAEncryption ::= {pkcs-1 15}
       SHA-512/256      sha512-256WithRSAEncryption ::= {pkcs-1 16}
 The EMSA-PKCS1-v1_5 encoding method includes an ASN.1 value of type
 DigestInfo, where the type DigestInfo has the syntax
     DigestInfo ::= SEQUENCE {
         digestAlgorithm DigestAlgorithm,
         digest OCTET STRING
     }
 digestAlgorithm identifies the hash function and SHALL be an
 algorithm ID with an OID in the set PKCS1-v1-5DigestAlgorithms.  For
 a discussion of supported hash functions, see Appendix B.1.

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     DigestAlgorithm ::= AlgorithmIdentifier {
        {PKCS1-v1-5DigestAlgorithms}
     }
     PKCS1-v1-5DigestAlgorithms    ALGORITHM-IDENTIFIER ::= {
         { OID id-md2        PARAMETERS NULL }|
         { OID id-md5        PARAMETERS NULL }|
         { OID id-sha1       PARAMETERS NULL }|
         { OID id-sha224     PARAMETERS NULL }|
         { OID id-sha256     PARAMETERS NULL }|
         { OID id-sha384     PARAMETERS NULL }|
         { OID id-sha512     PARAMETERS NULL }|
         { OID id-sha512-224 PARAMETERS NULL }|
         { OID id-sha512-256 PARAMETERS NULL }
     }

Appendix B. Supporting Techniques

 This section gives several examples of underlying functions
 supporting the encryption schemes in Section 7 and the encoding
 methods in Section 9.  A range of techniques is given here to allow
 compatibility with existing applications as well as migration to new
 techniques.  While these supporting techniques are appropriate for
 applications to implement, none of them is required to be
 implemented.  It is expected that profiles for PKCS #1 v2.2 will be
 developed that specify particular supporting techniques.
 This section also gives object identifiers for the supporting
 techniques.

B.1. Hash Functions

 Hash functions are used in the operations contained in Sections 7 and
 9.  Hash functions are deterministic, meaning that the output is
 completely determined by the input.  Hash functions take octet
 strings of variable length and generate fixed-length octet strings.
 The hash functions used in the operations contained in Sections 7 and
 9 should generally be collision-resistant.  This means that it is
 infeasible to find two distinct inputs to the hash function that
 produce the same output.  A collision-resistant hash function also
 has the desirable property of being one-way; this means that given an
 output, it is infeasible to find an input whose hash is the specified
 output.  In addition to the requirements, the hash function should
 yield a mask generation function (Appendix B.2) with pseudorandom
 output.

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 Nine hash functions are given as examples for the encoding methods in
 this document: MD2 [RFC1319] (which was retired by [RFC6149]), MD5
 [RFC1321], SHA-1, SHA-224, SHA-256, SHA-384, SHA-512, SHA-512/224,
 and SHA-512/256 [SHS].  For the RSAES-OAEP encryption scheme and
 EMSA-PSS encoding method, only SHA-1, SHA-224, SHA-256, SHA-384, SHA-
 512, SHA-512/224, and SHA-512/256 are RECOMMENDED.  For the EMSA-
 PKCS1-v1_5 encoding method, SHA-224, SHA-256, SHA-384, SHA-512, SHA-
 512/224, and SHA-512/256 are RECOMMENDED for new applications.  MD2,
 MD5, and SHA-1 are recommended only for compatibility with existing
 applications based on PKCS #1 v1.5.
 The object identifiers id-md2, id-md5, id-sha1, id-sha224, id-sha256,
 id-sha384, id-sha512, id-sha512/224, and id-sha512/256 identify the
 respective hash functions:
     id-md2      OBJECT IDENTIFIER ::= {
         iso (1) member-body (2) us (840) rsadsi (113549)
         digestAlgorithm (2) 2
     }
     id-md5      OBJECT IDENTIFIER ::= {
         iso (1) member-body (2) us (840) rsadsi (113549)
         digestAlgorithm (2) 5
     }
     id-sha1    OBJECT IDENTIFIER ::= {
         iso(1) identified-organization(3) oiw(14) secsig(3)
          algorithms(2) 26
     }
     id-sha224    OBJECT IDENTIFIER ::= {
         joint-iso-itu-t (2) country (16) us (840) organization (1)
         gov (101) csor (3) nistalgorithm (4) hashalgs (2) 4
     }
     id-sha256    OBJECT IDENTIFIER ::= {
         joint-iso-itu-t (2) country (16) us (840) organization (1)
         gov (101) csor (3) nistalgorithm (4) hashalgs (2) 1
     }
     id-sha384    OBJECT IDENTIFIER ::= {
         joint-iso-itu-t (2) country (16) us (840) organization (1)
         gov (101) csor (3) nistalgorithm (4) hashalgs (2) 2
     }

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     id-sha512    OBJECT IDENTIFIER ::= {
         joint-iso-itu-t (2) country (16) us (840) organization (1)
         gov (101) csor (3) nistalgorithm (4) hashalgs (2) 3
     }
     id-sha512-224    OBJECT IDENTIFIER ::= {
         joint-iso-itu-t (2) country (16) us (840) organization (1)
         gov (101) csor (3) nistalgorithm (4) hashalgs (2) 5
     }
     id-sha512-256    OBJECT IDENTIFIER ::= {
         joint-iso-itu-t (2) country (16) us (840) organization (1)
         gov (101) csor (3) nistalgorithm (4) hashalgs (2) 6
     }
 The parameters field associated with these OIDs in a value of type
 AlgorithmIdentifier SHALL have a value of type NULL.
 The parameters field associated with id-md2 and id-md5 in a value of
 type AlgorithmIdentifier shall have a value of type NULL.
 The parameters field associated with id-sha1, id-sha224, id-sha256,
 id-sha384, id-sha512, id-sha512/224, and id-sha512/256 should
 generally be omitted, but if present, it shall have a value of type
 NULL.
 This is to align with the definitions originally promulgated by NIST.
 For the SHA algorithms, implementations MUST accept
 AlgorithmIdentifier values both without parameters and with NULL
 parameters.
 Exception: When formatting the DigestInfoValue in EMSA-PKCS1-v1_5
 (see Section 9.2), the parameters field associated with id-sha1,
 id-sha224, id-sha256, id-sha384, id-sha512, id-sha512/224, and
 id-sha512/256 shall have a value of type NULL.  This is to maintain
 compatibility with existing implementations and with the numeric
 information values already published for EMSA-PKCS1-v1_5, which are
 also reflected in IEEE 1363a [IEEE1363A].
 Note: Version 1.5 of PKCS #1 also allowed for the use of MD4 in
 signature schemes.  The cryptanalysis of MD4 has progressed
 significantly in the intervening years.  For example, Dobbertin [MD4]
 demonstrated how to find collisions for MD4 and that the first two
 rounds of MD4 are not one-way [MD4FIRST].  Because of these results
 and others (e.g., [MD4LAST]), MD4 is NOT RECOMMENDED.
 Further advances have been made in the cryptanalysis of MD2 and MD5,
 especially after the findings of Stevens et al.  [PREFIX] on chosen-

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 prefix collisions on MD5.  MD2 and MD5 should be considered
 cryptographically broken and removed from existing applications.
 This version of the standard supports MD2 and MD5 just for backwards-
 compatibility reasons.
 There have also been advances in the cryptanalysis of SHA-1.
 Particularly, the results of Wang et al.  [SHA1CRYPT] (which have
 been independently verified by M.  Cochran in his analysis [COCHRAN])
 on using a differential path to find collisions in SHA-1, which
 conclude that the security strength of the SHA-1 hashing algorithm is
 significantly reduced.  However, this reduction is not significant
 enough to warrant the removal of SHA-1 from existing applications,
 but its usage is only recommended for backwards-compatibility
 reasons.
 To address these concerns, only SHA-224, SHA-256, SHA-384, SHA-512,
 SHA-512/224, and SHA-512/256 are RECOMMENDED for new applications.
 As of today, the best (known) collision attacks against these hash
 functions are generic attacks with complexity 2L/2, where L is the
 bit length of the hash output.  For the signature schemes in this
 document, a collision attack is easily translated into a signature
 forgery.  Therefore, the value L / 2 should be at least equal to the
 desired security level in bits of the signature scheme (a security
 level of B bits means that the best attack has complexity 2B).  The
 same rule of thumb can be applied to RSAES-OAEP; it is RECOMMENDED
 that the bit length of the seed (which is equal to the bit length of
 the hash output) be twice the desired security level in bits.

B.2. Mask Generation Functions

 A mask generation function takes an octet string of variable length
 and a desired output length as input and outputs an octet string of
 the desired length.  There may be restrictions on the length of the
 input and output octet strings, but such bounds are generally very
 large.  Mask generation functions are deterministic; the octet string
 output is completely determined by the input octet string.  The
 output of a mask generation function should be pseudorandom: Given
 one part of the output but not the input, it should be infeasible to
 predict another part of the output.  The provable security of
 RSAES-OAEP and RSASSA-PSS relies on the random nature of the output
 of the mask generation function, which in turn relies on the random
 nature of the underlying hash.
 One mask generation function is given here: MGF1, which is based on a
 hash function.  MGF1 coincides with the mask generation functions
 defined in IEEE 1363 [IEEE1363] and ANSI X9.44 [ANSIX944].  Future
 versions of this document may define other mask generation functions.

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B.2.1. MGF1

 MGF1 is a mask generation function based on a hash function.
 MGF1 (mgfSeed, maskLen)
 Options:
    Hash     hash function (hLen denotes the length in octets of
             the hash function output)
 Input:
    mgfSeed  seed from which mask is generated, an octet string
    maskLen  intended length in octets of the mask, at most 2^32 hLen
 Output:
    mask     mask, an octet string of length maskLen
 Error: "mask too long"
 Steps:
 1.  If maskLen > 2^32 hLen, output "mask too long" and stop.
 2.  Let T be the empty octet string.
 3.  For counter from 0 to \ceil (maskLen / hLen) - 1, do the
     following:
     A.  Convert counter to an octet string C of length 4 octets (see
         Section 4.1):
            C = I2OSP (counter, 4) .
     B.  Concatenate the hash of the seed mgfSeed and C to the octet
         string T:
            T = T || Hash(mgfSeed || C) .
 4.  Output the leading maskLen octets of T as the octet string mask.
 The object identifier id-mgf1 identifies the MGF1 mask generation
 function:
    id-mgf1    OBJECT IDENTIFIER ::= { pkcs-1 8 }

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 The parameters field associated with this OID in a value of type
 AlgorithmIdentifier shall have a value of type hashAlgorithm,
 identifying the hash function on which MGF1 is based.

Appendix C. ASN.1 Module

  1. - PKCS #1 v2.2 ASN.1 Module
  2. - Revised October 27, 2012
  1. - This module has been checked for conformance with the
  2. - ASN.1 standard by the OSS ASN.1 Tools
 PKCS-1 {
     iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) pkcs-1(1)
     modules(0) pkcs-1(1)
 }
 DEFINITIONS EXPLICIT TAGS ::=
 BEGIN
  1. - EXPORTS ALL
  2. - All types and values defined in this module are exported for use
  3. - in other ASN.1 modules.
 IMPORTS
 id-sha224, id-sha256, id-sha384, id-sha512, id-sha512-224,
 id-sha512-256
     FROM NIST-SHA2 {
         joint-iso-itu-t(2) country(16) us(840) organization(1)
         gov(101) csor(3) nistalgorithm(4) hashAlgs(2)
     };
  1. - ============================
  2. - Basic object identifiers
  3. - ============================
  1. - The DER encoding of this in hexadecimal is:
  2. - (0x)06 08
  3. - 2A 86 48 86 F7 0D 01 01
  4. -

pkcs-1 OBJECT IDENTIFIER ::= {

     iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) 1
 }
  1. -
  2. - When rsaEncryption is used in an AlgorithmIdentifier,

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  1. - the parameters MUST be present and MUST be NULL.
  2. -

rsaEncryption OBJECT IDENTIFIER ::= { pkcs-1 1 }

  1. -
  2. - When id-RSAES-OAEP is used in an AlgorithmIdentifier, the
  3. - parameters MUST be present and MUST be RSAES-OAEP-params.
  4. -

id-RSAES-OAEP OBJECT IDENTIFIER ::= { pkcs-1 7 }

  1. -
  2. - When id-pSpecified is used in an AlgorithmIdentifier, the
  3. - parameters MUST be an OCTET STRING.
  4. -

id-pSpecified OBJECT IDENTIFIER ::= { pkcs-1 9 }

  1. -
  2. - When id-RSASSA-PSS is used in an AlgorithmIdentifier, the
  3. - parameters MUST be present and MUST be RSASSA-PSS-params.
  4. -

id-RSASSA-PSS OBJECT IDENTIFIER ::= { pkcs-1 10 }

  1. -
  2. - When the following OIDs are used in an AlgorithmIdentifier,
  3. - the parameters MUST be present and MUST be NULL.
  4. -

md2WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 2 }

 md5WithRSAEncryption         OBJECT IDENTIFIER ::= { pkcs-1 4 }
 sha1WithRSAEncryption        OBJECT IDENTIFIER ::= { pkcs-1 5 }
 sha224WithRSAEncryption      OBJECT IDENTIFIER ::= { pkcs-1 14 }
 sha256WithRSAEncryption      OBJECT IDENTIFIER ::= { pkcs-1 11 }
 sha384WithRSAEncryption      OBJECT IDENTIFIER ::= { pkcs-1 12 }
 sha512WithRSAEncryption      OBJECT IDENTIFIER ::= { pkcs-1 13 }
 sha512-224WithRSAEncryption  OBJECT IDENTIFIER ::= { pkcs-1 15 }
 sha512-256WithRSAEncryption  OBJECT IDENTIFIER ::= { pkcs-1 16 }
  1. -
  2. - This OID really belongs in a module with the secsig OIDs.
  3. -

id-sha1 OBJECT IDENTIFIER ::= {

     iso(1) identified-organization(3) oiw(14) secsig(3) algorithms(2)
     26
 }
  1. -
  2. - OIDs for MD2 and MD5, allowed only in EMSA-PKCS1-v1_5.
  3. -

id-md2 OBJECT IDENTIFIER ::= {

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     iso(1) member-body(2) us(840) rsadsi(113549) digestAlgorithm(2) 2
 }
 id-md5 OBJECT IDENTIFIER ::= {
     iso(1) member-body(2) us(840) rsadsi(113549) digestAlgorithm(2) 5
 }
  1. -
  2. - When id-mgf1 is used in an AlgorithmIdentifier, the parameters
  3. - MUST be present and MUST be a HashAlgorithm, for example, sha1.
  4. -

id-mgf1 OBJECT IDENTIFIER ::= { pkcs-1 8 }

  1. - ================
  2. - Useful types
  3. - ================
 ALGORITHM-IDENTIFIER ::= CLASS {
     &id    OBJECT IDENTIFIER  UNIQUE,
     &Type  OPTIONAL
 }
     WITH SYNTAX { OID &id [PARAMETERS &Type] }
  1. - Note: the parameter InfoObjectSet in the following definitions
  2. - allows a distinct information object set to be specified for sets
  3. - of algorithms such as:
  4. - DigestAlgorithms ALGORITHM-IDENTIFIER ::= {
  5. - { OID id-md2 PARAMETERS NULL }|
  6. - { OID id-md5 PARAMETERS NULL }|
  7. - { OID id-sha1 PARAMETERS NULL }
  8. - }
  9. -
 AlgorithmIdentifier { ALGORITHM-IDENTIFIER:InfoObjectSet } ::=
     SEQUENCE {
       algorithm
           ALGORITHM-IDENTIFIER.&id({InfoObjectSet}),
       parameters
           ALGORITHM-IDENTIFIER.&Type({InfoObjectSet}{@.algorithm})
             OPTIONAL
 }
  1. - ==============
  2. - Algorithms
  3. - ==============
  1. -
  2. - Allowed EME-OAEP and EMSA-PSS digest algorithms.

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  1. -

OAEP-PSSDigestAlgorithms ALGORITHM-IDENTIFIER ::= {

     { OID id-sha1       PARAMETERS NULL }|
     { OID id-sha224     PARAMETERS NULL }|
     { OID id-sha256     PARAMETERS NULL }|
     { OID id-sha384     PARAMETERS NULL }|
     { OID id-sha512     PARAMETERS NULL }|
     { OID id-sha512-224 PARAMETERS NULL }|
     { OID id-sha512-256 PARAMETERS NULL },
     ...  -- Allows for future expansion --
 }
  1. -
  2. - Allowed EMSA-PKCS1-v1_5 digest algorithms.
  3. -

PKCS1-v1-5DigestAlgorithms ALGORITHM-IDENTIFIER ::= {

     { OID id-md2        PARAMETERS NULL }|
     { OID id-md5        PARAMETERS NULL }|
     { OID id-sha1       PARAMETERS NULL }|
     { OID id-sha224     PARAMETERS NULL }|
     { OID id-sha256     PARAMETERS NULL }|
     { OID id-sha384     PARAMETERS NULL }|
     { OID id-sha512     PARAMETERS NULL }|
     { OID id-sha512-224 PARAMETERS NULL }|
     { OID id-sha512-256 PARAMETERS NULL }
 }
  1. - When id-md2 and id-md5 are used in an AlgorithmIdentifier, the
  2. - parameters field shall have a value of type NULL.
  1. - When id-sha1, id-sha224, id-sha256, id-sha384, id-sha512,
  2. - id-sha512-224, and id-sha512-256 are used in an
  3. - AlgorithmIdentifier, the parameters (which are optional) SHOULD be
  4. - omitted, but if present, they SHALL have a value of type NULL.
  5. - However, implementations MUST accept AlgorithmIdentifier values
  6. - both without parameters and with NULL parameters.
  1. - Exception: When formatting the DigestInfoValue in EMSA-PKCS1-v1_5
  2. - (see Section 9.2), the parameters field associated with id-sha1,
  3. - id-sha224, id-sha256, id-sha384, id-sha512, id-sha512-224, and
  4. - id-sha512-256 SHALL have a value of type NULL. This is to
  5. - maintain compatibility with existing implementations and with the
  6. - numeric information values already published for EMSA-PKCS1-v1_5,
  7. - which are also reflected in IEEE 1363a.
 sha1    HashAlgorithm ::= {
     algorithm   id-sha1,
     parameters  SHA1Parameters : NULL

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 }
 HashAlgorithm ::= AlgorithmIdentifier { {OAEP-PSSDigestAlgorithms} }
 SHA1Parameters ::= NULL
  1. -
  2. - Allowed mask generation function algorithms.
  3. - If the identifier is id-mgf1, the parameters are a HashAlgorithm.
  4. -

PKCS1MGFAlgorithms ALGORITHM-IDENTIFIER ::= {

     { OID id-mgf1 PARAMETERS HashAlgorithm },
     ...  -- Allows for future expansion --
 }
  1. -
  2. - Default AlgorithmIdentifier for id-RSAES-OAEP.maskGenAlgorithm and
  3. - id-RSASSA-PSS.maskGenAlgorithm.
  4. -

mgf1SHA1 MaskGenAlgorithm ::= {

     algorithm   id-mgf1,
     parameters  HashAlgorithm : sha1
 }
 MaskGenAlgorithm ::= AlgorithmIdentifier { {PKCS1MGFAlgorithms} }
  1. -
  2. - Allowed algorithms for pSourceAlgorithm.
  3. -

PKCS1PSourceAlgorithms ALGORITHM-IDENTIFIER ::= {

     { OID id-pSpecified PARAMETERS EncodingParameters },
     ...  -- Allows for future expansion --
 }
 EncodingParameters ::= OCTET STRING(SIZE(0..MAX))
  1. -
  2. - This identifier means that the label L is an empty string, so the
  3. - digest of the empty string appears in the RSA block before
  4. - masking.
  5. -
 pSpecifiedEmpty    PSourceAlgorithm ::= {
     algorithm   id-pSpecified,
     parameters  EncodingParameters : emptyString
 }
 PSourceAlgorithm ::= AlgorithmIdentifier { {PKCS1PSourceAlgorithms} }

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 emptyString    EncodingParameters ::= ''H
  1. -
  2. - Type identifier definitions for the PKCS #1 OIDs.
  3. -

PKCS1Algorithms ALGORITHM-IDENTIFIER ::= {

     { OID rsaEncryption                PARAMETERS NULL } |
     { OID md2WithRSAEncryption         PARAMETERS NULL } |
     { OID md5WithRSAEncryption         PARAMETERS NULL } |
     { OID sha1WithRSAEncryption        PARAMETERS NULL } |
     { OID sha224WithRSAEncryption      PARAMETERS NULL } |
     { OID sha256WithRSAEncryption      PARAMETERS NULL } |
     { OID sha384WithRSAEncryption      PARAMETERS NULL } |
     { OID sha512WithRSAEncryption      PARAMETERS NULL } |
     { OID sha512-224WithRSAEncryption  PARAMETERS NULL } |
     { OID sha512-256WithRSAEncryption  PARAMETERS NULL } |
     { OID id-RSAES-OAEP   PARAMETERS RSAES-OAEP-params } |
     PKCS1PSourceAlgorithms                               |
     { OID id-RSASSA-PSS   PARAMETERS RSASSA-PSS-params },
     ...  -- Allows for future expansion --
 }
  1. - ===================
  2. - Main structures
  3. - ===================
 RSAPublicKey ::= SEQUENCE {
     modulus           INTEGER,  -- n
     publicExponent    INTEGER   -- e
 }
  1. -
  2. - Representation of RSA private key with information for the CRT
  3. - algorithm.
  4. -

RSAPrivateKey ::= SEQUENCE {

     version           Version,
     modulus           INTEGER,  -- n
     publicExponent    INTEGER,  -- e
     privateExponent   INTEGER,  -- d
     prime1            INTEGER,  -- p
     prime2            INTEGER,  -- q
     exponent1         INTEGER,  -- d mod (p-1)
     exponent2         INTEGER,  -- d mod (q-1)
     coefficient       INTEGER,  -- (inverse of q) mod p
     otherPrimeInfos   OtherPrimeInfos OPTIONAL
 }

Moriarty, et al. Informational [Page 73] RFC 8017 PKCS #1 v2.2 November 2016

 Version ::= INTEGER { two-prime(0), multi(1) }
     (CONSTRAINED BY
       {-- version MUST
  be multi if otherPrimeInfos present --})
 OtherPrimeInfos ::= SEQUENCE SIZE(1..MAX) OF OtherPrimeInfo
 OtherPrimeInfo ::= SEQUENCE {
     prime             INTEGER,  -- ri
     exponent          INTEGER,  -- di
     coefficient       INTEGER   -- ti
 }
  1. -
  2. - AlgorithmIdentifier.parameters for id-RSAES-OAEP.
  3. - Note that the tags in this Sequence are explicit.
  4. -

RSAES-OAEP-params ::= SEQUENCE {

     hashAlgorithm      [0] HashAlgorithm     DEFAULT sha1,
     maskGenAlgorithm   [1] MaskGenAlgorithm  DEFAULT mgf1SHA1,
     pSourceAlgorithm   [2] PSourceAlgorithm  DEFAULT pSpecifiedEmpty
 }
  1. -
  2. - Identifier for default RSAES-OAEP algorithm identifier.
  3. - The DER encoding of this is in hexadecimal:
  4. - (0x)30 0D
  5. - 06 09
  6. - 2A 86 48 86 F7 0D 01 01 07
  7. - 30 00
  8. - Notice that the DER encoding of default values is "empty".
  9. -
 rSAES-OAEP-Default-Identifier    RSAES-AlgorithmIdentifier ::= {
     algorithm   id-RSAES-OAEP,
     parameters  RSAES-OAEP-params : {
         hashAlgorithm       sha1,
         maskGenAlgorithm    mgf1SHA1,
         pSourceAlgorithm    pSpecifiedEmpty
     }
 }
 RSAES-AlgorithmIdentifier ::= AlgorithmIdentifier {
     {PKCS1Algorithms}
 }
  1. -

Moriarty, et al. Informational [Page 74] RFC 8017 PKCS #1 v2.2 November 2016

  1. - AlgorithmIdentifier.parameters for id-RSASSA-PSS.
  2. - Note that the tags in this Sequence are explicit.
  3. -

RSASSA-PSS-params ::= SEQUENCE {

     hashAlgorithm      [0] HashAlgorithm      DEFAULT sha1,
     maskGenAlgorithm   [1] MaskGenAlgorithm   DEFAULT mgf1SHA1,
     saltLength         [2] INTEGER            DEFAULT 20,
     trailerField       [3] TrailerField       DEFAULT trailerFieldBC
 }
 TrailerField ::= INTEGER { trailerFieldBC(1) }
  1. -
  2. - Identifier for default RSASSA-PSS algorithm identifier
  3. - The DER encoding of this is in hexadecimal:
  4. - (0x)30 0D
  5. - 06 09
  6. - 2A 86 48 86 F7 0D 01 01 0A
  7. - 30 00
  8. - Notice that the DER encoding of default values is "empty".
  9. -

rSASSA-PSS-Default-Identifier RSASSA-AlgorithmIdentifier ::= {

     algorithm   id-RSASSA-PSS,
     parameters  RSASSA-PSS-params : {
         hashAlgorithm       sha1,
         maskGenAlgorithm    mgf1SHA1,
         saltLength          20,
         trailerField        trailerFieldBC
     }
 }
 RSASSA-AlgorithmIdentifier ::= AlgorithmIdentifier {
     {PKCS1Algorithms}
 }
  1. -
  2. - Syntax for the EMSA-PKCS1-v1_5 hash identifier.
  3. -

DigestInfo ::= SEQUENCE {

     digestAlgorithm DigestAlgorithm,
     digest OCTET STRING
 }
 DigestAlgorithm ::= AlgorithmIdentifier {
     {PKCS1-v1-5DigestAlgorithms}
 }
 END

Moriarty, et al. Informational [Page 75] RFC 8017 PKCS #1 v2.2 November 2016

Appendix D. Revision History of PKCS #1

 Versions 1.0 - 1.5:
    Versions 1.0 - 1.3 were distributed to participants in RSA Data
    Security, Inc.'s Public-Key Cryptography Standards meetings in
    February and March 1991.
    Version 1.4 was part of the June 3, 1991 initial public release of
    PKCS.  Version 1.4 was published as NIST/OSI Implementors'
    Workshop document SEC-SIG-91-18.
    Version 1.5 incorporated several editorial changes, including
    updates to the references and the addition of a revision history.
    The following substantive changes were made:
  • Section 10: "MD4 with RSA" signature and verification processes

were added.

  • Section 11: md4WithRSAEncryption object identifier was added.
    Version 1.5 was republished as [RFC2313] (which was later
    obsoleted by [RFC2437]).
 Version 2.0:
    Version 2.0 incorporated major editorial changes in terms of the
    document structure and introduced the RSAES-OAEP encryption
    scheme.  This version continued to support the encryption and
    signature processes in version 1.5, although the hash algorithm
    MD4 was no longer allowed due to cryptanalytic advances in the
    intervening years.  Version 2.0 was republished as [RFC2437]
    (which was later obsoleted by [RFC3447]).
 Version 2.1:
    Version 2.1 introduced multi-prime RSA and the RSASSA-PSS
    signature scheme with appendix along with several editorial
    improvements.  This version continued to support the schemes in
    version 2.0.  Version 2.1 was republished as [RFC3447].

Moriarty, et al. Informational [Page 76] RFC 8017 PKCS #1 v2.2 November 2016

 Version 2.2:
    Version 2.2 updates the list of allowed hashing algorithms to
    align them with FIPS 180-4 [SHS], therefore adding SHA-224,
    SHA-512/224, and SHA-512/256.  The following substantive changes
    were made:
  • Object identifiers for sha224WithRSAEncryption,

sha512-224WithRSAEncryption, and sha512-256WithRSAEncryption

       were added.
  • This version continues to support the schemes in version 2.1.

Appendix E. About PKCS

 The Public-Key Cryptography Standards are specifications produced by
 RSA Laboratories in cooperation with secure systems developers
 worldwide for the purpose of accelerating the deployment of public-
 key cryptography.  First published in 1991 as a result of meetings
 with a small group of early adopters of public-key technology, the
 PKCS documents have become widely referenced and implemented.
 Contributions from the PKCS series have become part of many formal
 and de facto standards, including ANSI X9 and IEEE P1363 documents,
 PKIX, Secure Electronic Transaction (SET), S/MIME, SSL/TLS, and
 Wireless Application Protocol (WAP) / WAP Transport Layer Security
 (WTLS).
 Further development of most PKCS documents occurs through the IETF.
 Suggestions for improvement are welcome.

Moriarty, et al. Informational [Page 77] RFC 8017 PKCS #1 v2.2 November 2016

Acknowledgements

 This document is based on a contribution of RSA Laboratories, the
 research center of RSA Security Inc.

Authors' Addresses

 Kathleen M. Moriarty (editor)
 EMC Corporation
 176 South Street
 Hopkinton, MA  01748
 United States of America
 Email: kathleen.moriarty@emc.com
 Burt Kaliski
 Verisign
 12061 Bluemont Way
 Reston, VA  20190
 United States of America
 Email: bkaliski@verisign.com
 URI:   http://verisignlabs.com
 Jakob Jonsson
 Subset AB
 Munkbrogtan 4
 Stockholm  SE-11127
 Sweden
 Phone: +46 8 428 687 43
 Email: jakob.jonsson@subset.se
 Andreas Rusch
 RSA
 345 Queen Street
 Brisbane, QLD  4000
 Australia
 Email: andreas.rusch@rsa.com

Moriarty, et al. Informational [Page 78]

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