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Network Working Group J. Jonsson Request for Comments: 3447 B. Kaliski Obsoletes: 2437 RSA Laboratories Category: Informational February 2003

   Public-Key Cryptography Standards (PKCS) #1: RSA Cryptography
                    Specifications Version 2.1

Status of this Memo

 This memo provides information for the Internet community.  It does
 not specify an Internet standard of any kind.  Distribution of this
 memo is unlimited.

Copyright Notice

 Copyright (C) The Internet Society (2003).  All Rights Reserved.

Abstract

 This memo represents a republication of PKCS #1 v2.1 from RSA
 Laboratories' Public-Key Cryptography Standards (PKCS) series, and
 change control is retained within the PKCS process.  The body of this
 document is taken directly from the PKCS #1 v2.1 document, with
 certain corrections made during the publication process.

Table of Contents

 1.       Introduction...............................................2
 2.       Notation...................................................3
 3.       Key types..................................................6
    3.1      RSA public key..........................................6
    3.2      RSA private key.........................................7
 4.       Data conversion primitives.................................8
    4.1      I2OSP...................................................9
    4.2      OS2IP...................................................9
 5.       Cryptographic primitives..................................10
    5.1      Encryption and decryption primitives...................10
    5.2      Signature and verification primitives..................12
 6.       Overview of schemes.......................................14
 7.       Encryption schemes........................................15
    7.1      RSAES-OAEP.............................................16
    7.2      RSAES-PKCS1-v1_5.......................................23
 8.       Signature schemes with appendix...........................27
    8.1      RSASSA-PSS.............................................29
    8.2      RSASSA-PKCS1-v1_5......................................32
 9.       Encoding methods for signatures with appendix.............35

Jonsson & Kaliski Informational [Page 1] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

    9.1      EMSA-PSS...............................................36
    9.2      EMSA-PKCS1-v1_5........................................41
 Appendix A. ASN.1 syntax...........................................44
    A.1      RSA key representation.................................44
    A.2      Scheme identification..................................46
 Appendix B. Supporting techniques..................................52
    B.1      Hash functions.........................................52
    B.2      Mask generation functions..............................54
 Appendix C. ASN.1 module...........................................56
 Appendix D. Intellectual Property Considerations...................63
 Appendix E. Revision history.......................................64
 Appendix F. References.............................................65
 Appendix G. About PKCS.............................................70
 Appendix H. Corrections Made During RFC Publication Process........70
 Security Considerations............................................70
 Acknowledgements...................................................71
 Authors' Addresses.................................................71
 Full Copyright Statement...........................................72

1. Introduction

 This document provides recommendations for the implementation of
 public-key cryptography based on the RSA algorithm [42], covering the
 following aspects:
  • Cryptographic primitives
  • Encryption schemes
  • Signature schemes with appendix
  • 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 standard
 IEEE-1363-2000 [26] and draft standards currently being developed by
 the ANSI X9F1 [1] and IEEE P1363 [27] working groups.
 This document supersedes PKCS #1 version 2.0 [35][44] but includes
 compatible techniques.

Jonsson & Kaliski Informational [Page 2] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 The organization of this document is as follows:
  • Section 1 is an introduction.
  • Section 2 defines some notation used in this document.
  • Section 3 defines the RSA public and private key types.
  • Sections 4 and 5 define several primitives, or basic mathematical

operations. Data conversion primitives are in Section 4, and

    cryptographic primitives (encryption-decryption, signature-
    verification) are in Section 5.
  • 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
    OAEP-based [3] encryption scheme and Section 8 defines a PSS-based
    [4][5] signature scheme with appendix.
  • Section 9 defines the encoding methods for the signature schemes

in Section 8.

  • Appendix A defines the ASN.1 syntax for the keys defined in

Section 3 and the schemes in Sections 7 and 8.

  • Appendix B defines the hash functions and the mask generation

function used in this document, including ASN.1 syntax for the

    techniques.
  • Appendix C gives an ASN.1 module.
  • Appendices D, E, F and G cover intellectual property issues,

outline the revision history of PKCS #1, give references to other

    publications and standards, and provide general information about
    the Public-Key Cryptography Standards.

2. Notation

 c              ciphertext representative, an integer between 0 and
                n-1
 C              ciphertext, an octet string
 d              RSA private exponent

Jonsson & Kaliski Informational [Page 3] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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
 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

Jonsson & Kaliski Informational [Page 4] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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
 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
                value 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

Jonsson & Kaliski Informational [Page 5] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 \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 CRT can be applied in a non-recursive as well as a
 recursive way.  In this document a recursive approach following
 Garner's algorithm [22] 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 (Chinese Remainder
 Theorem) 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 [49].

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
 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.

Jonsson & Kaliski Informational [Page 6] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

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.
 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).

Jonsson & Kaliski Informational [Page 7] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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 [22] (see also Algorithm 14.71 in [37]). 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 [40] observed the benefit of applying the
    Chinese Remainder Theorem to RSA operations.

4. Data conversion primitives

 Two data conversion primitives are employed in the schemes defined in
 this document:
  • I2OSP - Integer-to-Octet-String primitive
  • 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.

Jonsson & Kaliski Informational [Page 8] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

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)).
 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

Jonsson & Kaliski Informational [Page 9] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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.
 One pair of encryption and decryption primitives is employed in the
 encryption schemes defined in this document and is specified here:
 RSAEP/RSADP.  RSAEP and RSADP involve the same mathematical
 operation, with different keys as input.
 The primitives defined here are the same as IFEP-RSA/IFDP-RSA in IEEE
 Std 1363-2000 [26] (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.

Jonsson & Kaliski Informational [Page 10] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

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)
          - 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

Jonsson & Kaliski Informational [Page 11] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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.

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: RSASP1/RSAVP1.
 The primitives defined here are the same as IFSP-RSA1/IFVP-RSA1 in
 IEEE 1363-2000 [26] (except that support for multi-prime RSA has
 been added) and are compatible with PKCS #1 v1.5.

Jonsson & Kaliski Informational [Page 12] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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
 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:
       i.    Let s_1 = m^dP mod p and s_2 = m^dQ mod q.
       ii.   If u > 2, let s_i = m^(d_i) mod r_i, i = 3, ..., u.
       iii.  Let h = (s_1 - s_2) * qInv mod p.
       iv.   Let s = s_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).

Jonsson & Kaliski Informational [Page 13] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

                2. Let h = (s_i - s) * t_i mod r_i.
                3. 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
 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 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

Jonsson & Kaliski Informational [Page 14] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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
 (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.

Jonsson & Kaliski Informational [Page 15] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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 [45] 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 recommended for new applications;
 RSAES-PKCS1-v1_5 is included only for compatibility with existing
 applications, and is not recommended for new applications.
 The encryption schemes given here follow a general model similar to
 that employed in IEEE Std 1363-2000 [26], 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
 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 [6] and [36]), 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 v2.0.

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 1.b 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 [3].  (OAEP stands for
 "Optimal Asymmetric Encryption Padding.").  It is compatible with the
 IFES scheme defined in IEEE Std 1363-2000 [26], 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.

Jonsson & Kaliski Informational [Page 16] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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 [21] 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.
 Note.  Recent results have helpfully clarified the security
 properties of the OAEP encoding method [3] (roughly the procedure
 described in step 1.b in Section 7.1.1).  The background is as
 follows.  In 1994, Bellare and Rogaway [3] 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 [2], came up
 with a new, stronger notion of plaintext awareness (PA98) that does
 imply security against CCA2.

Jonsson & Kaliski Informational [Page 17] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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 [3]
 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
 [48] 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 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 [21] 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 Adv against the CCA2 security of RSAES-OAEP to an algorithm
 Inv inverting RSA, the probability of success for Inv is only
 approximately \epsilon^2 / 2^18, where \epsilon is the probability of
 success for Adv.
 (Footnote: In [21] 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 [21] (Inv must apply
 Adv twice to invert RSA, and each application corresponds to a factor
 1 / 2^8).)

Jonsson & Kaliski Informational [Page 18] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 In addition, the running time for Inv 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-KEM key encapsulation
 paradigm offer tight proofs of security directly applicable to
 concrete parameters; see [30] for discussion.  Future versions of
 PKCS #1 may specify schemes based on this paradigm.

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.

Jonsson & Kaliski Informational [Page 19] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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 an octet string PS consisting of k - mLen - 2hLen - 2
       zero octets.  The length of PS may be zero.
    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).

Jonsson & Kaliski Informational [Page 20] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 4. Output the ciphertext C.
 Note.  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
 __________________________________________________________________
                           +----------+---------+-------+
                      DB = |  lHash   |    PS   |   M   |
                           +----------+---------+-------+
                                          |
                +----------+              V
                |   seed   |--> MGF ---> xor
                +----------+              |
                      |                   |
             +--+     V                   |
             |00|    xor <----- MGF <-----|
             +--+     |                   |
               |      |                   |
               V      V                   V
             +--+----------+----------------------------+
       EM =  |00|maskedSeed|          maskedDB          |
             +--+----------+----------------------------+
 __________________________________________________________________
 Figure 1: EME-OAEP encoding operation.  lHash is the hash of the
 optional label L.  Decoding operation follows reverse steps to
 recover M and verify lHash and PS.

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

Jonsson & Kaliski Informational [Page 21] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 Input:
 K        recipient's RSA private key (k denotes the length in octets
          of the RSA modulus n)
 C        ciphertext to be decrypted, an octet string of length k,
          where k = 2hLen + 2
 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).
       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).

Jonsson & Kaliski Informational [Page 22] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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.)
 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, or, more generally, 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 [36].

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 1 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

Jonsson & Kaliski Informational [Page 23] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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 [10]; [14] 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 [6].  Therefore, if RSAES-PKCS1-v1_5 is to be
 used, certain easily implemented countermeasures should be taken to
 thwart the attack found in [6].  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 PKCS #1 v1.5-based
 protocol.  See [7] for a further discussion of these and other
 countermeasures.  It has recently been shown that the security of the
 SSL/TLS handshake protocol [17], which uses RSAES-PKCS1-v1_5 and
 certain countermeasures, can be related to a variant of the RSA
 problem; see [32] for discussion.
 Note.  The following passages describe some security recommendations
 pertaining to the use of RSAES-PKCS1-v1_5.  Recommendations from
 version 1.5 of this document are included as well as new
 recommendations motivated by cryptanalytic advances made in the
 intervening years.
  • 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 [24] are one motivation for
    this recommendation.
  • 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.
  • The pseudorandom octets can also help thwart an attack due to

Coppersmith et al. [10] (see [14] 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

Jonsson & Kaliski Informational [Page 24] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

    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"
 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.

Jonsson & Kaliski Informational [Page 25] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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
 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).

Jonsson & Kaliski Informational [Page 26] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

    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.
 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 [6]; compare to
 Manger's attack [36].

8. Signature schemes 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.

Jonsson & Kaliski Informational [Page 27] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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 [28].  Related signature schemes could be employed in
 PKCS #7 [45], 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 recommended for eventual adoption in new applications.
 RSASSA-PKCS1-v1_5 is included for compatibility with existing
 applications, and while still appropriate for new applications, a
 gradual transition to RSASSA-PSS is encouraged.
 The signature schemes with appendix given here follow a general model
 similar to that employed in IEEE Std 1363-2000 [26], 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.
 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 [45] for an example format in
 the case of RSASSA-PKCS1-v1_5.

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8.1 RSASSA-PSS

 RSASSA-PSS combines the RSASP1 and RSAVP1 primitives with the EMSA-
 PSS encoding method.  It is compatible with the IFSSA scheme as
 amended in the IEEE P1363a draft [27], where the signature and
 verification primitives are IFSP-RSA1 and IFVP-RSA1 as defined in
 IEEE Std 1363-2000 [26] 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
 [4][13][31] 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
 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 [34] 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

Jonsson & Kaliski Informational [Page 29] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 of the scheme by affording a "tighter" security proof than
 deterministic alternatives such as Full Domain Hashing (FDH); see [4]
 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 [12].

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:
       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).

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    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.
 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.

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    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 Std 1363-2000 [26], 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 Std
 1363-2000, but is in the IEEE P1363a draft [27]).
 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.
 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 [34] 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

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 Desmedt and Odlyzko [16] 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 [15] 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 [11] 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
 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.

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 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
 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).

Jonsson & Kaliski Informational [Page 34] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

    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.
 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
 to 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.

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 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.

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 [31] 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)
 [4][5].  It is randomized and has an encoding operation and a
 verification operation.

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 Figure 2 illustrates the encoding operation.
 __________________________________________________________________
                                +-----------+
                                |     M     |
                                +-----------+
                                      |
                                      V
                                    Hash
                                      |
                                      V
                        +--------+----------+----------+
                   M' = |Padding1|  mHash   |   salt   |
                        +--------+----------+----------+
                                       |
             +--------+----------+     V
       DB =  |Padding2|maskedseed|   Hash
             +--------+----------+     |
                       |               |
                       V               |    +--+
                      xor <--- MGF <---|    |bc|
                       |               |    +--+
                       |               |      |
                       V               V      V
             +-------------------+----------+--+
       EM =  |    maskedDB       |maskedseed|bc|
             +-------------------+----------+--+
 __________________________________________________________________
 Figure 2: EMSA-PSS encoding operation.  Verification operation
 follows reverse steps to recover salt, then forward steps to
 recompute and compare H.
 Notes.
 1. The encoding method defined here differs from the one in Bellare
    and Rogaway's submission to IEEE P1363a [5] 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

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       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 Rabin-Williams IFSP-RW signature
       primitive in IEEE Std 1363-2000 [26] and the corresponding
       primitive in the draft ISO/IEC 9796-2 [29].
 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 [4][5] 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.
 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 [4] give a tight lower bound for the
    security of the original RSA-PSS scheme, which corresponds roughly
    to the former case, while Coron [12] gives a lower bound for the
    related Full Domain Hashing scheme, which corresponds roughly to
    the latter case.  In [13] Coron provides a general treatment with
    various salt lengths ranging from 0 to hLen; see [27] for
    discussion.  See also [31], which adapts the security proofs in
    [4][13] 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 P1363a [27], 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 [50].

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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"
 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.

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 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.

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.

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 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.
 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 DER
          encoding T of a certain value computed during the encoding
          operation

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 Output:
 EM       encoded message, an octet string of length emLen
 Errors:
 "message too long"; "intended encoded message length too short"
 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 Distinguished Encoding Rules (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.

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 Notes.
 1. For the six 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-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.
 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 implementer 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.

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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 [46], and PKCS #12 [47].
 The object identifier rsaEncryption identifies RSA public and private
 keys as defined in Appendices A.1.1 and A.1.2.  The parameters field
 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 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:
  • modulus is the RSA modulus n.
  • 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:
  • 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 --})
  • modulus is the RSA modulus n.
  • publicExponent is the RSA public exponent e.
  • privateExponent is the RSA private exponent d.
  • prime1 is the prime factor p of n.
  • prime2 is the prime factor q of n.
  • exponent1 is d mod (p - 1).
  • exponent2 is d mod (q - 1).
  • coefficient is the CRT coefficient q^(-1) mod p.
  • 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.

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       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:
  • prime is a prime factor r_i of n, where i >= 3.
  • exponent is d_i = d mod (r_i - 1).
  • 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.

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 sha256WithRSAEncryption    PARAMETERS NULL } |
        { OID sha384WithRSAEncryption    PARAMETERS NULL } |
        { OID sha512WithRSAEncryption    PARAMETERS NULL } |
        { OID id-RSAES-OAEP PARAMETERS RSAES-OAEP-params } |
        PKCS1PSourceAlgorithms                             |
        { OID id-RSASSA-PSS PARAMETERS RSASSA-PSS-params } ,
        ...  -- Allows for future expansion --
    }

Jonsson & Kaliski Informational [Page 46] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

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:
  • 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.
       HashAlgorithm ::= AlgorithmIdentifier {
          {OAEP-PSSDigestAlgorithms}
       }
       OAEP-PSSDigestAlgorithms    ALGORITHM-IDENTIFIER ::= {
           { OID id-sha1 PARAMETERS NULL   }|
           { OID id-sha256 PARAMETERS NULL }|
           { OID id-sha384 PARAMETERS NULL }|
           { OID id-sha512 PARAMETERS NULL },
           ...  -- Allows for future expansion --
       }
    The default hash function is SHA-1:
       sha1    HashAlgorithm ::= {
           algorithm   id-sha1,
           parameters  SHA1Parameters : NULL
       }
       SHA1Parameters ::= NULL
  • 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

Jonsson & Kaliski Informational [Page 47] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

    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 --
       }
    The default mask generation function is MGF1 with SHA-1:
       mgf1SHA1    MaskGenAlgorithm ::= {
           algorithm   id-mgf1,
           parameters  HashAlgorithm : sha1
       }
  • 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))

Jonsson & Kaliski Informational [Page 48] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

    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}
       }

A.2.2 RSAES-PKCS1-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 }

Jonsson & Kaliski Informational [Page 49] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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:
  • 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.
  • 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.
  • 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.
  • trailerField is the trailer field number, for compatibility with

the draft IEEE P1363a [27]. 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 P1363a) 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:

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       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 [45], 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 CMS [25] 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.

A.2.4 RSASSA-PKCS1-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-256, SHA-384, or SHA-512.  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          sha256WithRSAEncryption ::= {pkcs-1 11}

Jonsson & Kaliski Informational [Page 51] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

    SHA-384          sha384WithRSAEncryption ::= {pkcs-1 12}
    SHA-512          sha512WithRSAEncryption ::= {pkcs-1 13}
 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.
    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-sha256 PARAMETERS NULL }|
        { OID id-sha384 PARAMETERS NULL }|
        { OID id-sha512 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 v2.1 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.

Jonsson & Kaliski Informational [Page 52] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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.
 Six hash functions are given as examples for the encoding methods in
 this document: MD2 [33], MD5 [41], SHA-1 [38], and the proposed
 algorithms SHA-256, SHA-384, and SHA-512 [39].  For the RSAES-OAEP
 encryption scheme and EMSA-PSS encoding method, only SHA-1 and SHA-
 256/384/512 are recommended.  For the EMSA-PKCS1-v1_5 encoding
 method, SHA-1 or SHA-256/384/512 are recommended for new
 applications.  MD2 and MD5 are recommended only for compatibility
 with existing applications based on PKCS #1 v1.5.
 The object identifiers id-md2, id-md5, id-sha1, id-sha256, id-sha384,
 and id-sha512, 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-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
    }

Jonsson & Kaliski Informational [Page 53] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

    id-sha512    OBJECT IDENTIFIER ::= {
        joint-iso-itu-t(2) country(16) us(840) organization(1)
        gov(101) csor(3) nistalgorithm(4) hashalgs(2) 3
    }
 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-sha256, id-sha384,
 and id-sha512 should be omitted, but if present, shall have a value
 of type NULL.
 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 [18]
 demonstrated how to find collisions for MD4 and that the first two
 rounds of MD4 are not one-way [20].  Because of these results and
 others (e.g., [8]), MD4 is no longer recommended.  There have also
 been advances in the cryptanalysis of MD2 and MD5, although not
 enough to warrant removal from existing applications.  Rogier and
 Chauvaud [43] demonstrated how to find collisions in a modified
 version of MD2.  No one has demonstrated how to find collisions for
 the full MD5 algorithm, although partial results have been found
 (e.g., [9][19]).
 To address these concerns, SHA-1, SHA-256, SHA-384, or SHA-512 are
 recommended for new applications.  As of today, the best (known)
 collision attacks against these hash functions are generic attacks
 with complexity 2^(L/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 2^B).  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

Jonsson & Kaliski Informational [Page 54] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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 Std 1363-2000 [26] and the draft ANSI X9.44 [1].
 Future versions of this document may define other mask generation
 functions.

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) .

Jonsson & Kaliski Informational [Page 55] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 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 }
 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

PKCS-1 {

  iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) pkcs-1(1)
  modules(0) pkcs-1(1)

}

– $ Revision: 2.1r1 $

– This module has been checked for conformance with the ASN.1 – standard by the OSS ASN.1 Tools

DEFINITIONS EXPLICIT TAGS ::=

BEGIN

– EXPORTS ALL – All types and values defined in this module are exported for use – in other ASN.1 modules.

IMPORTS

id-sha256, id-sha384, id-sha512

  FROM NIST-SHA2 {
      joint-iso-itu-t(2) country(16) us(840) organization(1)
      gov(101) csor(3) nistalgorithm(4) modules(0) sha2(1)
  };

– Basic object identifiers –

– The DER encoding of this in hexadecimal is: – (0x)06 08 – 2A 86 48 86 F7 0D 01 01 – pkcs-1 OBJECT IDENTIFIER ::= {

Jonsson & Kaliski Informational [Page 56] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

  iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) 1

}

– – When rsaEncryption is used in an AlgorithmIdentifier the – parameters MUST be present and MUST be NULL. – rsaEncryption OBJECT IDENTIFIER ::= { pkcs-1 1 }

– – When id-RSAES-OAEP is used in an AlgorithmIdentifier the – parameters MUST be present and MUST be RSAES-OAEP-params. – id-RSAES-OAEP OBJECT IDENTIFIER ::= { pkcs-1 7 }

– – When id-pSpecified is used in an AlgorithmIdentifier the – parameters MUST be an OCTET STRING. – id-pSpecified OBJECT IDENTIFIER ::= { pkcs-1 9 }

– When id-RSASSA-PSS is used in an AlgorithmIdentifier the – parameters MUST be present and MUST be RSASSA-PSS-params. – id-RSASSA-PSS OBJECT IDENTIFIER ::= { pkcs-1 10 }

– – When the following OIDs are used in an AlgorithmIdentifier the – parameters MUST be present and MUST be NULL. – md2WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 2 } md5WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 4 } sha1WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 5 } sha256WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 11 } sha384WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 12 } sha512WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 13 }

– – This OID really belongs in a module with the secsig OIDs. – id-sha1 OBJECT IDENTIFIER ::= {

  iso(1) identified-organization(3) oiw(14) secsig(3)
  algorithms(2) 26

}

– – OIDs for MD2 and MD5, allowed only in EMSA-PKCS1-v1_5. –

Jonsson & Kaliski Informational [Page 57] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

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

}

– – When id-mgf1 is used in an AlgorithmIdentifier the parameters MUST – be present and MUST be a HashAlgorithm, for example sha1. – id-mgf1 OBJECT IDENTIFIER ::= { pkcs-1 8 }

– Useful types –

ALGORITHM-IDENTIFIER ::= CLASS {

  &id    OBJECT IDENTIFIER  UNIQUE,
  &Type  OPTIONAL

}

  WITH SYNTAX { OID &id [PARAMETERS &Type] }

– – Note: the parameter InfoObjectSet in the following definitions – allows a distinct information object set to be specified for sets – of algorithms such as: – DigestAlgorithms ALGORITHM-IDENTIFIER ::= { – { OID id-md2 PARAMETERS NULL }| – { OID id-md5 PARAMETERS NULL }| – { OID id-sha1 PARAMETERS NULL } – } –

AlgorithmIdentifier { ALGORITHM-IDENTIFIER:InfoObjectSet } ::= SEQUENCE {

  algorithm  ALGORITHM-IDENTIFIER.&id({InfoObjectSet}),
  parameters
      ALGORITHM-IDENTIFIER.&Type({InfoObjectSet}{@.algorithm})
          OPTIONAL

}

– Algorithms –

Jonsson & Kaliski Informational [Page 58] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

– Allowed EME-OAEP and EMSA-PSS digest algorithms. – OAEP-PSSDigestAlgorithms ALGORITHM-IDENTIFIER ::= {

  { OID id-sha1 PARAMETERS NULL   }|
  { OID id-sha256 PARAMETERS NULL }|
  { OID id-sha384 PARAMETERS NULL }|
  { OID id-sha512 PARAMETERS NULL },
  ...  -- Allows for future expansion --

}

– – Allowed EMSA-PKCS1-v1_5 digest algorithms. – PKCS1-v1-5DigestAlgorithms ALGORITHM-IDENTIFIER ::= {

  { OID id-md2 PARAMETERS NULL    }|
  { OID id-md5 PARAMETERS NULL    }|
  { OID id-sha1 PARAMETERS NULL   }|
  { OID id-sha256 PARAMETERS NULL }|
  { OID id-sha384 PARAMETERS NULL }|
  { OID id-sha512 PARAMETERS NULL }

}

– When id-md2 and id-md5 are used in an AlgorithmIdentifier the – parameters MUST be present and MUST be NULL.

– When id-sha1, id-sha256, id-sha384 and id-sha512 are used in an – AlgorithmIdentifier the parameters (which are optional) SHOULD – be omitted. However, an implementation MUST also accept – AlgorithmIdentifier values where the parameters are NULL.

sha1 HashAlgorithm ::= {

  algorithm   id-sha1,
  parameters  SHA1Parameters : NULL  -- included for compatibility
                                     -- with existing implementations

}

HashAlgorithm ::= AlgorithmIdentifier { {OAEP-PSSDigestAlgorithms} }

SHA1Parameters ::= NULL

– – Allowed mask generation function algorithms. – If the identifier is id-mgf1, the parameters are a HashAlgorithm. – PKCS1MGFAlgorithms ALGORITHM-IDENTIFIER ::= {

  { OID id-mgf1 PARAMETERS HashAlgorithm },
  ...  -- Allows for future expansion --

}

Jonsson & Kaliski Informational [Page 59] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

– – Default AlgorithmIdentifier for id-RSAES-OAEP.maskGenAlgorithm and – id-RSASSA-PSS.maskGenAlgorithm. – mgf1SHA1 MaskGenAlgorithm ::= {

  algorithm   id-mgf1,
  parameters  HashAlgorithm : sha1

}

MaskGenAlgorithm ::= AlgorithmIdentifier { {PKCS1MGFAlgorithms} }

– – Allowed algorithms for pSourceAlgorithm. – PKCS1PSourceAlgorithms ALGORITHM-IDENTIFIER ::= {

  { OID id-pSpecified PARAMETERS EncodingParameters },
  ...  -- Allows for future expansion --

}

EncodingParameters ::= OCTET STRING(SIZE(0..MAX))

– – This identifier means that the label L is an empty string, so the – digest of the empty string appears in the RSA block before – masking. – pSpecifiedEmpty PSourceAlgorithm ::= {

  algorithm   id-pSpecified,
  parameters  EncodingParameters : emptyString

}

PSourceAlgorithm ::= AlgorithmIdentifier { {PKCS1PSourceAlgorithms} }

emptyString EncodingParameters ::= ''H

– – 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 sha256WithRSAEncryption    PARAMETERS NULL } |
  { OID sha384WithRSAEncryption    PARAMETERS NULL } |
  { OID sha512WithRSAEncryption    PARAMETERS NULL } |
  { OID id-RSAES-OAEP PARAMETERS RSAES-OAEP-params } |
  PKCS1PSourceAlgorithms                             |

Jonsson & Kaliski Informational [Page 60] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

  { OID id-RSASSA-PSS PARAMETERS RSASSA-PSS-params } ,
  ...  -- Allows for future expansion --

}

– Main structures –

RSAPublicKey ::= SEQUENCE {

  modulus           INTEGER,  -- n
  publicExponent    INTEGER   -- e

}

– – Representation of RSA private key with information for the CRT – algorithm. – 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

}

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

}

– – AlgorithmIdentifier.parameters for id-RSAES-OAEP. – Note that the tags in this Sequence are explicit. – RSAES-OAEP-params ::= SEQUENCE {

Jonsson & Kaliski Informational [Page 61] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

  hashAlgorithm      [0] HashAlgorithm     DEFAULT sha1,
  maskGenAlgorithm   [1] MaskGenAlgorithm  DEFAULT mgf1SHA1,
  pSourceAlgorithm   [2] PSourceAlgorithm  DEFAULT pSpecifiedEmpty

}

– – Identifier for default RSAES-OAEP algorithm identifier. – The DER Encoding of this is in hexadecimal: – (0x)30 0D – 06 09 – 2A 86 48 86 F7 0D 01 01 07 – 30 00 – Notice that the DER encoding of default values is "empty". –

rSAES-OAEP-Default-Identifier RSAES-AlgorithmIdentifier ::= {

  algorithm   id-RSAES-OAEP,
  parameters  RSAES-OAEP-params : {
      hashAlgorithm       sha1,
      maskGenAlgorithm    mgf1SHA1,
      pSourceAlgorithm    pSpecifiedEmpty
  }

}

RSAES-AlgorithmIdentifier ::=

  AlgorithmIdentifier { {PKCS1Algorithms} }

– – AlgorithmIdentifier.parameters for id-RSASSA-PSS. – Note that the tags in this Sequence are explicit. – 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) }

– – Identifier for default RSASSA-PSS algorithm identifier – The DER Encoding of this is in hexadecimal: – (0x)30 0D – 06 09 – 2A 86 48 86 F7 0D 01 01 0A – 30 00 – Notice that the DER encoding of default values is "empty".

Jonsson & Kaliski Informational [Page 62] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

– 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} }

– – Syntax for the EMSA-PKCS1-v1_5 hash identifier. – DigestInfo ::= SEQUENCE {

  digestAlgorithm DigestAlgorithm,
  digest OCTET STRING

}

DigestAlgorithm ::=

  AlgorithmIdentifier { {PKCS1-v1-5DigestAlgorithms} }

END – PKCS1Definitions

Appendix D. Intellectual Property Considerations

 The RSA public-key cryptosystem is described in U.S. Patent
 4,405,829, which expired on September 20, 2000.  RSA Security Inc.
 makes no other patent claims on the constructions described in this
 document, although specific underlying techniques may be covered.
 Multi-prime RSA is described in U.S. Patent 5,848,159.
 The University of California has indicated that it has a patent
 pending on the PSS signature scheme [5].  It has also provided a
 letter to the IEEE P1363 working group stating that if the PSS
 signature scheme is included in an IEEE standard, "the University of
 California will, when that standard is adopted, FREELY license any
 conforming implementation of PSS as a technique for achieving a
 digital signature with appendix" [23].  The PSS signature scheme is
 specified in the IEEE P1363a draft [27], which was in ballot
 resolution when this document was published.

Jonsson & Kaliski Informational [Page 63] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 License to copy this document is granted provided that it is
 identified as "RSA Security Inc.  Public-Key Cryptography Standards
 (PKCS)" in all material mentioning or referencing this document.
 RSA Security Inc. makes no other representations regarding
 intellectual property claims by other parties.  Such determination is
 the responsibility of the user.

Appendix E. Revision history

 Versions 1.0 - 1.3
    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
    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
    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:
  1. Section 10: "MD4 with RSA" signature and verification processes

were added.

  1. Section 11: md4WithRSAEncryption object identifier was added.
    Version 1.5 was republished as IETF RFC 2313.
 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 IETF RFC 2437
    [35].

Jonsson & Kaliski Informational [Page 64] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 Version 2.1
    Version 2.1 introduces multi-prime RSA and the RSASSA-PSS
    signature scheme with appendix along with several editorial
    improvements.  This version continues to support the schemes in
    version 2.0.

Appendix F: References

 [1]   ANSI X9F1 Working Group.  ANSI X9.44 Draft D2: Key
       Establishment Using Integer Factorization Cryptography.
       Working Draft, March 2002.
 [2]   M. Bellare, A. Desai, D. Pointcheval and P. Rogaway.  Relations
       Among Notions of Security for Public-Key Encryption Schemes.
       In H. Krawczyk, editor, Advances in Cryptology - Crypto '98,
       volume 1462 of Lecture Notes in Computer Science, pp. 26 - 45.
       Springer Verlag, 1998.
 [3]   M. Bellare and P. Rogaway.  Optimal Asymmetric Encryption - How
       to Encrypt with RSA.  In A. De Santis, editor, Advances in
       Cryptology - Eurocrypt '94, volume 950 of Lecture Notes in
       Computer Science, pp. 92 - 111.  Springer Verlag, 1995.
 [4]   M. Bellare and P. Rogaway.  The Exact Security of Digital
       Signatures - How to Sign with RSA and Rabin.  In U. Maurer,
       editor, Advances in Cryptology - Eurocrypt '96, volume 1070 of
       Lecture Notes in Computer Science, pp. 399 - 416.  Springer
       Verlag, 1996.
 [5]   M. Bellare and P. Rogaway.  PSS: Provably Secure Encoding
       Method for Digital Signatures.  Submission to IEEE P1363
       working group, August 1998.  Available from
       http://grouper.ieee.org/groups/1363/.
 [6]   D. Bleichenbacher.  Chosen Ciphertext Attacks Against Protocols
       Based on the RSA Encryption Standard PKCS #1.  In H. Krawczyk,
       editor, Advances in Cryptology - Crypto '98, volume 1462 of
       Lecture Notes in Computer Science, pp. 1 - 12.  Springer
       Verlag, 1998.
 [7]   D. Bleichenbacher, B. Kaliski and J. Staddon.  Recent Results
       on PKCS #1: RSA Encryption Standard.  RSA Laboratories'
       Bulletin No. 7, June 1998.

Jonsson & Kaliski Informational [Page 65] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 [8]   B. den Boer and A. Bosselaers.  An Attack on the Last Two
       Rounds of MD4.  In J.  Feigenbaum, editor, Advances in
       Cryptology - Crypto '91, volume 576 of Lecture Notes in
       Computer Science, pp. 194 - 203.  Springer Verlag, 1992.
 [9]   B. den Boer and A. Bosselaers.  Collisions for the Compression
       Function of MD5.  In T. Helleseth, editor, Advances in
       Cryptology - Eurocrypt '93, volume 765 of Lecture Notes in
       Computer Science, pp. 293 - 304.  Springer Verlag, 1994.
 [10]  D. Coppersmith, M. Franklin, J. Patarin and M. Reiter.  Low-
       Exponent RSA with Related Messages.  In U. Maurer, editor,
       Advances in Cryptology - Eurocrypt '96, volume 1070 of Lecture
       Notes in Computer Science, pp. 1 - 9.  Springer Verlag, 1996.
 [11]  D. Coppersmith, S. Halevi and C. Jutla.  ISO 9796-1 and the New
       Forgery Strategy.  Presented at the rump session of Crypto '99,
       August 1999.
 [12]  J.-S. Coron.  On the Exact Security of Full Domain Hashing.  In
       M. Bellare, editor, Advances in Cryptology - Crypto 2000,
       volume 1880 of Lecture Notes in Computer Science, pp. 229 -
       235.  Springer Verlag, 2000.
 [13]  J.-S. Coron.  Optimal Security Proofs for PSS and Other
       Signature Schemes.   In L. Knudsen, editor, Advances in
       Cryptology - Eurocrypt 2002, volume 2332 of Lecture Notes in
       Computer Science, pp. 272 - 287.  Springer Verlag, 2002.
 [14]  J.-S. Coron, M. Joye, D. Naccache and P. Paillier.  New Attacks
       on PKCS #1 v1.5 Encryption.  In B. Preneel, editor, Advances in
       Cryptology - Eurocrypt 2000, volume 1807 of Lecture Notes in
       Computer Science, pp. 369 - 379.  Springer Verlag, 2000.
 [15]  J.-S. Coron, D. Naccache and J. P. Stern.  On the Security of
       RSA Padding.  In M. Wiener, editor, Advances in Cryptology -
       Crypto '99, volume 1666 of Lecture Notes in Computer Science,
       pp. 1 - 18.  Springer Verlag, 1999.
 [16]  Y. Desmedt and A.M. Odlyzko.  A Chosen Text Attack on the RSA
       Cryptosystem and Some Discrete Logarithm Schemes.  In H.C.
       Williams, editor, Advances in Cryptology - Crypto '85, volume
       218 of Lecture Notes in Computer Science, pp. 516 - 522.
       Springer Verlag, 1986.
 [17]  Dierks, T. and C. Allen, "The TLS Protocol, Version 1.0", RFC
       2246, January 1999.

Jonsson & Kaliski Informational [Page 66] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 [18]  H. Dobbertin.  Cryptanalysis of MD4.  In D. Gollmann, editor,
       Fast Software Encryption '96, volume 1039 of Lecture Notes in
       Computer Science, pp. 55 - 72.  Springer Verlag, 1996.
 [19]  H. Dobbertin.  Cryptanalysis of MD5 Compress.  Presented at the
       rump session of Eurocrypt '96, May 1996.
 [20]  H. Dobbertin.  The First Two Rounds of MD4 are Not One-Way.  In
       S. Vaudenay, editor, Fast Software Encryption '98, volume 1372
       in Lecture Notes in Computer Science, pp. 284 - 292.  Springer
       Verlag, 1998.
 [21]  E. Fujisaki, T. Okamoto, D. Pointcheval and J. Stern.  RSA-OAEP
       is Secure under the RSA Assumption.  In J. Kilian, editor,
       Advances in Cryptology - Crypto 2001, volume 2139 of Lecture
       Notes in Computer Science, pp. 260 - 274.  Springer Verlag,
       2001.
 [22]  H. Garner.  The Residue Number System.  IRE Transactions on
       Electronic Computers, EC-8 (6), pp. 140 - 147, June 1959.
 [23]  M.L. Grell.  Re: Encoding Methods PSS/PSS-R.  Letter to IEEE
       P1363 working group, University of California, June 15, 1999.
       Available from
       http://grouper.ieee.org/groups/1363/P1363/patents.html.
 [24]  J. Haastad.  Solving Simultaneous Modular Equations of Low
       Degree.  SIAM Journal of Computing, volume 17, pp. 336 - 341,
       1988.
 [25]  Housley, R., "Cryptographic Message Syntax (CMS)", RFC 3369,
       August 2002.  Housley, R., "Cryptographic Message Syntax (CMS)
       Algorithms", RFC 3370, August 2002.
 [26]  IEEE Std 1363-2000: Standard Specifications for Public Key
       Cryptography.  IEEE, August 2000.
 [27]  IEEE P1363 working group.  IEEE P1363a D11: Draft Standard
       Specifications for Public Key Cryptography -- Amendment 1:
       Additional Techniques. December 16, 2002.  Available from
       http://grouper.ieee.org/groups/1363/.
 [28]  ISO/IEC 9594-8:1997: Information technology - Open Systems
       Interconnection - The Directory: Authentication Framework.
       1997.

Jonsson & Kaliski Informational [Page 67] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 [29]  ISO/IEC FDIS 9796-2: Information Technology - Security
       Techniques - Digital Signature Schemes Giving Message Recovery
       - Part 2: Integer Factorization Based Mechanisms.  Final Draft
       International Standard, December 2001.
 [30]  ISO/IEC 18033-2: Information Technology - Security Techniques -
       Encryption Algorithms - Part 2: Asymmetric Ciphers.  V. Shoup,
       editor, Text for 2nd Working Draft, January 2002.
 [31]  J. Jonsson.  Security Proof for the RSA-PSS Signature Scheme
       (extended abstract).  Second Open NESSIE Workshop.  September
       2001.  Full version available from
       http://eprint.iacr.org/2001/053/.
 [32]  J. Jonsson and B. Kaliski.  On the Security of RSA Encryption
       in TLS.  In M. Yung, editor, Advances in Cryptology - CRYPTO
       2002, vol. 2442 of Lecture Notes in Computer Science, pp. 127 -
       142.  Springer Verlag, 2002.
 [33]  Kaliski, B., "The MD2 Message-Digest Algorithm", RFC 1319,
       April 1992.
 [34]  B. Kaliski.  On Hash Function Identification in Signature
       Schemes.  In B. Preneel, editor, RSA Conference 2002,
       Cryptographers' Track, volume 2271 of Lecture Notes in Computer
       Science, pp. 1 - 16.  Springer Verlag, 2002.
 [35]  Kaliski, B. and J. Staddon, "PKCS #1: RSA Cryptography
       Specifications Version 2.0", RFC 2437, October 1998.
 [36]  J. Manger.  A Chosen Ciphertext Attack on RSA Optimal
       Asymmetric Encryption Padding (OAEP) as Standardized in PKCS #1
       v2.0. In J. Kilian, editor, Advances in Cryptology - Crypto
       2001, volume 2139 of Lecture Notes in Computer Science, pp. 260
       - 274.  Springer Verlag, 2001.
 [37]  A. Menezes, P. van Oorschot and S. Vanstone.  Handbook of
       Applied Cryptography.  CRC Press, 1996.
 [38]  National Institute of Standards and Technology (NIST).  FIPS
       Publication 180-1: Secure Hash Standard.  April 1994.
 [39]  National Institute of Standards and Technology (NIST).  Draft
       FIPS 180-2: Secure Hash Standard.  Draft, May 2001.  Available
       from http://www.nist.gov/sha/.

Jonsson & Kaliski Informational [Page 68] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

 [40]  J.-J. Quisquater and C. Couvreur.  Fast Decipherment Algorithm
       for RSA Public-Key Cryptosystem.  Electronics Letters, 18 (21),
       pp. 905 - 907, October 1982.
 [41]  Rivest, R., "The MD5 Message-Digest Algorithm", RFC 1321, April
       1992.
 [42]  R. Rivest, A. Shamir and L. Adleman.  A Method for Obtaining
       Digital Signatures and Public-Key Cryptosystems.
       Communications of the ACM, 21 (2), pp. 120-126, February 1978.
 [43]  N. Rogier and P. Chauvaud.  The Compression Function of MD2 is
       not Collision Free.  Presented at Selected Areas of
       Cryptography '95.  Carleton University, Ottawa, Canada.  May
       1995.
 [44]  RSA Laboratories.  PKCS #1 v2.0: RSA Encryption Standard.
       October 1998.
 [45]  RSA Laboratories.  PKCS #7 v1.5: Cryptographic Message Syntax
       Standard.  November 1993.  (Republished as IETF RFC 2315.)
 [46]  RSA Laboratories.  PKCS #8 v1.2: Private-Key Information Syntax
       Standard.  November 1993.
 [47]  RSA Laboratories.  PKCS #12 v1.0: Personal Information Exchange
       Syntax Standard.  June 1999.
 [48]  V. Shoup.  OAEP Reconsidered.  In J. Kilian, editor, Advances
       in Cryptology - Crypto 2001, volume 2139 of Lecture Notes in
       Computer Science, pp. 239 - 259.  Springer Verlag, 2001.
 [49]  R. D. Silverman.  A Cost-Based Security Analysis of Symmetric
       and Asymmetric Key Lengths.  RSA Laboratories Bulletin No. 13,
       April 2000.  Available from
       http://www.rsasecurity.com.rsalabs/bulletins/.
 [50]  G. J. Simmons.  Subliminal communication is easy using the DSA.
       In T. Helleseth, editor, Advances in Cryptology - Eurocrypt
       '93, volume 765 of Lecture Notes in Computer Science, pp. 218-
       232.  Springer-Verlag, 1993.

Jonsson & Kaliski Informational [Page 69] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

Appendix G: 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, SET, S/MIME, SSL/TLS, and WAP/WTLS.
 Further development of PKCS occurs through mailing list discussions
 and occasional workshops, and suggestions for improvement are
 welcome.  For more information, contact:
    PKCS Editor
    RSA Laboratories
    174 Middlesex Turnpike
    Bedford, MA  01730 USA
    pkcs-editor@rsasecurity.com
    http://www.rsasecurity.com/rsalabs/pkcs

Appendix H: Corrections Made During RFC Publication Process

 The following corrections were made in converting the PKCS #1 v2.1
 document to this RFC:
  • The requirement that the parameters in an AlgorithmIdentifier

value for id-sha1, id-sha256, id-sha384, and id-sha512 be NULL was

    changed to a recommendation that the parameters be omitted (while
    still allowing the parameters to be NULL). This is to align with
    the definitions originally promulgated by NIST. Implementations
    MUST accept AlgorithmIdentifier values both without parameters and
    with NULL parameters.
  • The notes after RSADP and RSASP1 (Secs. 5.1.2 and 5.2.1) were

corrected to refer to step 2.b rather than 2.a.

  • References [25], [27] and [32] were updated to reflect new

publication data.

 These corrections will be reflected in future editions of PKCS #1
 v2.1.

Security Considerations

 Security issues are discussed throughout this memo.

Jonsson & Kaliski Informational [Page 70] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

Acknowledgements

 This document is based on a contribution of RSA Laboratories, the
 research center of RSA Security Inc.  Any substantial use of the text
 from this document must acknowledge RSA Security Inc.  RSA Security
 Inc. requests that all material mentioning or referencing this
 document identify this as "RSA Security Inc. PKCS #1 v2.1".

Authors' Addresses

 Jakob Jonsson
 Philipps-Universitaet Marburg
 Fachbereich Mathematik und Informatik
 Hans Meerwein Strasse, Lahnberge
 DE-35032 Marburg
 Germany
 Phone: +49 6421 28 25672
 EMail: jonsson@mathematik.uni-marburg.de
 Burt Kaliski
 RSA Laboratories
 174 Middlesex Turnpike
 Bedford, MA 01730 USA
 Phone: +1 781 515 7073
 EMail: bkaliski@rsasecurity.com

Jonsson & Kaliski Informational [Page 71] RFC 3447 PKCS #1: RSA Cryptography Specifications February 2003

Full Copyright Statement

 Copyright (C) The Internet Society 2003.  All Rights Reserved.
 This document and translations of it may be copied and furnished to
 others provided that the above copyright notice and this paragraph
 are included on all such copies.  However, this document itself may
 not be modified in any way, such as by removing the copyright notice
 or references to the Internet Society or other Internet
 organizations, except as required to translate it into languages
 other than English.
 The limited permissions granted above are perpetual and will not be
 revoked by the Internet Society or its successors or assigns.
 This document and the information contained herein is provided on an
 "AS IS" basis and THE INTERNET SOCIETY AND THE INTERNET ENGINEERING
 TASK FORCE DISCLAIMS ALL WARRANTIES, EXPRESS OR IMPLIED, INCLUDING
 BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE INFORMATION
 HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED WARRANTIES OF
 MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE.

Acknowledgement

 Funding for the RFC Editor function is currently provided by the
 Internet Society.

Jonsson & Kaliski Informational [Page 72]

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