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rfc:rfc5091

Network Working Group X. Boyen Request for Comments: 5091 L. Martin Category: Informational Voltage Security

                                                         December 2007
          Identity-Based Cryptography Standard (IBCS) #1:
Supersingular Curve Implementations of the BF and BB1 Cryptosystems

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.

IESG Note

 This document specifies two mathematical algorithms for identity
 based encryption (IBE).  Due to its specialized nature, this document
 experienced limited review within the IETF.  Readers of this RFC
 should carefully evaluate its value for implementation and
 deployment.

Abstract

 This document describes the algorithms that implement Boneh-Franklin
 (BF) and Boneh-Boyen (BB1) Identity-based Encryption.  This document
 is in part based on IBCS #1 v2 of Voltage Security's Identity-based
 Cryptography Standards (IBCS) documents, from which some irrelevant
 sections have been removed to create the content of this document.

Boyen & Martin Informational [Page 1] RFC 5091 IBCS #1 December 2007

Table of Contents

 1. Introduction ....................................................4
    1.1. Sending a Message That Is Encrypted Using IBE ..............5
         1.1.1. Sender Obtains Recipient's Public Parameters ........6
         1.1.2. Construct and Send an IBE-Encrypted Message .........6
    1.2. Receiving and Viewing an IBE-Encrypted Message .............7
         1.2.1. Recipient Obtains Public Parameters from PPS ........8
         1.2.2. Recipient Obtains IBE Private Key from PKG ..........8
         1.2.3. Recipient Decrypts IBE-Encrypted Message ............9
 2. Notation and Definitions ........................................9
    2.1. Notation ...................................................9
    2.2. Definitions ...............................................12
 3. Basic Elliptic Curve Algorithms ................................12
    3.1. The Group Action in Affine Coordinates ....................13
         3.1.1. Implementation for Type-1 Curves ...................13
    3.2. Point Multiplication ......................................14
    3.3. Operations in Jacobian Projective Coordinates .............17
         3.3.1. Implementation for Type-1 Curves ...................17
    3.4. Divisors on Elliptic Curves ...............................19
         3.4.1. Implementation in F_p^2 for Type-1 Curves ..........19
    3.5. The Tate Pairing ..........................................21
         3.5.1. Tate Pairing Calculation ...........................21
         3.5.2. The Miller Algorithm for Type-1 Curves .............21
 4. Supporting Algorithms ..........................................24
    4.1. Integer Range Hashing .....................................24
         4.1.1. Hashing to an Integer Range ........................24
    4.2. Pseudo-Random Byte Generation by Hashing ..................25
         4.2.1. Keyed Pseudo-Random Bytes Generator ................25
    4.3. Canonical Encodings of Extension Field Elements ...........26
         4.3.1. Encoding an Extension Element as a String ..........26
         4.3.2. Type-1 Curve Implementation ........................27
    4.4. Hashing onto a Subgroup of an Elliptic Curve ..............28
         4.4.1. Hashing a String onto a Subgroup of an
                Elliptic Curve .....................................28
         4.4.2. Type-1 Curve Implementation ........................29
    4.5. Bilinear Mapping ..........................................29
         4.5.1. Regular or Modified Tate Pairing ...................29
         4.5.2. Type-1 Curve Implementation ........................30
    4.6. Ratio of Bilinear Pairings ................................31
         4.6.1. Ratio of Regular or Modified Tate Pairings .........31
         4.6.2. Type-1 Curve Implementation ........................32
 5. The Boneh-Franklin BF Cryptosystem .............................32
    5.1. Setup .....................................................32
         5.1.1. Master Secret and Public Parameter Generation ......32
         5.1.2. Type-1 Curve Implementation ........................33
    5.2. Public Key Derivation .....................................34

Boyen & Martin Informational [Page 2] RFC 5091 IBCS #1 December 2007

         5.2.1. Public Key Derivation from an Identity and
                Public Parameters ..................................34
    5.3. Private Key Extraction ....................................35
         5.3.1. Private Key Extraction from an Identity, a
                Set of Public ......................................35
    5.4. Encryption ................................................36
         5.4.1. Encrypt a Session Key Using an Identity and
                Public Parameters ..................................36
    5.5. Decryption ................................................37
         5.5.1. Decrypt an Encrypted Session Key Using
                Public Parameters, a Private Key ...................37
 6. The Boneh-Boyen BB1 Cryptosystem ...............................38
    6.1. Setup .....................................................38
         6.1.1. Generate a Master Secret and Public Parameters .....38
         6.1.2. Type-1 Curve Implementation ........................39
    6.2. Public Key Derivation .....................................41
         6.2.1. Derive a Public Key from an Identity and
                Public Parameters ..................................41
    6.3. Private Key Extraction ....................................41
         6.3.1. Extract a Private Key from an Identity,
                Public Parameters and a Master Secret ..............41
    6.4. Encryption ................................................42
         6.4.1. Encrypt a Session Key Using an Identity and
                Public Parameters ..................................42
    6.5. Decryption ................................................45
         6.5.1. Decrypt Using Public Parameters and Private Key ....45
 7. Test Data ......................................................47
    7.1. Algorithm 3.2.2 (PointMultiply) ...........................47
    7.2. Algorithm 4.1.1 (HashToRange) .............................48
    7.3. Algorithm 4.5.1 (Pairing) .................................48
    7.4. Algorithm 5.2.1 (BFderivePubl) ............................49
    7.5. Algorithm 5.3.1 (BFextractPriv) ...........................49
    7.6. Algorithm 5.4.1 (BFencrypt) ...............................50
    7.7. Algorithm 6.3.1 (BBextractPriv) ...........................51
    7.8. Algorithm 6.4.1 (BBencrypt) ...............................52
 8. ASN.1 Module ...................................................53
 9. Security Considerations ........................................58
 10. Acknowledgments ...............................................60
 11. References ....................................................60
    11.1. Normative References .....................................60
    11.2. Informative References ...................................60

Boyen & Martin Informational [Page 3] RFC 5091 IBCS #1 December 2007

1. Introduction

 This document provides a set of specifications for implementing
 identity-based encryption (IBE) systems based on bilinear pairings.
 Two cryptosystems are described: the IBE system proposed by Boneh and
 Franklin (BF) [BF], and the IBE system proposed by Boneh and Boyen
 (BB1) [BB1].  Fully secure and practical implementations are
 described for each system, comprising the core IBE algorithms as well
 as ancillary hybrid components used to achieve security against
 active attacks.  These specifications are restricted to a family of
 supersingular elliptic curves over finite fields of large prime
 characteristic, referred to as "type-1" curves (see Section 2.1).
 Implementations based on other types of curves currently fall outside
 the scope of this document.
 IBE is a public-key technology, but one which varies from other
 public-key technologies in a slight, yet significant way.  In
 particular, IBE keys are calculated instead of being generated
 randomly, which leads to a different architecture for a system using
 IBE than for a system using other public-key technologies.  An
 overview of these differences and how a system using IBE works is
 given in [IBEARCH].
 Identity-based encryption (IBE) is a public-key encryption technology
 that allows a public key to be calculated from an identity, and the
 corresponding private key to be calculated from the public key.
 Calculation of both the public and private keys in an IBE-based
 system can occur as needed, resulting in just-in-time key material.
 This contrasts with other public-key systems [P1363], in which keys
 are generated randomly and distributed prior to secure communication
 commencing.  The ability to calculate a recipient's public key, in
 particular, eliminates the need for the sender and receiver in an
 IBE-based messaging system to interact with each other, either
 directly or through a proxy such as a directory server, before
 sending secure messages.
 This document describes an IBE-based messaging system and how the
 components of the system work together.  The components required for
 a complete IBE messaging system are the following:
 o  a Private-key Generator (PKG).  The PKG contains the cryptographic
    material, known as a master secret, for generating an individual's
    IBE private key.  A PKG accepts an IBE user's private key request,
    and after successfully authenticating them in some way, returns
    the IBE private key.

Boyen & Martin Informational [Page 4] RFC 5091 IBCS #1 December 2007

 o  a Public Parameter Server (PPS).  IBE System Parameters include
    publicly sharable cryptographic material, known as IBE public
    parameters, and policy information for the PKG.  A PPS provides a
    well-known location for secure distribution of IBE public
    parameters and policy information for the IBE PKG.
 A logical architecture would be to have a PKG/PPS per name space,
 such as a DNS zone.  The organization that controls the DNS zone
 would also control the PKG/PPS and thus the determination of which
 PKG/PSS to use when creating public and private keys for the
 organization's members.  In this case the PPS URI can be uniquely
 created by the form of the identity that it supports.  This
 architecture would make it clear which set of public parameters to
 use and where to retrieve them for a given identity.
 IBE-encrypted messages can use standard message formats, such as the
 Cryptographic Message Syntax (CMS) [CMS].  How to use IBE with CMS is
 described in [IBECMS].
 Note that IBE algorithms are used only for encryption, so if digital
 signatures are required, they will need to be provided by an
 additional mechanism.
 The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
 "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
 document are to be interpreted as described in [KEYWORDS].

1.1. Sending a Message That Is Encrypted Using IBE

 In order to send an encrypted message, an IBE user must perform the
 following steps:
    1.  Obtain the recipient's public parameters.
       The recipient's IBE public parameters allow the creation of
       unique public and private keys.  A user of an IBE system is
       capable of calculating the public key of a recipient after he
       obtains the public parameters for their IBE system.  Once the
       public parameters are obtained, IBE-encrypted messages can be
       sent.
    2.  Construct and send an IBE-encrypted message.
       All that is needed, in addition to the IBE public parameters,
       is the recipient's identity in order to generate their public
       key for use in encrypting messages to them.  When this identity
       is the same as the identity that a message would be addressed
       to, then no more information is needed from a user to send

Boyen & Martin Informational [Page 5] RFC 5091 IBCS #1 December 2007

       someone a secure message than is needed to send them an
       unsecured message.  This is one of the major benefits of an
       IBE-based secure messaging system.  Examples of identities can
       be an individual, group, or role identifiers.

1.1.1. Sender Obtains Recipient's Public Parameters

 The sender of a message obtains the IBE public parameters that he
 needs for calculating the IBE public key of the recipient from a PPS
 that is hosted at a well-known URI.  The IBE public parameters
 contain all of the information that the sender needs to create an
 IBE-encrypted message except for the identity of the recipient.
 [IBEARCH] describes the URI where a PPS is located, the format of IBE
 public parameters, and how to obtain them.  The URI from which users
 obtain IBE public parameters MUST be authenticated in some way; PPS
 servers MUST support Transport Layer Security (TLS) 1.1 [TLS] to
 satisfy this requirement and MUST verify that the subject name in the
 server certificate matches the URI of the PPS.  [IBEARCH] also
 describes the way in which identity formats are defined and a minimum
 interoperable format that all PPSs and PKGs MUST support.  This step
 is shown below in Figure 1.
                IBE Public Parameter Request
               ----------------------------->
        Sender                                PPS
               <-----------------------------
                    IBE Public Parameters
       Figure 1.  Requesting IBE Public Parameters
 The sender of an IBE-encrypted message selects the PPS and
 corresponding PKG based on his local security policy.  Different PPSs
 may provide public parameters that specify different IBE algorithms
 or different key strengths, for example, or require the use of PKGs
 that require different levels of authentication before granting IBE
 private keys.

1.1.2. Construct and Send an IBE-Encrypted Message

 To IBE-encrypt a message, the sender chooses a content encryption key
 (CEK) and uses it to encrypt his message and then encrypts the CEK
 with the recipient's IBE public key (for example, as described in
 [CMS]).  This operation is shown below in Figure 2.  This document
 describes the algorithms needed to implement two forms of IBE.
 [IBECMS] describes how to use the Cryptographic Message Syntax (CMS)
 to encapsulate the encrypted message along with the IBE information
 that the recipient needs to decrypt the message.

Boyen & Martin Informational [Page 6] RFC 5091 IBCS #1 December 2007

                CEK ----> Sender ----> IBE-encrypted CEK
                            ^
                            |
                            |
                   Recipient's Identity
                 and IBE Public Parameters
       Figure 2.  Using an IBE Public-Key Algorithm to Encrypt

1.2. Receiving and Viewing an IBE-Encrypted Message

 In order to read an encrypted message, a recipient of an
 IBE-encrypted message parses the message (for example, as described
 in [IBECMS]).  This gives him the URI he needs to obtain the IBE
 public parameters required to perform IBE calculations as well as the
 identity that was used to encrypt the message.  Next, the recipient
 must carry out the following steps:
    1.  Obtain the recipient's public parameters.
       An IBE system's public parameters allow it to uniquely create
       public and private keys.  The recipient of an IBE-encrypted
       message can decrypt an IBE-encrypted message if he has both the
       IBE public parameters and the necessary IBE private key.  The
       PPS can also provide the URI of the PKG where the recipient of
       an IBE-encrypted message can obtain the IBE private keys.
    2.  Obtain the IBE private key from the PKG.
       To decrypt an IBE-encrypted message, in addition to the IBE
       public parameters, the recipient needs to obtain the private
       key that corresponds to the public key that the sender used.
       The IBE private key is obtained after successfully
       authenticating to a private key generator (PKG), a trusted
       third party that calculates private keys for users.  The
       recipient receives the IBE private key over an HTTPS
       connection.  The URI of a PKG MUST be authenticated in some
       way; PKG servers MUST support TLS 1.1 [TLS] to satisfy this
       requirement.
    3.  Decrypt the IBE-encrypted message.
       The IBE private key decrypts the CEK, which is then used to
       decrypt encrypted message.

Boyen & Martin Informational [Page 7] RFC 5091 IBCS #1 December 2007

       The PKG may allow users other than the intended recipient to
       receive some IBE private keys.  Giving a mail filtering
       appliance permission to obtain IBE private keys on behalf of
       users, for example, can allow the appliance to decrypt and scan
       encrypted messages for viruses or other malicious features.

1.2.1. Recipient Obtains Public Parameters from PPS

 Before he can perform any IBE calculations related to the message
 that he has received, the recipient of an IBE-encrypted message needs
 to obtain the IBE public parameters that were used in the encryption
 operation.  This operation is shown below in Figure 3.
               IBE Public Parameter Request
              ----------------------------->
    Recipient                                PPS
              <-----------------------------
                   IBE Public Parameters
         Figure 3.  Requesting IBE Public Parameters

1.2.2. Recipient Obtains IBE Private Key from PKG

 To obtain an IBE private key, the recipient of an IBE-encrypted
 message provides the IBE public key used to encrypt the message and
 their authentication credentials to a PKG and requests the private
 key that corresponds to the IBE public key.  Section 4 of this
 document defines the protocol for communicating with a PKG as well as
 a minimum interoperable way to authenticate to a PKG that all IBE
 implementations MUST support.  Because the security of IBE private
 keys is vital to the overall security of an IBE system, IBE private
 keys MUST be transported to recipients over a secure protocol.  PKGs
 MUST support TLS 1.1 [TLS] for transport of IBE private keys.  This
 operation is shown below in Figure 4.
                 IBE Private Key Request
              ---------------------------->
    Recipient                                PKG
              <----------------------------
                     IBE Private Key
         Figure 4.  Obtaining an IBE Private Key

Boyen & Martin Informational [Page 8] RFC 5091 IBCS #1 December 2007

1.2.3. Recipient Decrypts IBE-Encrypted Message

 After obtaining the necessary IBE private key, the recipient uses
 that IBE private key, and the corresponding IBE public parameters, to
 decrypt the CEK.  This operation is shown below in Figure 5.  He then
 uses the CEK to decrypt the encrypted message content (for example,
 as specified in [IBECMS]).
    IBE-encrypted CEK ----> Recipient ----> CEK
                                ^
                                |
                                |
                        IBE Private Key
                    and IBE Public Parameters
       Figure 5.  Using an IBE Public-Key Algorithm to Decrypt

2. Notation and Definitions

2.1. Notation

 This section summarizes the notions and definitions regarding
 identity-based cryptosystems on elliptic curves.  The reader is
 referred to [ECC] for the mathematical background and to [BF],
 [IBEARCH] regarding all notions pertaining to identity-based
 encryption.
 F_p denotes finite field of prime characteristic p; F_p^2 denotes its
 extension field of degree 2.
 Let E/F_p: y^2 = x^3 + a * x + b be an elliptic curve over F_p.  For
 an extension of degree 2, the curve E/F_p defines a group (E(F_p^2),
 +), which is the additive group of points of affine coordinates (x,
 y) in (F_p^2)^2 satisfying the curve equation over F_p^2, with null
 element, or point at infinity, denoted as 0.
 Let q be a prime such that E(F_p) has a cyclic subgroup G1' of order
 q.
 Let G1'' be a cyclic subgroup of E(F_p^2) of order q, and G2 be a
 cyclic subgroup of (F_p^2)* of order p.
 Under these conditions, a mathematical construction known as the Tate
 pairing provides an efficiently computable map e: G1' x G1'' -> G2
 that is linear in both arguments and believed hard to invert [BF].
 If an efficiently computable non-rational endomorphism phi: G1' ->

Boyen & Martin Informational [Page 9] RFC 5091 IBCS #1 December 2007

 G1'' is available for the selected elliptic curve on which the Tate
 pairing is computed, then we can construct a function e': G1' x G1''
 -> G2, defined as e'(A, B) = e(A, phi(B)), called the modified Tate
 pairing.  We generically call a pairing either the Tate pairing e or
 the modified Tate pairing e', depending on the chosen elliptic curve
 used in a particular implementation.
 The following additional notation is used throughout this document.
 p - A 512-bit to 7680-bit prime, which is the order of the finite
 field F_p.
 F_p - The base finite field of order p over which the elliptic curve
 of interest E/F_p is defined.
 #G - The size of the set G.
 F* - The multiplicative group of the non-zero elements in the field
 F; e.g., (F_p)* is the multiplicative group of the finite field F_p.
 E/F_p - The equation of an elliptic curve over the field F_p, which,
 when p is neither 2 nor 3, is of the form E/F_p: y^2 = x^3 + a * x +
 b, for specified a, b in F_p.
 0 - The null element of any additive group of points on an elliptic
 curve, also called the point at infinity.
 E(F_p) - The additive group of points of affine coordinates (x, y),
 with x, y in F_p, that satisfy the curve equation E/F_p, including
 the point at infinity 0.
 q - A 160-bit to 512-bit prime that is the order of the cyclic
 subgroup of interest in E(F_p).
 k - The embedding degree of the cyclic subgroup of order q in E(F_p).
 For type-1 curves this is always equal to 2.
 F_p^2 - The extension field of degree 2 of the field F_p.
 E(F_p^2) - The group of points of affine coordinates in F_p^2
 satisfying the curve equation E/F_p, including the point at infinity
 0.
 Z_p - The additive group of integers modulo p.
 lg - The base 2 logarithm function, so that 2^lg(x) = x.
 The term "object identifier" will be abbreviated "OID."

Boyen & Martin Informational [Page 10] RFC 5091 IBCS #1 December 2007

 A Solinas prime is a prime of the form 2^a (+/-) 2^b (+/-) 1.
 The following conventions are assumed for curve operations.
 Point addition - If A and B are two points on a curve E, their sum is
 denoted as A + B.
 Point multiplication - If A is a point on a curve, and n an integer,
 the result of adding A to itself a total of n times is denoted [n]A.
 The following class of elliptic curves is exclusively considered for
 pairing operations in the present version of this document, which are
 referred to as "type-1" curves.
 Type-1 curves - The class of curves of type-1 is defined as the class
 of all elliptic curves of equation E/F_p: y^2 = x^3 + 1 for all
 primes p congruent to 11 modulo 12.  This class forms a subclass of
 the class of supersingular curves.  These curves satisfy #E(F_p) = p
 + 1, and the p points (x, y) in E(F_p) \ {0} have the property that x
 = (y^2 - 1)^(1/3) (mod p).  Type-1 curves always have an embedding
 degree k = 2.
 Groups of points on type-1 curves are plentiful and easy to construct
 by random selection of a prime p of the appropriate form.  Therefore,
 rather than to standardize upon a small set of common values of p, it
 is henceforth assumed that all type-1 curves are freshly generated at
 random for the given cryptographic application (an example of such
 generation will be given in Algorithm 5.1.2 (BFsetup1) or Algorithm
 6.1.2 (BBsetup1)).  Implementations based on different classes of
 curves are currently unsupported.
 We assume that the following concrete representations of mathematical
 objects are used.
 Base field elements - The p elements of the base field F_p are
 represented directly using the integers from 0 to p - 1.
 Extension field elements - The p^2 elements of the extension field
 F_p^2 are represented as ordered pairs of elements of F_p.  An
 ordered pair (a_0, a_1) is interpreted as the complex number a_0 +
 a_1 * i, where i^2 = -1.  This allows operations on elements of F_p^2
 to be implemented as follows.  Suppose that a = (a_0, a_1) and b =
 (b_0, b_1) are elements of F_p^2.  Then a + b = ((a_0 + b_0)(mod p),
 (a_1 + b_1)(mod p)) and a * b = ((a_1 * b_1 - a_0 * b_0)(mod p), (a_1
 * b_0 + a_0 * b_1)(mod p)).

Boyen & Martin Informational [Page 11] RFC 5091 IBCS #1 December 2007

 Elliptic curve points - Points in E(F_p^2) with the point P = (x, y)
 in F_p^2 x F_p^2 satisfying the curve equation E/F_p.  Points not
 equal to 0 are internally represented using the affine coordinates
 (x, y), where x and y are elements of F_p^2.

2.2. Definitions

 The following terminology is used to describe an IBE system.
 Public parameters - The public parameters are a set of common,
 system-wide parameters generated and published by the private key
 generator (PKG).
 Master secret - The master secret is the master key generated and
 privately kept by the key server and used to generate the private
 keys of the users.
 Identity - An identity is an arbitrary string, usually a
 human-readable unambiguous designator of a system user, possibly
 augmented with a time stamp and other attributes.
 Public key - A public key is a string that is algorithmically derived
 from an identity.  The derivation may be performed by anyone,
 autonomously.
 Private key - A private key is issued by the key server to correspond
 to a given identity (and the public key that derives from it) under
 the published set of public parameters.
 Plaintext - Plaintext is an unencrypted representation, or in the
 clear, of any block of data to be transmitted securely.  For the
 present purposes, plaintexts are typically session keys, or sets of
 session keys, for further symmetric encryption and authentication
 purposes.
 Ciphertext - Ciphertext is an encrypted representation of any block
 of data, including plaintext, to be transmitted securely.

3. Basic Elliptic Curve Algorithms

 This section describes algorithms for performing all needed basic
 arithmetic operations on elliptic curves.  The presentation is
 specialized to the type of curves under consideration for simplicity
 of implementation.  General algorithms may be found in [ECC].

Boyen & Martin Informational [Page 12] RFC 5091 IBCS #1 December 2007

3.1. The Group Action in Affine Coordinates

3.1.1. Implementation for Type-1 Curves

 Algorithm 3.1.1 (PointDouble1): adds a point to itself on a type-1
 elliptic curve.
 Input:
 o  A point A in E(F_p^2), with A = (x, y) or 0
 o  An elliptic curve E/F_p: y^2 = x^3 + 1
 Output:
 o  The point [2]A = A + A
 Method:
 1. If A = 0 or y = 0, then return 0
 2. Let lambda = (3 * x^2) / (2 * y)
 3. Let x' = lambda^2 - 2 * x
 4. Let y' = (x - x') * lambda - y
 5. Return (x', y')
 Algorithm 3.1.2 (PointAdd1): adds two points on a type-1 elliptic
 curve.
 Input:
 o  A point A in E(F_p^2), with A = (x_A, y_A) or 0
 o  A point B in E(F_p^2), with B = (x_B, y_B) or 0
 o  An elliptic curve E/F_p: y^2 = x^3 + 1
 Output:
 o  The point A + B
 Method:
 1. If A = 0, return B

Boyen & Martin Informational [Page 13] RFC 5091 IBCS #1 December 2007

 2. If B = 0, return A
 3. If x_A = x_B:
    (a) If y_A = -y_B, return 0
    (b) Else return [2]A computed using Algorithm 3.1.1 (PointDouble1)
 4.  Otherwise:
    (a) Let lambda = (y_B - y_A) / (x_B - x_A)
    (b) Let x' = lambda^2 - x_A - x_B
    (c) Let y' = (x_A - x') * lambda - y_A
    (d) Return (x', y')

3.2. Point Multiplication

 Algorithm 3.2.1 (SignedWindowDecomposition): computes the signed
 m-ary window representation of a positive integer [ECC].
 Input:
 o  An integer k > 0, where k has the binary representation k =
    {Sum(k_j * 2^j, for j = 0 to l} where each k_j is either 0 or 1
    and k_l = 0
 o  An integer window bit-size r > 0
 Output:
 o  An integer d and the unique d-element sequence {(b_i, e_i), for i
    = 0 to d - 1} such that k = {Sum(b_i * 2^(e_i), for i = 0 to d -
    1}, each b_i = +/- 2^j for some 0 < j <= r - 1 and each e_i is a
    non-negative integer
 Method:
 1. Let d = 0
 2. Let j = 0
 3. While j <= l, do:
    (a) If k_j = 0, then:

Boyen & Martin Informational [Page 14] RFC 5091 IBCS #1 December 2007

       i. Let j = j + 1
    (b) Else:
       i. Let t = min{l, j + r - 1}
      ii. Let h_d = (k_t, k_(t - 1), ..., k_j) (base 2)
     iii. If h_d > 2^(r - 1), then:
          A. Let b_d = h_d - 2^r
          B. Increment the number (k_l, k_(l-1),...,k_j) (base 2) by 1
      iv.  Else:
          A.  Let b_d = h_d
       v.  Let e_d = j
      vi.  Let d = d + 1
     vii.  Let j = t + 1
 4.  Return d and the sequence {(b_0, e_0), ...,
     (b_(d - 1), e_(d - 1))}
 Algorithm 3.2.2 (PointMultiply): scalar multiplication on an elliptic
 curve using the signed m-ary window method.
 Input:
 o  A point A in E(F_p^2)
 o  An integer l > 0
 o  An elliptic curve E/F_p: y^2 = x^3 + a * x + b
 Output:
 o  The point [l]A
 Method:
 1.  (Window decomposition)
    (a) Let r > 0 be an integer (fixed) bit-wise window size,
        e.g., r = 5

Boyen & Martin Informational [Page 15] RFC 5091 IBCS #1 December 2007

    (b) Let l' = l where l = {Sum(l_j * 2^j), for j = 0 to
        len_l} is the binary expansion of l, where len_l =
        Ceiling(lg(l))
    (c) Compute (d, {(b_i, e_i), for i = 0 to d - 1} =
        SignedWindowDecomposition(l, r), the signed 2^r-ary window
        representation of l using Algorithm 3.2.1
        (SignedWindowDecomposition)
 2.  (Precomputation)
    (a) Let A_1 = A
    (b) Let A_2 = [2]A, using Algorithm 3.1.1 (PointDouble1)
    (c) For i = 1 to 2^(r - 2) - 1, do:
       i.  Let A_(2 * i + 1) = A_(2 * i - 1) + A_2 using
           Algorithm 3.1.2 (PointAdd1)
    (d) Let Q = A_(b_(d - 1))
 3.  Main loop
    (a) For i = d - 2 to 0 by -1, do:
       i. Let Q = [2^(e_(i + 1) - e_i)]Q, using repeated
          applications of Algorithm 3.1.1 (PointDouble1)
          e_(i + 1) - e_i times
      ii. If b_i > 0, then:
          A. Let Q = Q + A_(b_i) using Algorithm 3.1.2
             (PointAdd1)
     iii. Else:
          A. Let Q = Q - A_(-(b_i)) using Algorithm 3.1.2
             (PointAdd1)
    (b) Calculate Q = [2^(e_0)]Q using repeated applications of
        Algorithm 3.1.1 (PointDouble1) e_0 times
 4.  Return Q.

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3.3. Operations in Jacobian Projective Coordinates

3.3.1. Implementation for Type-1 Curves

 Algorithm 3.3.1 (ProjectivePointDouble1): adds a point to itself in
 Jacobian projective coordinates for type-1 curves.
 Input:
 o  A point (x, y, z) = A in E(F_p^2) in Jacobian projective
    coordinates
 o  An elliptic curve E/F_p: y^2 = x^3 + 1
 Output:
 o  The point [2]A in Jacobian projective coordinates
 Method:
 1. If z = 0 or y = 0, return (0, 1, 0) = 0, otherwise:
 2. Let lambda_1 = 3 * x^2
 3. Let z' = 2 * y * z
 4. Let lambda_2 = y^2
 5. Let lambda_3 = 4 * lambda_2 * x
 6. Let x' = lambda_1^2 - 2 * lambda_3
 7. Let lambda_4 = 8 * lambda_2^2
 8. Let y' = lambda_1 * (lambda_3 - x') - lambda_4
 9. Return (x', y', z')
 Algorithm 3.3.2 (ProjectivePointAccumulate1): adds a point in affine
 coordinates to an accumulator in Jacobian projective coordinates, for
 type-1 curves.
 Input:
 o  A point (x_A, y_A, z_A) = A in E(F_p^2) in Jacobian
    projective coordinates

Boyen & Martin Informational [Page 17] RFC 5091 IBCS #1 December 2007

 o  A point (x_B, y_B) = B in E(F_p^2) \ {0} in affine
    coordinates
 o  An elliptic curve E/F_p: y^2 = x^3 + 1
 Output:
 o  The point A + B in Jacobian projective coordinates
 Method:
 1. If z_A = 0, return (x_B, y_B, 1) = B, otherwise:
 2. Let lambda_1 = z_A^2
 3. Let lambda_2 = lambda_1 * x_B
 4. Let lambda_3 = x_A - lambda_2
 5. If lambda_3 = 0, then return (0, 1, 0), otherwise:
 6. Let lambda_4 = lambda_3^2
 7. Let lambda_5 = lambda_1 * y_B * z_A
 8. Let lambda_6 = lambda_4 - lambda_5
 9. Let lambda_7 = x_A + lambda_2
 10. Let lambda_8 = y_A + lambda_5
 11. Let x' = lambda_6^2 - lambda_7 * lambda_4
 12. Let lambda_9 = lambda_7 * lambda_4 - 2 * x'
 13. Let y' = (lambda_9 * lambda_6 -
     lambda_8 * lambda_3 * lambda_4) / 2
 14. Let z' = lambda_3 * z_A
 15. Return (x', y', z')

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3.4. Divisors on Elliptic Curves

3.4.1. Implementation in F_p^2 for Type-1 Curves

 Algorithm 3.4.1 (EvalVertical1): evaluates the divisor of a vertical
 line on a type-1 elliptic curve.
 Input:
 o  A point B in E(F_p^2) with B != 0
 o  A point A in E(F_p)
 o  A description of a type-1 elliptic curve E/F_p
 Output:
 o  An element of F_p^2 that is the divisor of the vertical line going
    through A evaluated at B
 Method:
 1. Let r = x_B - x_A
 2. Return r
 Algorithm 3.4.2 (EvalTangent1): evaluates the divisor of a tangent on
 a type-1 elliptic curve.
 Input:
 o  A point B in E(F_p^2) with B != 0
 o  A point A in E(F_p)
 o  A description of a type-1 elliptic curve E/F_p
 Output:
 o  An element of F_p^2 that is the divisor of the line tangent to A
    evaluated at B
 Method:
 1. (Special cases)
    (a) If A = 0, return 1

Boyen & Martin Informational [Page 19] RFC 5091 IBCS #1 December 2007

    (b) If y_A = 0, return EvalVertical1(B, A) using Algorithm 3.4.1
        (EvalVertical1)
 2. (Line computation)
    (a) Let a = -3 * (x_A)^2
    (b) Let b = 2 * y_A
    (c) Let c = -b * y_A - a * x_A
 3. (Evaluation at B)
    (a) Let r = a * x_B + b * y_B + c
 4. Return r
 Algorithm 3.4.3 (EvalLine1): evaluates the divisor of a line on a
 type-1 elliptic curve.
 Input:
 o  A point B in E(F_p^2) with B != 0
 o  Two points A', A'' in E(F_p)
 o  A description of a type-1 elliptic curve E/F_p
 Output:
 o  An element of F_p^2 that is the divisor of the line going through
    A' and A'' evaluated at B
 Method:
 1. (Special cases)
    (a) If A' = 0, return EvalVertical1(B, A'') using Algorithm 3.4.1
       (EvalVertical1)
    (b) If A'' = 0, return EvalVertical1(B, A') using Algorithm 3.4.1
       (EvalVertical1)
    (c) If A' = -A'', return EvalVertical1(B, A') using Algorithm
       3.4.1 (EvalVertical1)
    (d) If A' = A'', return EvalTangent1(B, A') using Algorithm 3.4.2
       (EvalTangent1)

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 2. (Line computation)
       (a) Let a = y_A' - y_A''
       (b) Let b = x_A'' - x_A'
       (c) Let c = -b * y_A' - a * x_A'
 3. (Evaluation at B)
       (a) Let r = a * x_B + b * y_B + c
 4. Return r

3.5. The Tate Pairing

3.5.1. Tate Pairing Calculation

 Algorithm 3.5.1 (Tate): computes the Tate pairing on an elliptic
 curve.
 Input:
 o  A point A of order q in E(F_p)
 o  A point B of order q in E(F_p^2)
 o  A description of an elliptic curve E/F_p such that E(F_p) and
    E(F_p^2) have a subgroup of order q
 Output:
 o  The value e(A, B) in F_p^2, computed using the Miller algorithm
 Method:
 1. For a type-1 curve E, execute Algorithm 3.5.2 (TateMillerSolinas)

3.5.2. The Miller Algorithm for Type-1 Curves

 Algorithm 3.5.2 (TateMillerSolinas): computes the Tate pairing on a
 type-1 elliptic curve.
 Input:
 o  A point A of order q in E(F_p)
 o  A point B of order q in E(F_p^2)

Boyen & Martin Informational [Page 21] RFC 5091 IBCS #1 December 2007

 o  A description of a type-1 supersingular elliptic curve E/F_p such
    that E(F_p) and E(F_p^2) have a subgroup of Solinas prime order q
    where q = 2^a + s * 2^b + c, where c and s are limited to the
    values +/-1
 Output:
 o  The value e(A, B) in F_p^2, computed using the Miller algorithm
 Method:
 1. (Initialization)
    (a) Let v_num = 1 in F_p^2
    (b) Let v_den = 1 in F_p^2
    (c) Let V = (x_V , y_V , z_V ) = (x_A, y_A, 1) in (F_p)^3, being
        the representation of (x_A, y_A) = A using Jacobian projective
        coordinates
    (d) Let t_num = 1 in F_p^2
    (e) Let t_den = 1 in F_p^2
 2. (Calculation of the (s * 2^b) contribution)
    (a) (Repeated doublings) For n = 0 to b - 1:
       i. Let t_num = t_num^2
      ii. Let t_den = t_den^2
     iii. Let t_num = t_num * EvalTangent1(B, (x_V / z_V^2, y_V /
          z_V^3)) using Algorithm 3.4.2 (EvalTangent1)
      iv. Let V = (x_V , y_V , z_V ) = [2]V  using Algorithm 3.3.1
          (ProjectivePointDouble1)
       v. Let t_den = t_den * EvalVertical1(B, (x_V / z_V^2, y_V /
          z_V^3)using Algorithm 3.4.1 (EvalVertical1)

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    (b) (Normalization)
       i. Let V_b = (x_(V_b) , y_(V_b))
          = (x_V / z_V^2, s * y_V / z_V^3) in (F_p)^2,
          resulting in a point V_b in E(F_p)
    (c) (Accumulation) Selecting on s:
       i. If s = -1:
          A. Let v_num = v_num * t_den
          B. Let v_den = v_den * t_num * EvalVertical1(B, (x_V /
             z_V^2, y_V / z_V^3))) using Algorithm 3.4.1
             (EvalVertical1)
       ii. If s = 1:
          A. Let v_num = v_num * t_num
          B. Let v_den = v_den * t_den
 3. (Calculation of the 2^a contribution)
    (a) (Repeated doublings) For n = b to a - 1:
       i. Let t_num = t_num^2
      ii. Let t_den = t_den^2
     iii. Let t_num = t_num * EvalTangent1(B, (x_V / z_V^2, y_V /
          z_V^3))) using Algorithm 3.4.2 (EvalTangent1)
      iv. Let V = (x_V , y_V , z_V) = [2]V  using Algorithm 3.3.1
          (ProjectivePointDouble1)
       v. Let t_den = t_den * EvalVertical1(B, (x_V / z_V^2, y_V /
          z_V^3))) using Algorithm 3.4.1 (EvalVertical1)
    (b) (Normalization)
       i. Let V_a = (x_(V_a) , y_(V_a)) =
          (x_V /z_V^2, s * x_V / z_V^3) in (F_p)^2,
          resulting in a point V_a in E(F_p)

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    (c) (Accumulation)
       i. Let v_num = v_num * t_num
      ii. Let v_den = v_den * t_den
 4. (Correction for the (s * 2^b) and (c) contributions)
    (a) Let v_num = v_num * EvalLine1(B, V_a, V_b) using Algorithm
        3.4.3 (EvalLine1)
    (b) Let v_den = v_den * EvalVertical1(B, V_a + V_b) using
        Algorithm 3.4.1 (EvalVertical1)
    (c) If c = -1, then:
       i. Let v_den = v_den * EvalVertical1(B, A) using Algorithm
          3.4.1 (EvalVertical1)
 5. (Correcting exponent)
    (a) Let eta = (p^2 - 1) / q
 6. (Final result)
    (a) Return (v_num / v_den)^eta

4. Supporting Algorithms

 This section describes a number of supporting algorithms for encoding
 and hashing.

4.1. Integer Range Hashing

4.1.1. Hashing to an Integer Range

 HashToRange(s, n, hashfcn) takes a string s, an integer n, and a
 cryptographic hash function hashfcn as input and returns an integer
 in the range 0 to n - 1 by cryptographic hashing.  The input n MUST
 be less than 2^(hashlen), where hashlen is the number of octets
 comprising the output of the hash function hashfcn.  HashToRange is
 based on Merkle's method for hashing [MERKLE], which is provably as
 secure as the underlying hash function hashfcn.
 Algorithm 4.1.1 (HashToRange): cryptographically hashes strings to
 integers in a range.

Boyen & Martin Informational [Page 24] RFC 5091 IBCS #1 December 2007

 Input:
 o  A string s of length |s| octets
 o  A positive integer n represented as Ceiling(lg(n) / 8) octets.
 o  A cryptographic hash function hashfcn
 Output:
 o  A positive integer v in the range 0 to n - 1
 Method:
 1. Let hashlen be the number of octets comprising the output of
    hashfcn
 2. Let v_0 = 0
 3. Let h_0 = 0x00...00, a string of null octets with a length of
    hashlen
 4. For i = 1 to 2, do:
    (a) Let t_i = h_(i - 1) || s, which is the (|s| + hashlen)- octet
        string concatenation of the strings h_(i - 1) and s
    (b) Let h_i = hashfcn(t_i), which is a hashlen-octet string
        resulting from the hash algorithm hashfcn on the input t_i
    (c) Let a_i = Value(h_i) be the integer in the range 0 to
        256^hashlen - 1 denoted by the raw octet string h_i
        interpreted in the unsigned big-endian convention
    (d) Let v_i = 256^hashlen * v_(i - 1) + a_i
 5. Let v = v_l (mod n)

4.2. Pseudo-Random Byte Generation by Hashing

4.2.1. Keyed Pseudo-Random Bytes Generator

 HashBytes(b, p, hashfcn) takes an integer b, a string p, and a
 cryptographic hash function hashfcn as input and returns a b-octet
 pseudo-random string r as output.  The value of b MUST be less than
 or equal to the number of bytes in the output of hashfcn.  HashBytes
 is based on Merkle's method for hashing [MERKLE], which is provably
 as secure as the underlying hash function hashfcn.

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 Algorithm 4.2.1 (HashBytes): keyed cryptographic pseudo-random bytes
 generator.
 Input:
 o  An integer b
 o  A string p
 o  A cryptographic hash function hashfcn
 Output:
 o  A string r comprising b octets
 Method:
 1. Let hashlen be the number of octets comprising the output of
    hashfcn
 2. Let K = hashfcn(p)
 3. Let h_0 = 0x00...00, a string of null octets with a length of
    hashlen
 4. Let l = Ceiling(b / hashlen)
 5. For each i in 1 to l, do:
    (a) Let h_i = hashfcn(h_(i - 1))
    (b) Let r_i = hashfcn(h_i || K), where h_i || K is the (2 *
        hashlen)-octet concatenation of h_i and K
 6. Let r = LeftmostOctets(b, r_1 || ... || r_l), i.e., r is formed as
    the concatenation of the r_i, truncated to the desired number of
    octets

4.3. Canonical Encodings of Extension Field Elements

4.3.1. Encoding an Extension Element as a String

 Canonical(p, k, o, v) takes an element v in F_p^k, and returns a
 canonical octet string of fixed length representing v.  The parameter
 o MUST be either 0 or 1, and specifies the ordering of the encoding.
 Algorithm 4.3.1 (Canonical): encodes elements of an extension field
 F_p^2 as strings.

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 Input:
 o  An element v in F_p^2
 o  A description of F_p^2
 o  An ordering parameter o, either 0 or 1
 Output:
 o  A fixed-length string s representing v
 Method:
 1. For a type-1 curve, execute Algorithm 4.3.2 (Canonical1)

4.3.2. Type-1 Curve Implementation

 Canonical1(p, o, v) takes an element v in F_p^2 and returns a
 canonical representation of v as an octet string s of fixed size.
 The parameter o MUST be either 0 or 1, and specifies the ordering of
 the encoding.
 Algorithm 4.3.2 (Canonical1): canonically represents elements of an
 extension field F_p^2.
 Input:
 o  An element v in F_p^2
 o  A description of p, where p is congruent to 3 modulo 4
 o  A ordering parameter o, either 0 or 1
 Output:
 o  A string s of size 2 * Ceiling(lg(p) / 8) octets
 Method:
 1. Let l = Ceiling(lg(p) / 8), the number of octets needed to
    represent integers in Z_p
 2. Let v = a + b * i, where i^2 = -1
 3. Let a_(256^l) be the big-endian zero-padded fixed-length octet
    string representation of a in Z_p

Boyen & Martin Informational [Page 27] RFC 5091 IBCS #1 December 2007

 4. Let b_(256^l) be the big-endian zero-padded fixed-length octet
    string representation of b in Z_p
 5. Depending on the choice of ordering o:
    (a) If o = 0, then let s = a_(256^l) || b_(256^l), which is the
        concatenation of a_(256^l) followed by b_(256^l)
    (b) If o = 1, then let s = b_(256^l) || a_(256^l), which is the
        concatenation of b_(256^l) followed by a_(256^l)
 6. Return s

4.4. Hashing onto a Subgroup of an Elliptic Curve

4.4.1. Hashing a String onto a Subgroup of an Elliptic Curve

 HashToPoint(E, p, q, id, hashfcn) takes an identity string id, the
 description of a subgroup of prime order q in E(F_p) or E(F_p^2), and
 a cryptographic hash function hashfcn and returns a point Q_id of
 order q in E(F_p) or E(F_p^2).
 Algorithm 4.4.1 (HashToPoint): cryptographically hashes strings to
 points on elliptic curves.
 Input:
 o  An elliptic curve E
 o  A prime p
 o  A prime q
 o  A string id
 o  A cryptographic hash function hashfcn
 Output:
 o  A point Q_id = (x, y) of order q n E(F_p)
 Method:
 1. For a type-1 curve E, execute Algorithm 4.4.2 (HashToPoint1)

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4.4.2. Type-1 Curve Implementation

 HashToPoint1(p, q, id, hashfcn) takes an identity string id and the
 description of a subgroup of order q in E(F_p), where E: y^2 = x^3 +
 1 with p congruent to 11 modulo 12, and returns a point Q_id of order
 q in E(F_p) that is calculated using the cryptographic hash function
 hashfcn.  The parameters p, q and hashfcn MUST be part of a valid set
 of public parameters as defined in Section 5.1.2 or Section 6.1.2.
 Algorithm 4.4.2 (HashToPoint1): cryptographically hashes strings to
 points on type-1 curves.
 Input:
 o  A prime p
 o  A prime q
 o  A string id
 o  A cryptographic hash function hashfcn
 Output:
 o  A point Q_id of order q in E(F_p)
 Method:
 1. Let y = HashToRange(id, p, hashfcn), using Algorithm 4.1.1
    (HashToRange), an element of F_p
 2. Let x = (y^2 - 1)^((2 * p - 1) / 3) modulo p, an element of F_p
 3. Let Q' = (x, y), a non-zero point in E(F_p)
 4. Let Q = [(p + 1) / q ]Q', a point of order q in E(F_p)

4.5. Bilinear Mapping

4.5.1. Regular or Modified Tate Pairing

 Pairing(E, p, q, A, B) takes two points A and B, both of order q,
 and, in the type-1 case, returns the modified pairing e'(A, phi(B))
 in F_p^2 where A and B are both in E(F_p).
 Algorithm 4.5.1 (Pairing): computes the regular or modified Tate
 pairing depending on the curve type.

Boyen & Martin Informational [Page 29] RFC 5091 IBCS #1 December 2007

 Input:
 o  A description of an elliptic curve E/F_p such that E(F_p) and
    E(F_p^2) have a subgroup of order q
 o  Two points A and B of order q in E(F_p) or E(F_p^2)
 Output:
 o  On supersingular curves, the value of e'(A, B) in F_p^2 where A
    and B are both in E(F_p)
 Method:
 1. If E is a type-1 curve, execute Algorithm 4.5.2 (Pairing1)

4.5.2. Type-1 Curve Implementation

 Algorithm 4.5.2 (Pairing1): computes the modified Tate pairing on
 type-1 curves.  The values of p and q MUST be part of a valid set of
 public parameters as defined in Section 5.1.2 or Section 6.1.2.
 Input:
 o  A curve E/F_p: y^2 = x^3 + 1 where p is congruent to 11 modulo 12
    and E(F_p) has a subgroup of order q
 o  Two points A and B of order q in E(F_p)
 Output:
 o  The value of e'(A, B) = e(A, phi(B)) in F_p^2
 Method:
 1. Compute B' = phi(B), as follows:
    (a) Let (x, y) in F_p x F_p be the coordinates of B in E(F_p)
    (b) Let zeta = (a_zeta , b_zeta), where a_zeta = (p - 1) / 2 and
        b_zeta = 3^((p + 1) / 4) (mod p), an element of F_p^2
    (c) Let x' =  x * zeta in F_p^2
    (d) Let B' = (x', y) in F_p^2 x F_p

Boyen & Martin Informational [Page 30] RFC 5091 IBCS #1 December 2007

 2. Compute the Tate pairing e(A, B') = e(A, phi(B)) in F_p^2 using
    the Miller method, as in Algorithm 3.5.1 (Tate) described in
    Section 3.5

4.6. Ratio of Bilinear Pairings

4.6.1. Ratio of Regular or Modified Tate Pairings

 PairingRatio(E, p, q, A, B, C, D) takes four points as input and
 computes the ratio of the two bilinear pairings, Pairing(E, p, q, A,
 B) / Pairing(E, p, q, C, D), or, equivalently, the product,
 Pairing(E, p, q, A, B) * Pairing(E, p, q, C, -D).
 On type-1 curves, all four points are of order q in E(F_p), and the
 result is an element of order q in the extension field F_p^2 .
 The motivation for this algorithm is that the ratio of two pairings
 can be calculated more efficiently than by computing each pairing
 separately and dividing one into the other, since certain
 calculations that would normally appear in each of the two pairings
 can be combined and carried out at once.  Such calculations include
 the repeated doublings in steps 2(a)i, 2(a)ii, 3(a)i, and 3(a)ii of
 Algorithm 3.5.2 (TateMillerSolinas), as well as the final
 exponentiation in step 6(a) of Algorithm 3.5.2 (TateMillerSolinas).
 Algorithm 4.6.1 (PairingRatio): computes the ratio of two regular or
 modified Tate pairings depending on the curve type.
 Input:
 o  A description of an elliptic curve E/F_p such that E(F_p) and
    E(F_p^2) have a subgroup of order q
 o  Four points A, B, C, and D, of order q in E(F_p) or E(F_p^2)
 Output:
 o  On supersingular curves, the value of e'(A, B) / e'(C, D) in F_p^2
    where A, B, C, D are all in E(F_p)
 Method:
 1. If E is a type-1 curve, execute Algorithm 4.6.2 (PairingRatio1)

Boyen & Martin Informational [Page 31] RFC 5091 IBCS #1 December 2007

4.6.2. Type-1 Curve Implementation

 Algorithm 4.6.2 (PairingRatio1): computes the ratio of two modified
 Tate pairings on type-1 curves.  The values of p and q MUST be part
 of a valid set of public parameters as defined in Section 5.1.2 or
 Section 6.1.2.
 Input:
 o  A curve E/F_p: y^2 = x^3 + 1, where p is congruent to 11 modulo 12
    and E(F_p) has a subgroup of order q
 o  Four points A, B, C, and D of order q in E(F_p)
 Output:
 o  The value of e'(A, B) / e'(C, D) = e(A, phi(B)) / e(C, phi(D)) =
    e(A, phi(B)) * e(-C, phi(D)), in F_p^2
 Method:
 1. The step-by-step description of the optimized algorithm is omitted
    in this normative specification
 The correct result can always be obtained, although more slowly, by
 computing the product of pairings Pairing1(E, p, q, A, B) *
 Pairing1(E, p, q, -C, D) by using two invocations of Algorithm 4.5.2
 (Pairing1).

5. The Boneh-Franklin BF Cryptosystem

 This chapter describes the algorithms constituting the Boneh-Franklin
 identity-based cryptosystem as described in [BF].

5.1. Setup

5.1.1. Master Secret and Public Parameter Generation

 Algorithm 5.1.1 (BFsetup): randomly selects a master secret and the
 associated public parameters.
 Input:
 o  An integer version number
 o  A security parameter n (MUST take values either 1024, 2048, 3072,
    7680, 15360)

Boyen & Martin Informational [Page 32] RFC 5091 IBCS #1 December 2007

 Output:
 o  A set of public parameters (version, E, p, q, P, P_pub, hashfcn)
 o  A corresponding master secret s
 Method:
 1. Depending on the selected type t:
    (a) If version = 2, then execute Algorithm 5.1.2 (BFsetup1)
 2. The resulting master secret and public parameters are separately
    encoded as per the application protocol requirements

5.1.2. Type-1 Curve Implementation

 BFsetup1 takes a security parameter n as input.  For type-1 curves,
 the scale of n corresponds to the modulus bit-size believed [BF] of
 comparable security in the classical Diffie-Hellman or RSA public-key
 cryptosystems.
 Algorithm 5.1.2 (BFsetup1): establishes a master secret and public
 parameters for type-1 curves.
 Input:
 o  A security parameter n, which MUST be either 1024, 2048, 3072,
    7680 or 15360
 Output:
 o  A set of common public parameters (version, p, q, P, Ppub,
    hashfcn)
 o  A corresponding master secret s
 Method:
 1. Set the version to version = 2.
 2. Determine the subordinate security parameters n_p and n_q as
    follows:
    (a) If n = 1024, then let n_p = 512, n_q = 160, hashfcn =
        1.3.14.3.2.26 (SHA-1 [SHA]

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    (b) If n = 2048, then let n_p = 1024, n_q = 224, hashfcn =
        2.16.840.1.101.3.4.2.4 (SHA-224 [SHA])
    (c) If n = 3072, then let n_p = 1536, n_q = 256, hashfcn =
        2.16.840.1.101.3.4.2.1 (SHA-256 [SHA])
    (d) If n = 7680, then let n_p = 3840, n_q = 384, hashfcn =
        2.16.840.1.101.3.4.2.2 (SHA-384 [SHA])
    (e) If n = 15360, then let n_p = 7680, n_q = 512, hashfcn =
        2.16.840.1.101.3.4.2.3 (SHA-512 [SHA])
 3. Construct the elliptic curve and its subgroup of interest, as
    follows:
    (a) Select an arbitrary n_q-bit Solinas prime q
    (b) Select a random integer r such that p = 12 * r * q - 1 is an
        n_p-bit prime
 4. Select a point P of order q in E(F_p), as follows:
    (a) Select a random point P' of coordinates (x', y') on the curve
        E/F_p: y^2 = x^3 + 1 (mod p)
    (b) Let P = [12 * r]P'
    (c) If P = 0, then start over in step 3a
 5. Determine the master secret and the public parameters as follows:
    (a) Select a random integer s in the range 2 to q - 1
    (b) Let P_pub = [s]P
 6. (version, E, p, q, P, P_pub) are the public parameters where E:
    y^2 = x^3 + 1 is represented by the OID 2.16.840.1.114334.1.1.1.1.
 7. The integer s is the master secret

5.2. Public Key Derivation

5.2.1. Public Key Derivation from an Identity and Public Parameters

 BFderivePubl takes an identity string id and a set of public
 parameters, and it returns a point Q_id.  The public parameters used
 MUST be a valid set of public parameters as defined by Section 5.1.2.

Boyen & Martin Informational [Page 34] RFC 5091 IBCS #1 December 2007

 Algorithm 5.2.1 (BFderivePubl): derives the public key corresponding
 to an identity string.
 Input:
 o  An identity string id
 o  A set of public parameters (version, E, p, q, P, P_pub, hashfcn)
 Output:
 o  A point Q_id of order q in E(F_p) or E(F_p^2)
 Method:
 1. Q_id = HashToPoint(E, p, q, id, hashfcn), using Algorithm 4.4.1
    (HashToPoint)

5.3. Private Key Extraction

5.3.1. Private Key Extraction from an Identity, a Set of Public

      Parameters and a Master Secret
 BFextractPriv takes an identity string id, a set of public
 parameters, and corresponding master secret, and it returns a point
 S_id.  The public parameters used MUST be a valid set of public
 parameters as defined by Section 5.1.2.
 Algorithm 5.3.1 (BFextractPriv): extracts the private key
 corresponding to an identity string.
 Input:
 o  An identity string id
 o  A set of public parameters (version, E, p, q, P, P_pub, hashfcn)
 Output:
 o  A point S_id of order q in E(F_p)
 Method:
 1. Let Q_id = HashToPoint(E, p, q, id, hashfcn) using Algorithm 4.4.1
    (HashToPoint)
 2. Let S_id = [s]Q_id

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5.4. Encryption

5.4.1. Encrypt a Session Key Using an Identity and Public Parameters

 BFencrypt takes three inputs: a public parameter block, an identity
 id, and a plaintext m.  The plaintext MUST be a random symmetric
 session key.  The public parameters used MUST be a valid set of
 public parameters as defined by Section 5.1.2.
 Algorithm 5.4.1 (BFencrypt): encrypts a random session key for an
 identity string.
 Input:
 o  A plaintext string m of size |m| octets
 o  A recipient identity string id
 o  A set of public parameters (version, E, p, q, P, P_pub, hashfcn)
 Output:
 o  A ciphertext tuple (U, V, W) in E(F_p) x {0, ... , 255}^hashlen x
    {0, ... , 255}^|m|
 Method:
 1. Let hashlen be the length of the output of the cryptographic hash
    function hashfcn from the public parameters.
 2. Q_id = HashToPoint(E, p, q, id, hashfcn), using Algorithm 4.4.1
    (HashToPoint), which results in a point of order q in E(F_p)
 3. Select a random hashlen-bit vector rho, represented as (hashlen /
    8)-octet string in big-endian convention
 4. Let t = hashfcn(m), a hashlen-octet string resulting from applying
    the hashfcn algorithm to the input m
 5. Let l = HashToRange(rho || t, q, hashfcn), an integer in the range
    0 to q - 1 resulting from applying Algorithm 4.1.1 (HashToRange)
    to the (2 * hashlen)-octet concatenation of rho and t
 6. Let U = [l]P, which is a point of order q in E(F_p)
 7. Let theta = Pairing(E, p, q, P_pub, Q_id), which is an element of
    the extension field F_p^2 obtained using the modified Tate pairing
    of Algorithm 4.5.1 (Pairing)

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 8. Let theta' = theta^l, which is theta raised to the power of l in
    F_p^2
 9. Let z = Canonical(p, k, 0, theta'), using Algorithm 4.3.1
    (Canonical), the result of which is a canonical string
    representation of theta'
 10. Let w = hashfcn(z) using the hashfcn hashing algorithm, the
     result of which is a hashlen-octet string
 11. Let V = w XOR rho, which is the hashlen-octet long bit-wise XOR
     of w and rho
 12. Let W = HashBytes(|m|, rho, hashfcn) XOR m, which is the bit-wise
     XOR of m with the first |m| octets of the pseudo-random bytes
     produced by Algorithm 4.2.1 (HashBytes) with seed rho
 13. The ciphertext is the triple (U, V, W)

5.5. Decryption

5.5.1. Decrypt an Encrypted Session Key Using Public Parameters,

      a Private Key
 BFdecrypt takes three inputs: a public parameter block, a private key
 block key, and a ciphertext parsed as (U', V', W').  The public
 parameters used MUST be a valid set of public parameters as defined
 by Section 5.1.2.
 Algorithm 5.5.1 (BFdecrypt): decrypts an encrypted session key using
 a private key.
 Input:
 o  A private key point S_id of order q in E(F_p)
 o  A ciphertext triple (U, V, W) in E(F_p) x {0, ... , 255}^hashlen x
    {0, ... , 255}*
 o  A set of public parameters (version, E, p, q, P, P_pub, hashfcn)
 Output:
 o  A decrypted plaintext m, or an invalid ciphertext flag

Boyen & Martin Informational [Page 37] RFC 5091 IBCS #1 December 2007

 Method:
 1. Let hashlen be the length of the output of the hash function
    hashlen measured in octets
 2. Let theta = Pairing(E, p ,q, U, S_id) by applying the modified
    Tate pairing of Algorithm 4.5.1 (Pairing)
 3. Let z = Canonical(p, k, 0, theta) using Algorithm 4.3.1
    (Canonical), the result of which is a canonical string
    representation of theta
 4. Let w = hashfcn(z) using the hashfcn hashing algorithm, the result
    of which is a hashlen-octet string
 5. Let rho = w XOR V, the bit-wise XOR of w and V
 6. Let m = HashBytes(|W|, rho, hashfcn) XOR W, which is the bit-wise
    XOR of m with the first |W| octets of the pseudo-random bytes
    produced by Algorithm 4.2.1 (HashBytes) with seed rho
 7. Let t = hashfcn(m) using the hashfcn algorithm
 8. Let l = HashToRange(rho || t, q, hashfcn) using Algorithm 4.1.1
    (HashToRange) on the (2 * hashlen)-octet concatenation of rho and
    t
 9. Verify that U = [l]P:
    (a) If this is the case, then the decrypted plaintext m is
    returned
    (b) Otherwise, the ciphertext is rejected and no plaintext is
    returned

6. The Boneh-Boyen BB1 Cryptosystem

 This section describes the algorithms constituting the first of the
 two Boneh-Boyen identity-based cryptosystems proposed in [BB1].  The
 description follows the practical implementation given in [BB1].

6.1. Setup

6.1.1. Generate a Master Secret and Public Parameters

 Algorithm 6.1.1 (BBsetup).  Randomly selects a set of master secrets
 and the associated public parameters.

Boyen & Martin Informational [Page 38] RFC 5091 IBCS #1 December 2007

 Input:
 o  An integer version number
 o  An integer security parameter n (MUST take values either 1024,
    2048, 3072, 7680, or 15360)
 Output:
 o  A set of public parameters
 o  A corresponding master secret
 Method:
 1. Depending on the version:
    (a) If version = 2, then execute Algorithm 6.1.2 (BBsetup1)

6.1.2. Type-1 Curve Implementation

 BBsetup1 takes a security parameter n as input.  For type-1 curves, n
 corresponds to the modulus bit-size believed [BF] of comparable
 security in the classical Diffie-Hellman or RSA public-key
 cryptosystems.  For this implementation, n MUST be one of 1024, 2048,
 3072, 7680 or 15360, which correspond to the equivalent bit security
 levels of 80, 112, 128, 192 and 256 bits respectively.
 Algorithm 6.1.2 (BBsetup1): randomly establishes a master secret and
 public parameters for type-1 curves.
 Input:
 o  A security parameter n, either 1024, 2048, 3072, 7680, or 15360
 Output:
 o  A set of public parameters (version, k, E, p, q, P, P_1, P_2, P_3,
    v, hashfcn)
 o  A corresponding triple of master secrets (alpha, beta, gamma)
 Method:
 1. Determine the subordinate security parameters n_p and n_q as
    follows:

Boyen & Martin Informational [Page 39] RFC 5091 IBCS #1 December 2007

    (a) If n = 1024, then let n_p = 512, n_q = 160, hashfcn =
        1.3.14.3.2.26 (SHA-1 [SHA]
    (b) If n = 2048, then let n_p = 1024, n_q = 224, hashfcn =
        2.16.840.1.101.3.4.2.4 (SHA-224 [SHA])
    (c) If n = 3072, then let n_p = 1536, n_q = 256, hashfcn =
        2.16.840.1.101.3.4.2.1 (SHA-256 [SHA])
    (d) If n = 7680, then let n_p = 3840, n_q = 384, hashfcn =
        2.16.840.1.101.3.4.2.2 (SHA-384 [SHA])
    (e) If n = 15360, then let n_p = 7680, n_q = 512, hashfcn =
        2.16.840.1.101.3.4.2.3 (SHA-512 [SHA])
 2. Construct the elliptic curve and its subgroup of interest as
    follows:
    (a) Select a random n_q-bit Solinas prime q
    (b) Select a random integer r, such that p = 12 * r * q - 1 is an
        n_p-bit prime
 3. Select a point P of order q in E(F_p), as follows:
    (a) Select a random point P' of coordinates (x', y') on the curve
        E/F_p: y^2 = x^3 + 1 (mod p)
    (b) Let P = [12 * r]P'
    (c) If P = 0, then start over in step 3a
 4. Determine the master secret and the public parameters as follows:
    (a) Select three random integers alpha, beta, gamma, each of them
        in the range 1 to q - 1
    (b) Let P_1 = [alpha]P
    (c) Let P_2 = [beta]P
    (d) Let P_3 = [gamma]P
    (e) Let v = Pairing(E, p, q, P_1, P_2), which is an element of the
        extension field F_p^2 obtained using the modified Tate pairing
        of Algorithm 4.5.1 (Pairing)

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 5. (version, E, p, q, P, P_1, P_2, P_3, v, hashfcn) are the public
    parameters
 6. (alpha, beta, gamma) constitute the master secret

6.2. Public Key Derivation

6.2.1. Derive a Public Key from an Identity and Public Parameters

 Takes an identity string id and a set of public parameters and
 returns an integer h_id.  The public parameters used MUST be a valid
 set of public parameters as defined by Section 6.1.2.
 Algorithm 6.2.1 (BBderivePubl): derives the public key corresponding
 to an identity string.  The public parameters used MUST be a valid
 set of public parameters as defined by Section 6.1.2.
 Input:
 o  An identity string id
 o  A set of common public parameters (version, k, E, p, q, P, P_1,
    P_2, P_3, v, hashfcn)
 Output:
 o  An integer h_id modulo q
 Method:
 1. Let h_id = HashToRange(id, q, hashfcn), using Algorithm 4.1.1
    (HashToRange)

6.3. Private Key Extraction

6.3.1. Extract a Private Key from an Identity, Public Parameters and a

      Master Secret
 BBextractPriv takes an identity string id, a set of public
 parameters, and corresponding master secrets, and it returns a
 private key consisting of two points D_0 and D_1.  The public
 parameters used MUST be a valid set of public parameters as defined
 by Section 6.1.2.
 Algorithm 6.3.1 (BBextractPriv): extracts the private key
 corresponding to an identity string.

Boyen & Martin Informational [Page 41] RFC 5091 IBCS #1 December 2007

 Input:
 o  An identity string id
 o  A set of public parameters (version, k, E, p, q, P, P_1, P_2, P_3,
    v, hashfcn)
 Output:
 o  A pair of points (D_0, D_1), each of which has order q in E(F_p)
 Method:
 1. Select a random integer r in the range 1 to q - 1
 2. Calculate the point D_0 as follows:
    (a) Let hid = HashToRange(id, q, hashfcn) using Algorithm 4.1.1
        (HashToRange)
    (b) Let y = alpha * beta + r * (alpha * h_id + gamma) in F_q
    (c) Let D_0 = [y]P
 3. Calculate the point D_1 as follows:
    (a) Let D_1 = [r]P
 4. The pair of points (D_0, D_1) constitutes the private key for id

6.4. Encryption

6.4.1. Encrypt a Session Key Using an Identity and Public Parameters

 BBencrypt takes three inputs: a set of public parameters, an identity
 id, and a plaintext m.  The plaintext MUST be a random session key.
 The public parameters used MUST be a valid set of public parameters
 as defined by Section 6.1.2.
 Algorithm 6.4.1 (BBencrypt): encrypts a session key for an identity
 string.
 Input:
 o  A plaintext string m of size |m| octets
 o  A recipient identity string id

Boyen & Martin Informational [Page 42] RFC 5091 IBCS #1 December 2007

 o  A set of public parameters (version, k, E, p, q, P, P_1, P_2, P_3,
    v, hashfcn)
 Output:
 o  A ciphertext tuple (u, C_0, C_1, y) in F_q x E(F_p) x E(F_p) x
    {0, ... , 255}^|m|
 Method:
 1. Select a random integer s in the range 1 to q - 1
 2. Let w = v^s, which is v raised to the power of s in F_p^2, the
    result is an element of order q in F_p^2
 3. Calculate the point C_0 as follows:
    (a) Let C_0 = [s]P
 4. Calculate the point C_1 as follows:
    (a) Let _hid = HashToRange(id, q, hashfcn) using Algorithm 4.1.1
        (HashToRange)
    (b) Let y = s * h_id in F_q
    (c) Let C_1 = [y]P_1 + [s]P_3
 5. Obtain canonical string representations of certain elements:
    (a) Let psi = Canonical(p, k, 1, w) using Algorithm 4.3.1
        (Canonical), the result of which is a canonical octet string
        representation of w
    (b) Let l = Ceiling(lg(p) / 8), the number of octets needed to
        represent integers in F_p, and represent each of these F_p
        elements as a big-endian zero-padded octet string of fixed
        length l:
        (x_0)_(256^l) to represent the x coordinate of C_0
        (y_0)_(256^l) to represent the y coordinate of C_0
        (x_1)_(256^l) to represent the x coordinate of C_1
        (y_1)_(256^l) to represent the y coordinate of C_1

Boyen & Martin Informational [Page 43] RFC 5091 IBCS #1 December 2007

 6. Encrypt the message m into the string y as follows:
    (a) Compute an encryption key h_0 as a two-pass hash of w via its
        representation psi:
          i. Let zeta = hashfcn(psi) using the hashing algorithm
             hashfcn
         ii. Let xi = hashfcn(zeta || psi) using the hashing algorithm
             hashfcn
        iii. Let h' = xi || zeta, the concatenation of the previous
             two hashfcn outputs
    (b) Let y = HashBytes(|m|, h', hashfcn) XOR m, which is the
        bit-wise XOR of m with the first |m| octets of the pseudo-
        random bytes produced by Algorithm 4.2.1 (HashBytes) with seed
        h'
 7. Create the integrity check tag u as follows:
    (a) Compute a one-time pad h'' as a dual-pass hash of the
        representation of (w, C_0, C_1, y):
          i. Let sigma = (y_1)_(256^l) || (x_1)_(256^l) ||
             (y_0)_(256^l) || (x_0)_(256^l) || y || psi be the
             concatenation of y and the five indicated strings in the
             specified order
         ii. Let eta = hashfcn(sigma) using the hashing algorithm
             hashfcn
        iii. Let mu = hashfcn(eta || sigma) using the hashfcn hashing
             algorithm
         iv. Let h'' = mu || eta, the concatenation of the previous
             two outputs of hashfcn
    (b) Build the tag u as the encryption of the integer s with the
        one-time pad h'':
       i. Let rho = HashToRange(h'', q, hashfcn) to get an integer in
          Z_q
      ii. Let u = s + rho (mod q)
 8. The complete ciphertext is given by the quadruple (u, C_0, C_1, y)

Boyen & Martin Informational [Page 44] RFC 5091 IBCS #1 December 2007

6.5. Decryption

6.5.1. Decrypt Using Public Parameters and Private Key

 BBdecrypt takes three inputs: a set of public parameters (version, k,
 E, p, q, P, P_1, P_2, P_3, v, hashfcn), a private key (D_0, D_1), and
 a ciphertext (u, C_0, C_1, y).  It outputs a message m, or signals an
 error if the ciphertext is invalid for the given key.  The public
 parameters used MUST be a valid set of public parameters as defined
 by Section 6.1.2.
 Algorithm 6.5.1 (BBdecrypt): decrypts a ciphertext using public
 parameters and a private key.
 Input:
 o  A private key given as a pair of points (D_0, D_1) of order q in
    E(F_p)
 o  A ciphertext quadruple (u, C_0, C_1, y) in Z_q x E(F_p) x E(F_p) x
    {0, ... , 255}*
 o  A set of public parameters (version, k, E, p, q, P, P_1, P_2, P_3,
    v, hashfcn)
 Output:
 o  A decrypted plaintext m, or an invalid ciphertext flag
 Method:
 1. Let w = PairingRatio(E, p, q, C_0, D_0, C_1, D_1), which computes
    the ratio of two Tate pairings (modified, for type-1 curves) as
    specified in Algorithm 4.6.1 (PairingRatio)
 2. Obtain canonical string representations of certain elements:
    (a) Let psi = Canonical(p, k, 1, w) using Algorithm 4.3.1
        (Canonical); the result is a canonical octet string
        representation of w
    (b) Let l = Ceiling(lg(p) / 8), the number of octets needed to
        represent integers in F_p, and represent each of these F_p
        elements as a big-endian zero-padded octet string of fixed
        length l:
        (x_0)_(256^l) to represent the x coordinate of C_0

Boyen & Martin Informational [Page 45] RFC 5091 IBCS #1 December 2007

        (y_0)_(256^l) to represent the y coordinate of C_0
        (x_1)_(256^l) to represent the x coordinate of C_1
        (y_1)_(256^l) to represent the y coordinate of C_1
 3. Decrypt the message m from the string y as follows:
    (a) Compute the decryption key h' as a dual-pass hash of w via its
        representation psi:
       i. Let zeta = hashfcn(psi) using the hashing algorithm hashfcn
      ii. Let xi = hashfcn(zeta || psi) using the hashing algorithm
          hashfcn
     iii. Let h' = xi || zeta, the concatenation of the previous two
          hashfcn outputs
    (b) Let m = HashBytes(|y|, h', hashfcn)_XOR y, which is the
        bit-wise XOR of y with the first |y| octets of the pseudo-
        random bytes produced by Algorithm 4.2.1 (HashBytes) with seed
        h'
 4. Obtain the integrity check tag u as follows:
    (a) Recover the one-time pad h'' as a dual-pass hash of the
        representation of (w, C_0, C_1, y):
       i. Let sigma = (y_1)_(256^l) || (x_1)_(256^l) || (y_0)_(256^l)
          || (x_0)_(256^l) || y || psi be the concatenation of y and
          the five indicated strings in the specified order
      ii. Let eta = hashfcn(sigma) using the hashing algorithm hashfcn
     iii. Let mu = hashfcn(eta || sigma) using the hashing algorithm
          hashfcn
      iv. Let h'' = mu || eta, the concatenation of the previous two
          hashfcn outputs
    (b) Unblind the encryption randomization integer s from the tag u
        using h'':
       i. Let rho = HashToRange(h'', q, hashfcn) to get an integer in
          Z_q
      ii. Let s = u - rho (mod q)

Boyen & Martin Informational [Page 46] RFC 5091 IBCS #1 December 2007

 5. Verify the ciphertext consistency according to the decrypted
    values:
    (a) Test whether the equality w = v^s holds
    (b) Test whether the equality C_0 = [s]P holds
 6. Adjudication and final output:
    (a) If either of the tests performed in step 5 fails, the
        ciphertext is rejected, and no decryption is output
    (b) Otherwise, i.e., when both tests performed in step 5 succeed,
        the decrypted message is the output

7. Test Data

 The following data can be used to verify the correct operation of
 selected algorithms that are defined in this document.

7.1. Algorithm 3.2.2 (PointMultiply)

 Input:
 q = 0xfffffffffffffffffffffffffffbffff
 p = 0xbffffffffffffffffffffffffffcffff3
 E/F_p: y^2 = x^3 + 1
 A = (0x489a03c58dcf7fcfc97e99ffef0bb4634,
 0x510c6972d795ec0c2b081b81de767f808)
 l = 0xb8bbbc0089098f2769b32373ade8f0daf
 Output:
 [l]A = (0x073734b32a882cc97956b9f7e54a2d326,
 0x9c4b891aab199741a44a5b6b632b949f7)

Boyen & Martin Informational [Page 47] RFC 5091 IBCS #1 December 2007

7.2. Algorithm 4.1.1 (HashToRange)

 Input:
 s =
 54:68:69:73:20:41:53:43:49:49:20:73:74:72:69:6e:67:20:77:69:74
 :68:6f:75:74:20:6e:75:6c:6c:2d:74:65:72:6d:69:6e:61:74:6f:72
 ("This ASCII string without null-terminator")
 n = 0xffffffffffffffffffffefffffffffffffffffff
 hashfcn = 1.3.14.3.2.16 (SHA-1)
 Output:
 v = 0x79317c1610c1fc018e9c53d89d59c108cd518608

7.3. Algorithm 4.5.1 (Pairing)

 Input:
 q = 0xfffffffffffffffffffffffffffbffff
 p = 0xbffffffffffffffffffffffffffcffff3
 E/F_p: y^2 = x^3 + 1
 A = (0x489a03c58dcf7fcfc97e99ffef0bb4634,
 0x510c6972d795ec0c2b081b81de767f808)
 B = (0x40e98b9382e0b1fa6747dcb1655f54f75,
 0xb497a6a02e7611511d0db2ff133b32a3f)
 Output:
 e'(A, B) = (0x8b2cac13cbd422658f9e5757b85493818,
 0xbc6af59f54d0a5d83c8efd8f5214fad3c)

Boyen & Martin Informational [Page 48] RFC 5091 IBCS #1 December 2007

7.4. Algorithm 5.2.1 (BFderivePubl)

 Input:
 id = 6f:42:62 ("Bob")
 version = 2
 p = 0xa6a0ffd016103ffffffffff595f002fe9ef195f002fe9efb
 q = 0xffffffffffffffffffffffeffffffffffff
 P = (0x6924c354256acf5a0ff7f61be4f0495b54540a5bf6395b3d,
 0x024fd8e2eb7c09104bca116f41c035219955237c0eac19ab)
 P_pub = (0xa68412ae960d1392701066664d20b2f4a76d6ee715621108,
 0x9e7644e75c9a131d075752e143e3f0435ff231b6745a486f)
 Output:
 Q_id = (0x22fa1207e0d19e1a4825009e0e88e35eb57ba79391498f59,
 0x982d29acf942127e0f01c881b5ec1b5fe23d05269f538836)

7.5. Algorithm 5.3.1 (BFextractPriv)

 Input:
 s = 0x749e52ddb807e0220054417e514742b05a0
 version = 2
 p = 0xa6a0ffd016103ffffffffff595f002fe9ef195f002fe9efb
 q = 0xffffffffffffffffffffffeffffffffffff
 P = (0x6924c354256acf5a0ff7f61be4f0495b54540a5bf6395b3d,
 0x024fd8e2eb7c09104bca116f41c035219955237c0eac19ab)
 P_pub = (0xa68412ae960d1392701066664d20b2f4a76d6ee715621108,
 0x9e7644e75c9a131d075752e143e3f0435ff231b6745a486f)
 Output:
 Q_id = (0x8212b74ea75c841a9d1accc914ca140f4032d191b5ce5501,
 0x950643d940aba68099bdcb40082532b6130c88d317958657)

Boyen & Martin Informational [Page 49] RFC 5091 IBCS #1 December 2007

7.6. Algorithm 5.4.1 (BFencrypt)

    Note: the following values can also be used to test
    Algorithm 5.5.1 (BFdecrypt).
 Input:
 m = 48:69:20:74:68:65:72:65:21 ("Hi there!")
 id = 6f:42:62 ("Bob")
 version = 2
 p = 0xa6a0ffd016103ffffffffff595f002fe9ef195f002fe9efb
 q = 0xffffffffffffffffffffffeffffffffffff
 P = (0x6924c354256acf5a0ff7f61be4f0495b54540a5bf6395b3d,
 0x024fd8e2eb7c09104bca116f41c035219955237c0eac19ab)
 P_pub = (0xa68412ae960d1392701066664d20b2f4a76d6ee715621108,
 0x9e7644e75c9a131d075752e143e3f0435ff231b6745a486f)
 Output:
 Using the random value rho =
 0xed5397ff77b567ba5ecb644d7671d6b6f2082968, we get the
 following output:
 U =
 (0x1b5f6c461497acdfcbb6d6613ad515430c8b3fa23b61c585e9a541b199e
 2a6cb,
 0x9bdfbed1ae664e51e3d4533359d733ac9a600b61048a7d899104e826a0ec
 4fa4)
 V =
 e0:1d:ad:81:32:6c:b1:73:af:c2:8d:72:2e:7a:32:1a:7b:29:8a:aa
 W = f9:04:ba:40:30:e9:ce:6e:ff

Boyen & Martin Informational [Page 50] RFC 5091 IBCS #1 December 2007

7.7. Algorithm 6.3.1 (BBextractPriv)

 Input:
 alpha = 0xa60c395285ded4d70202c8283d894bad4f0
 beta = 0x48bf012da19f170b13124e5301561f45053
 gamma = 0x226fba82bc38e2ce4e28e56472ccf94a499
 version = 2
 p = 0x91bbe2be1c8950750784befffffffffffff6e441d41e12fb
 q = 0xfffffffffbfffffffffffffffffffffffff
 P = (0x13cc538fe950411218d7f5c17ae58a15e58f0877b29f2fe1,
 0x8cf7bab1a748d323cc601fabd8b479f54a60be11e28e18cf)
 P_1 = (0x0f809a992ed2467a138d72bc1d8931c6ccdd781bedc74627,
 0x11c933027beaaf73aa9022db366374b1c68d6bf7d7a888c2)
 P_2 = (0x0f8ac99a55e575bf595308cfea13edb8ec673983919121b0,
 0x3febb7c6369f5d5f18ee3ea6ca0181448a4f3c4f3385019c)
 P_3 = (0x2c10b43991052e78fac44fdce639c45824f5a3a2550b2a45,
 0x6d7c12d8a0681426a5bbc369c9ef54624356e2f6036a064f)
 v = (0x38f91032de6847a89fc3c83e663ed0c21c8f30ce65c0d7d3,
 0x44b9aa10849cc8d8987ef2421770a340056745da8b99fba2)
 id = 6f:42:62 ("Bob")
 Output:
 Using the random value r =
 0x695024c25812112187162c08aa5f65c7a2c, we get the following
 output:
 D_0 = (0x3264e13feeb7c506493888132964e79ad657a952334b9e53,
 0x3eeaefc14ba1277a1cd6fdea83c7c882fe6d85d957055c7b)
 D_1 = (0x8d7a72ad06909bb3bb29b67676d935018183a905e7e8cb18,
 0x2b346c6801c1db638f270af915a21054f16044ab67f6c40e)

Boyen & Martin Informational [Page 51] RFC 5091 IBCS #1 December 2007

7.8. Algorithm 6.4.1 (BBencrypt)

    Note: the following values can also be used to test
    Algorithm 5.5.1 (BFdecrypt).
 Input:
 m = 48:69:20:74:68:65:72:65:21 ("Hi there!")
 id = 6f:42:62 ("Bob")
 version = 2
 E: y^2 = x^3 + 1
 p = 0x91bbe2be1c8950750784befffffffffffff6e441d41e12fb
 q = 0xfffffffffbfffffffffffffffffffffffff
 P = (0x13cc538fe950411218d7f5c17ae58a15e58f0877b29f2fe1,
 0x8cf7bab1a748d323cc601fabd8b479f54a60be11e28e18cf)
 P_1 = (0x0f809a992ed2467a138d72bc1d8931c6ccdd781bedc74627,
 0x11c933027beaaf73aa9022db366374b1c68d6bf7d7a888c2)
 P_2 = (0x0f8ac99a55e575bf595308cfea13edb8ec673983919121b0,
 0x3febb7c6369f5d5f18ee3ea6ca0181448a4f3c4f3385019c)
 P_3 = (0x2c10b43991052e78fac44fdce639c45824f5a3a2550b2a45,
 0x6d7c12d8a0681426a5bbc369c9ef54624356e2f6036a064f)
 v = (0x38f91032de6847a89fc3c83e663ed0c21c8f30ce65c0d7d3,
 0x44b9aa10849cc8d8987ef2421770a340056745da8b99fba2)
 hashfcn = 1.3.14.3.2.26 (SHA-1)
 Output:
 Using the random value s =
 0x62759e95ce1af248040e220263fb41b965e, we get the following
 output:
 u = 0xad1ebfa82edf0bcb5111e9dc08ff0737c68
 C_0 = (0x79f8f35904579f1aaf51897b1e8f1d84e1c927b8994e81f9,
 0x1cf77bb2516606681aba2e2dc14764aa1b55a45836014c62)

Boyen & Martin Informational [Page 52] RFC 5091 IBCS #1 December 2007

 C_1 = (0x410cfeb0bccf1fa4afc607316c8b12fe464097b20250d684,
 0x8bb76e7195a7b1980531b0a5852ce710cab5d288b2404e90)
 y = 82:a6:42:b9:bb:e9:82:c4:57

8. ASN.1 Module

 This section defines the ASN.1 module for the encodings discussed in
 this document.
 IBCS { joint-iso-itu-t(2) country(16) us(840) organization(1)
    identicrypt(114334) ibcs(1) module(5) version(1) }
 DEFINITIONS IMPLICIT TAGS ::= BEGIN
  1. -
  2. - Identity-based cryptography standards (IBCS):
  3. - supersingular curve implementations of
  4. - the BF and BB1 cryptosystems
  5. -
  6. - This version only supports IBE using
  7. - type-1 curves, i.e., the curve y^2 = x^3 + 1.
  8. -
 ibcs OBJECT IDENTIFIER ::= {
    joint-iso-itu-t(2) country(16) us(840) organization(1)
       identicrypt(114334) ibcs(1)
 }
  1. -
  2. - IBCS1
  3. -
  4. - IBCS1 defines the algorithms used to implement IBE
  5. -
 ibcs1 OBJECT IDENTIFIER ::= {
    ibcs ibcs1(1)
 }
  1. -
  2. - An elliptic curve is specified by an OID.
  3. - A type1curve is defined by the equation y^2 = x^3 + 1.
  4. -
 type1curve OBJECT IDENTIFIER ::= {
    ibcs1 curve-types(1) type1-curve(1)
 }

Boyen & Martin Informational [Page 53] RFC 5091 IBCS #1 December 2007

  1. -
  2. - Supporting types
  3. -
  1. -
  2. - Encoding of a point on an elliptic curve E/F_p
  3. - An FpPoint can either represent an element of
  4. - F_p^2 or an element of (F_p)^2.
  FpPoint ::= SEQUENCE {
    x  INTEGER,
    y  INTEGER
 }
  1. -
  2. - The following hash functions are supported:
  3. -
  4. - SHA-1
  5. -
  6. - id-sha1 OBJECT IDENTIFIER ::= {
  7. - iso(1) identified-organization(3) oiw(14)
  8. - secsig(3) algorithms(2) hashAlgorithmIdentifier(26)
  9. - }
  10. -
  11. - SHA-224
  12. -
  13. - id-sha224 OBJECT IDENTIFIER ::= {
  14. - joint-iso-itu-t(2)country(16) us(840)
  15. - organization(1) gov(101)
  16. - csor(3) nistAlgorithm(4) hashAlgs(2) sha224(4)
  17. - }
  18. -
  19. - SHA-256
  20. -
  21. - id-sha256 OBJECT IDENTIFIER ::= {
  22. - joint-iso-itu-t(2)country(16) us(840)
  23. - organization(1) gov(101)
  24. - csor(3) nistAlgorithm(4) hashAlgs(2) sha256(1)
  25. - }
  26. -
  27. - SHA-384
  28. -
  29. - id-sha384 OBJECT IDENTIFIER ::= {
  30. - joint-iso-itu-t(2)country(16) us(840)
  31. - organization(1) gov(101)
  32. - csor(3) nistAlgorithm(4) hashAlgs(2) sha384(2)
  33. - }
  34. -

Boyen & Martin Informational [Page 54] RFC 5091 IBCS #1 December 2007

  1. - SHA-512
  2. -
  3. - id-sha512 OBJECT IDENTIFIER ::= {
  4. - joint-iso-itu-t(2) country(16) us(840)
  5. - organization(1) gov(101)
  6. - csor(3) nistAlgorithm(4) hashAlgs(2) sha512(3)
  7. - }
  8. -
  9. -
  10. - Algorithms
  11. -
 ibe-algorithms OBJECT IDENTIFIER ::= {
    ibcs1 ibe-algorithms(2)
 }
  1. – Boneh-Franklin IBE
 bf OBJECT IDENTIFIER ::= { ibe-algorithms bf(1) }
  1. -
  2. - Encoding of a BF public parameters block.
  3. - The only version currently supported is version 2.
  4. - The values p and q define a subgroup of E(F_p) of order q.
  5. -
 BFPublicParameters ::= SEQUENCE {
    version     INTEGER { v2(2) },
    curve       OBJECT IDENTIFIER,
    p           INTEGER,
    q           INTEGER,
    pointP      FpPoint,
    pointPpub   FpPoint,
    hashfcn     OBJECT IDENTIFIER
 }
  1. -
  2. - A BF private key is a point on an elliptic curve,
  3. - which is an FpPoint.
  4. - The only version supported is version 2.
  5. -
 BFPrivateKeyBlock ::= SEQUENCE {
    version     INTEGER { v2(2) },
    privateKey  FpPoint
 }

Boyen & Martin Informational [Page 55] RFC 5091 IBCS #1 December 2007

  1. -
  2. - A BF master secret is an integer.
  3. - The only version supported is version 2.
  4. -
 BFMasterSecret ::= SEQUENCE {
    version        INTEGER {v2(2) },
    masterSecret   INTEGER
 }
  1. -
  2. - BF ciphertext block
  3. - The only version supported is version 2.
  4. -
 BFCiphertextBlock ::= SEQUENCE {
    version  INTEGER { v2(2) },
    u        FpPoint,
    v        OCTET STRING,
    w        OCTET STRING
 }
  1. -
  2. - Boneh-Boyen (BB1) IBE
  3. -
 bb1 OBJECT IDENTIFIER ::= { ibe-algorithms bb1(2) }
  1. -
  2. - Encoding of a BB1 public parameters block.
  3. - The version is currently fixed to 2.
  4. -
  5. -
 BB1PublicParameters ::= SEQUENCE {
    version     INTEGER { v2(2) },
    curve       OBJECT IDENTIFIER,
    p           INTEGER,
    q           INTEGER,
    pointP      FpPoint,
    pointP1     FpPoint,
    pointP2     FpPoint,
    pointP3     FpPoint,
    v           FpPoint,
    hashfcn     OBJECT IDENTIFIER
 }

Boyen & Martin Informational [Page 56] RFC 5091 IBCS #1 December 2007

  1. -
  2. - BB1 master secret block
  3. - The only version supported is version 2.
  4. -
 BB1MasterSecret ::= SEQUENCE {
    version  INTEGER { v2(2) },
    alpha    INTEGER,
    beta     INTEGER,
    gamma    INTEGER
 }
  1. -
  2. - BB1 private Key block
  3. - The only version supported is version 2.
  4. -
 BB1PrivateKeyBlock ::= SEQUENCE {
    version  INTEGER { v2(2) },
    pointD0  FpPoint,
    pointD1  FpPoint
 }
  1. -
  2. - BB1 ciphertext block
  3. - The only version supported is version 2.
  4. -
 BB1CiphertextBlock ::= SEQUENCE {
    version     INTEGER {v2(2) },
    pointChi0   FpPoint,
    pointChi1   FpPoint,
    nu          INTEGER,
    y           OCTET STRING
 }
 END

Boyen & Martin Informational [Page 57] RFC 5091 IBCS #1 December 2007

9. Security Considerations

 This document describes cryptographic algorithms.  We assume that the
 security provided by such algorithms depends entirely on the secrecy
 of the relevant private key, and for an adversary to defeat the
 security provided by the algorithms, he will need to perform
 computationally-intensive cryptanalytic attacks to recover the
 private key.
 We assume that users of the algorithms described in this document
 will require one of five levels of cryptographic strength: the
 equivalent of 80 bits, 112 bits, 128 bits, 192 bits or, 256 bits.
 The 80-bit level is suitable for legacy applications and SHOULD NOT
 be used to protect information whose useful life extends past the
 year 2010.  The 112-bit level is suitable for use in key transport of
 Triple-DES keys and should be adequate to protect information whose
 useful life extends up to the year 2030.  The 128-bit levels and
 higher are suitable for use in the transport of Advanced Encryption
 Standard (AES) keys of the corresponding length or less and are
 adequate to protect information whose useful life extends past the
 year 2030.
 Table 1 summarizes the security parameters for the BF and BB1
 algorithms that will attain these levels of security.  In this table,
 |p| represents the number of bits in a prime number p, and |q|
 represents the number of bits in a subprime q.  This table assumes
 that a Type-1 supersingular curve is used.
 Bits of Security   |p|    |q|
 80                 512    160
 112                1024   224
 128                1536   256
 192                3840   384
 256                7680   512
 Table 1: Sizes of BF and BB1 Parameters Required to Attain Standard
 Levels of Bit Security [SP800-57].
 If an IBE key is used to transport a symmetric key that provides more
 bits of security than the bit strength of the IBE key, users should
 understand that the security of the system is then limited by the
 strength of the weaker IBE key. So if an IBE key that provides 112
 bits of security is used to transport a 128-bit AES key, then the
 security provided is limited by the 112 bits of security of the IBE
 key.

Boyen & Martin Informational [Page 58] RFC 5091 IBCS #1 December 2007

 Note that this document specifies the use of the National Institute
 of Standards and Technology (NIST) hashing algorithms [SHA] to hash
 identities to either a point on an elliptic curve or an integer.
 Recent attacks on SHA-1 [SHA] have discovered ways to find collisions
 with less work than the expected 2^80 hashes required based on the
 size of the output of the hash function alone.  If an attacker can
 find a collision, then they could use the colliding preimages to
 create two identities that have the same IBE private key.  The
 practical use of such a SHA-1 [SHA] collision is extremely unlikely,
 however.
 Identities are typically not random strings like the preimages of a
 hash collision would be.  In particular, this is true if IBE is used
 as described in [IBECMS], in which components of an identity are
 defined to be an e-mail address, a validity period, and a URI.  In
 this case, the unpredictable results of a collision are extremely
 unlikely to fit the format of a valid identity, and thus, are of no
 use to an attacker.  Any protocol using IBE MUST define an identity
 in a way that makes collisions in a hash function essentially useless
 to an attacker.  Because random strings are rarely used as
 identities, this requirement should not be unduly difficult to
 fulfill.
 The randomness of the random values that are required by the
 cryptographic algorithms is vital to the security provided by the
 algorithms.  Any implementation of these algorithms MUST use a source
 of random values that provides an adequate level of security.
 Appropriate algorithms to generate such values include [FIPS186-2]
 and [X9.62].  This will ensure that the random values used to mask
 plaintext messages in Sections 5.4 and 6.4 are not reused with a
 significant probability.
 The strength of a system using the algorithms described in this
 document relies on the strength of the mechanism used to authenticate
 a user requesting a private key from a PKG, as described in step 2 of
 Section 1.2 of this document.  This is analogous to the way in which
 the strength of a system using digital certificates [X.509] is
 limited by the strength of the authentication required of users
 before certificates are granted to them.  In either case, a weak
 mechanism for authenticating users will result in a weak system that
 relies on the technology.  A system that uses the algorithms
 described in this document MUST require users to authenticate in a
 way that is suitably strong, particularly if IBE private keys will be
 used for authentication.
 Note that IBE systems have different properties than other asymmetric
 cryptographic schemes when it comes to key recovery.  If a master
 secret is maintained on a secure PKG, then the PKG and any

Boyen & Martin Informational [Page 59] RFC 5091 IBCS #1 December 2007

 administrator with the appropriate level of access will be able to
 create arbitrary private keys, so that controls around such
 administrators and logging of all actions performed by such
 administrators SHOULD be part of a functioning IBE system.
 On the other hand, it is also possible to create IBE private keys
 using a master secret and to then destroy the master secret, making
 any key recovery impossible.  If this property is not desired, an
 administrator of an IBE system SHOULD require that the format of the
 identity used by the system contain a component that is short-lived.
 The format of identity that is defined in [IBECMS], for example,
 contains information about the time period of validity of the key
 that will be calculated from the identity.  Such an identity can
 easily be changed to allow the rekeying of users if their IBE private
 key is somehow compromised.

10. Acknowledgments

 This document is based on the IBCS #1 v2 document of Voltage
 Security, Inc.  Any substantial use of material from this document
 should acknowledge Voltage Security, Inc.  as the source of the
 information.

11. References

11.1. Normative References

 [KEYWORDS]   Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119, March 1997.
 [TLS]        Dierks, T. and E. Rescorla, "The Transport Layer
              Security (TLS) Protocol Version 1.1", RFC 4346, April
              2006.

11.2. Informative References

 [BB1]        D. Boneh and X. Boyen, "Efficient selective-ID secure
              identity based encryption without random oracles," In
              Proc. of EUROCRYPT 04, LNCS 3027, pp. 223-238, 2004.
 [BF]         D. Boneh and M. Franklin, "Identity-based encryption
              from the Weil pairing," in Proc. of CRYPTO 01, LNCS
              2139, pp. 213-229, 2001.
 [CMS]        Housley, R., "Cryptographic Message Syntax (CMS)", RFC
              3852, July 2004.

Boyen & Martin Informational [Page 60] RFC 5091 IBCS #1 December 2007

 [ECC]        I. Blake, G. Seroussi, and N. Smart, "Elliptic Curves in
              Cryptography", Cambridge University Press, 1999.
 [FIPS186-2]  National Institute of Standards and Technology, "Digital
              Signature Standard," Federal Information Processing
              Standard 186-2, August 2002.
 [IBEARCH]    G. Appenzeller, L. Martin, and M. Schertler, "Identity-
              based Encryption Architecture", Work in Progress.
 [IBECMS]     L. Martin and M. Schertler, "Using the Boneh-Franklin
              and Boneh-Boyen identity-based encryption algorithms
              with the Cryptographic Message Syntax (CMS)", Work in
              Progress.
 [MERKLE]     R. Merkle, "A fast software one-way hash function,"
              Journal of Cryptology, Vol. 3 (1990), pp. 43-58.
 [P1363]      IEEE P1363-2000, "Standard Specifications for Public Key
              Cryptography," 2001.
 [SP800-57]   E. Barker, W. Barker, W. Burr, W. Polk and M. Smid,
              "Recommendation for Key Management - Part 1: General
              (Revised)," NIST Special Publication 800-57, March 2007.
 [SHA]        National Institute for Standards and Technology, "Secure
              Hash Standard," Federal Information Processing Standards
              Publication 180-2, August 2002, with Change Notice 1,
              February 2004.
 [X9.62]      American National Standards Institute, "Public Key
              Cryptography for the Financial Services Industry: The
              Elliptic Curve Digital Signature Algorithm (ECDSA),"
              American National Standard for Financial Services
              X9.62-2005, November 2005.
 [X.509]      ITU-T Recommendation X.509 (2000) | ISO/IEC 9594-8:2001,
              Information Technology - Open Systems Interconnection -
              The Directory:  Public-key and Attribute Certificate
              Frameworks.

Boyen & Martin Informational [Page 61] RFC 5091 IBCS #1 December 2007

Authors' Addresses

 Xavier Boyen
 Voltage Security
 1070 Arastradero Rd Suite 100
 Palo Alto, CA 94304
 EMail: xavier@voltage.com
 Luther Martin
 Voltage Security
 1070 Arastradero Rd Suite 100
 Palo Alto, CA 94304
 EMail: martin@voltage.com

Boyen & Martin Informational [Page 62] RFC 5091 IBCS #1 December 2007

Full Copyright Statement

 Copyright (C) The IETF Trust (2007).
 This document is subject to the rights, licenses and restrictions
 contained in BCP 78 and at www.rfc-editor.org/copyright.html, and
 except as set forth therein, the authors retain all their rights.
 This document and the information contained herein are provided on an
 "AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS
 OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY, THE IETF TRUST AND
 THE INTERNET ENGINEERING TASK FORCE DISCLAIM 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.

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 The IETF takes no position regarding the validity or scope of any
 Intellectual Property Rights or other rights that might be claimed to
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 this document or the extent to which any license under such rights
 might or might not be available; nor does it represent that it has
 made any independent effort to identify any such rights.  Information
 on the procedures with respect to rights in RFC documents can be
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 Copies of IPR disclosures made to the IETF Secretariat and any
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 The IETF invites any interested party to bring to its attention any
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 rights that may cover technology that may be required to implement
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Boyen & Martin Informational [Page 63]

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