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rfc:rfc6090

Internet Engineering Task Force (IETF) D. McGrew Request for Comments: 6090 Cisco Systems Category: Informational K. Igoe ISSN: 2070-1721 M. Salter

                                              National Security Agency
                                                         February 2011
         Fundamental Elliptic Curve Cryptography Algorithms

Abstract

 This note describes the fundamental algorithms of Elliptic Curve
 Cryptography (ECC) as they were defined in some seminal references
 from 1994 and earlier.  These descriptions may be useful for
 implementing the fundamental algorithms without using any of the
 specialized methods that were developed in following years.  Only
 elliptic curves defined over fields of characteristic greater than
 three are in scope; these curves are those used in Suite B.

Status of This Memo

 This document is not an Internet Standards Track specification; it is
 published for informational purposes.
 This document is a product of the Internet Engineering Task Force
 (IETF).  It represents the consensus of the IETF community.  It has
 received public review and has been approved for publication by the
 Internet Engineering Steering Group (IESG).  Not all documents
 approved by the IESG are a candidate for any level of Internet
 Standard; see Section 2 of RFC 5741.
 Information about the current status of this document, any errata,
 and how to provide feedback on it may be obtained at
 http://www.rfc-editor.org/info/rfc6090.

McGrew, et al. Informational [Page 1] RFC 6090 Fundamental ECC February 2011

Copyright Notice

 Copyright (c) 2011 IETF Trust and the persons identified as the
 document authors.  All rights reserved.
 This document is subject to BCP 78 and the IETF Trust's Legal
 Provisions Relating to IETF Documents
 (http://trustee.ietf.org/license-info) in effect on the date of
 publication of this document.  Please review these documents
 carefully, as they describe your rights and restrictions with respect
 to this document.  Code Components extracted from this document must
 include Simplified BSD License text as described in Section 4.e of
 the Trust Legal Provisions and are provided without warranty as
 described in the Simplified BSD License.

Table of Contents

 1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  3
   1.1.  Conventions Used in This Document  . . . . . . . . . . . .  4
 2.  Mathematical Background  . . . . . . . . . . . . . . . . . . .  4
   2.1.  Modular Arithmetic . . . . . . . . . . . . . . . . . . . .  4
   2.2.  Group Operations . . . . . . . . . . . . . . . . . . . . .  5
   2.3.  The Finite Field Fp  . . . . . . . . . . . . . . . . . . .  6
 3.  Elliptic Curve Groups  . . . . . . . . . . . . . . . . . . . .  7
   3.1.  Homogeneous Coordinates  . . . . . . . . . . . . . . . . .  8
   3.2.  Other Coordinates  . . . . . . . . . . . . . . . . . . . .  9
   3.3.  ECC Parameters . . . . . . . . . . . . . . . . . . . . . .  9
     3.3.1.  Discriminant . . . . . . . . . . . . . . . . . . . . . 10
     3.3.2.  Security . . . . . . . . . . . . . . . . . . . . . . . 10
 4.  Elliptic Curve Diffie-Hellman (ECDH) . . . . . . . . . . . . . 10
   4.1.  Data Types . . . . . . . . . . . . . . . . . . . . . . . . 11
   4.2.  Compact Representation . . . . . . . . . . . . . . . . . . 11
 5.  Elliptic Curve ElGamal Signatures  . . . . . . . . . . . . . . 11
   5.1.  Background . . . . . . . . . . . . . . . . . . . . . . . . 11
   5.2.  Hash Functions . . . . . . . . . . . . . . . . . . . . . . 12
   5.3.  KT-IV Signatures . . . . . . . . . . . . . . . . . . . . . 12
     5.3.1.  Keypair Generation . . . . . . . . . . . . . . . . . . 12
     5.3.2.  Signature Creation . . . . . . . . . . . . . . . . . . 13
     5.3.3.  Signature Verification . . . . . . . . . . . . . . . . 13
   5.4.  KT-I Signatures  . . . . . . . . . . . . . . . . . . . . . 14
     5.4.1.  Keypair Generation . . . . . . . . . . . . . . . . . . 14
     5.4.2.  Signature Creation . . . . . . . . . . . . . . . . . . 14
     5.4.3.  Signature Verification . . . . . . . . . . . . . . . . 14
   5.5.  Converting KT-IV Signatures to KT-I Signatures . . . . . . 15
   5.6.  Rationale  . . . . . . . . . . . . . . . . . . . . . . . . 15
 6.  Converting between Integers and Octet Strings  . . . . . . . . 16
   6.1.  Octet-String-to-Integer Conversion . . . . . . . . . . . . 17
   6.2.  Integer-to-Octet-String Conversion . . . . . . . . . . . . 17

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 7.  Interoperability . . . . . . . . . . . . . . . . . . . . . . . 17
   7.1.  ECDH . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
   7.2.  KT-I and ECDSA . . . . . . . . . . . . . . . . . . . . . . 18
 8.  Validating an Implementation . . . . . . . . . . . . . . . . . 18
   8.1.  ECDH . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
   8.2.  KT-I . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
 9.  Intellectual Property  . . . . . . . . . . . . . . . . . . . . 20
   9.1.  Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . 20
 10. Security Considerations  . . . . . . . . . . . . . . . . . . . 21
   10.1. Subgroups  . . . . . . . . . . . . . . . . . . . . . . . . 21
   10.2. Diffie-Hellman . . . . . . . . . . . . . . . . . . . . . . 22
   10.3. Group Representation and Security  . . . . . . . . . . . . 22
   10.4. Signatures . . . . . . . . . . . . . . . . . . . . . . . . 23
 11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 23
 12. References . . . . . . . . . . . . . . . . . . . . . . . . . . 23
   12.1. Normative References . . . . . . . . . . . . . . . . . . . 23
   12.2. Informative References . . . . . . . . . . . . . . . . . . 25
 Appendix A.  Key Words . . . . . . . . . . . . . . . . . . . . . . 29
 Appendix B.  Random Integer Generation . . . . . . . . . . . . . . 29
 Appendix C.  Why Compact Representation Works  . . . . . . . . . . 30
 Appendix D.  Example ECC Parameter Set . . . . . . . . . . . . . . 31
 Appendix E.  Additive and Multiplicative Notation  . . . . . . . . 32
 Appendix F.  Algorithms  . . . . . . . . . . . . . . . . . . . . . 32
   F.1.  Affine Coordinates . . . . . . . . . . . . . . . . . . . . 32
   F.2.  Homogeneous Coordinates  . . . . . . . . . . . . . . . . . 33

1. Introduction

 ECC is a public-key technology that offers performance advantages at
 higher security levels.  It includes an elliptic curve version of the
 Diffie-Hellman key exchange protocol [DH1976] and elliptic curve
 versions of the ElGamal Signature Algorithm [E1985].  The adoption of
 ECC has been slower than had been anticipated, perhaps due to the
 lack of freely available normative documents and uncertainty over
 intellectual property rights.
 This note contains a description of the fundamental algorithms of ECC
 over finite fields with characteristic greater than three, based
 directly on original references.  Its intent is to provide the
 Internet community with a summary of the basic algorithms that
 predate any specialized or optimized algorithms.  The summary is
 detailed enough for use as a normative reference.  The original
 descriptions and notations were followed as closely as possible.
 There are several standards that specify or incorporate ECC
 algorithms, including the Internet Key Exchange (IKE), ANSI X9.62,
 and IEEE P1363.  The algorithms in this note can interoperate with

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 some of the algorithms in these standards, with a suitable choice of
 parameters and options.  The specifics are itemized in Section 7.
 The rest of the note is organized as follows.  Sections 2.1, 2.2, and
 2.3 furnish the necessary terminology and notation from modular
 arithmetic, group theory, and the theory of finite fields,
 respectively.  Section 3 defines the groups based on elliptic curves
 over finite fields of characteristic greater than three.  Section 4
 presents the fundamental Elliptic Curve Diffie-Hellman (ECDH)
 algorithm.  Section 5 presents elliptic curve versions of the ElGamal
 signature method.  The representation of integers as octet strings is
 specified in Section 6.  Sections 2 through 6, inclusive, contain all
 of the normative text (the text that defines the norm for
 implementations conforming to this specification), and all of the
 following sections are purely informative.  Interoperability is
 discussed in Section 7.  Validation testing is described in
 Section 8.  Section 9 reviews intellectual property issues.
 Section 10 summarizes security considerations.  Appendix B describes
 random number generation, and other appendices provide clarifying
 details.

1.1. Conventions Used in This Document

 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 Appendix A.

2. Mathematical Background

 This section reviews mathematical preliminaries and establishes
 terminology and notation that are used below.

2.1. Modular Arithmetic

 This section reviews modular arithmetic.  Two integers x and y are
 said to be congruent modulo n if x - y is an integer multiple of n.
 Two integers x and y are coprime when their greatest common divisor
 is 1; in this case, there is no third number z > 1 such that z
 divides x and z divides y.
 The set Zq = { 0, 1, 2, ..., q-1 } is closed under the operations of
 modular addition, modular subtraction, modular multiplication, and
 modular inverse.  These operations are as follows.
    For each pair of integers a and b in Zq, a + b mod q is equal to
    a + b if a + b < q, and is equal to a + b - q otherwise.

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    For each pair of integers a and b in Zq, a - b mod q is equal to
    a - b if a - b >= 0, and is equal to a - b + q otherwise.
    For each pair of integers a and b in Zq, a * b mod q is equal to
    the remainder of a * b divided by q.
    For each integer x in Zq that is coprime with q, the inverse of x
    modulo q is denoted as 1/x mod q, and can be computed using the
    extended Euclidean algorithm (see Section 4.5.2 of [K1981v2], for
    example).
 Algorithms for these operations are well known; for instance, see
 Chapter 4 of [K1981v2].

2.2. Group Operations

 This section establishes some terminology and notation for
 mathematical groups, which are needed later on.  Background
 references abound; see [D1966], for example.
 A group is a set of elements G together with an operation that
 combines any two elements in G and returns a third element in G.  The
 operation is denoted as * and its application is denoted as a * b,
 for any two elements a and b in G.  The operation is associative,
 that is, for all a, b, and c in G, a * (b * c) is identical to (a *
 b) * c.  Repeated application of the group operation N-1 times to the
 element a is denoted as a^N, for any element a in G and any positive
 integer N.  That is, a^2 = a * a, a^3 = a * a * a, and so on.  The
 associativity of the group operation ensures that the computation of
 a^n is unambiguous; any grouping of the terms gives the same result.
 The above definition of a group operation uses multiplicative
 notation.  Sometimes an alternative called additive notation is used,
 in which a * b is denoted as a + b, and a^N is denoted as N * a.  In
 multiplicative notation, a^N is called exponentiation, while the
 equivalent operation in additive notation is called scalar
 multiplication.  In this document, multiplicative notation is used
 throughout for consistency.  Appendix E elucidates the correspondence
 between the two notations.
 Every group has a special element called the identity element, which
 we denote as e.  For each element a in G, e * a = a * e = a.  By
 convention, a^0 is equal to the identity element for any a in G.
 Every group element a has a unique inverse element b such that
 a * b = b * a = e.  The inverse of a is denoted as a^-1 in
 multiplicative notation.  (In additive notation, the inverse of a is
 denoted as -a.)

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 For any positive integer X, a^(-X) is defined to be (a^-1)^(X).
 Using this convention, exponentiation behaves as one would expect,
 namely for any integers X and Y:
    a^(X+Y) = (a^X)*(a^Y)
    (a^X)^Y = a^(XY) = (a^Y)^X.
 In cryptographic applications, one typically deals with finite groups
 (groups with a finite number of elements), and for such groups, the
 number of elements of the group is also called the order of the
 group.  A group element a is said to have finite order if a^X = e for
 some positive integer X, and the order of a is the smallest such X.
 If no such X exists, a is said to have infinite order.  All elements
 of a finite group have a finite order, and the order of an element is
 always a divisor of the group order.
 If a group element a has order R, then for any integers X and Y,
    a^X = a^(X mod R),
    a^X = a^Y if and only if X is congruent to Y mod R,
    the set H = { a, a^2, a^3, ... , a^R=e } forms a subgroup of G,
    called the cyclic subgroup generated by a, and a is said to be a
    generator of H.
 Typically, there are several group elements that generate H.  Any
 group element of the form a^M, with M relatively prime to R, also
 generates H.  Note that a^M is equal to g^(M modulo R) for any non-
 negative integer M.
 Given the element a of order R, and an integer i between 1 and R-1,
 inclusive, the element a^i can be computed by the "square and
 multiply" method outlined in Section 2.1 of [M1983] (see also Knuth,
 Vol. 2, Section 4.6.3), or other methods.

2.3. The Finite Field Fp

 This section establishes terminology and notation for finite fields
 with prime characteristic.
 When p is a prime number, then the set Zp, with the addition,
 subtraction, multiplication, and division operations, is a finite
 field with characteristic p.  Each nonzero element x in Zp has an
 inverse 1/x.  There is a one-to-one correspondence between the
 integers between zero and p-1, inclusive, and the elements of the
 field.  The field Zp is sometimes denoted as Fp or GF(p).

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 Equations involving field elements do not explicitly denote the "mod
 p" operation, but it is understood to be implicit.  For example, the
 statement that x, y, and z are in Fp and
    z = x + y
 is equivalent to the statement that x, y, and z are in the set
 { 0, 1, ..., p-1 } and
    z = x + y mod p.

3. Elliptic Curve Groups

 This note only covers elliptic curves over fields with characteristic
 greater than three; these are the curves used in Suite B [SuiteB].
 For other fields, the definition of the elliptic curve group would be
 different.
 An elliptic curve over a field Fp is defined by the curve equation
    y^2 = x^3 + a*x + b,
 where x, y, a, and b are elements of the field Fp [M1985], and the
 discriminant is nonzero (as described in Section 3.3.1).  A point on
 an elliptic curve is a pair (x,y) of values in Fp that satisfies the
 curve equation, or it is a special point (@,@) that represents the
 identity element (which is called the "point at infinity").  The
 order of an elliptic curve group is the number of distinct points.
 Two elliptic curve points (x1,y1) and (x2,y2) are equal whenever
 x1=x2 and y1=y2, or when both points are the point at infinity.  The
 inverse of the point (x1,y1) is the point (x1,-y1).  The point at
 infinity is its own inverse.
 The group operation associated with the elliptic curve group is as
 follows [BC1989].  To an arbitrary pair of points P and Q specified
 by their coordinates (x1,y1) and (x2,y2), respectively, the group
 operation assigns a third point P*Q with the coordinates (x3,y3).
 These coordinates are computed as follows:
    (x3,y3) = (@,@) when P is not equal to Q and x1 is equal to x2.
    x3 = ((y2-y1)/(x2-x1))^2 - x1 - x2 and
    y3 = (x1-x3)*(y2-y1)/(x2-x1) - y1 when P is not equal to Q and
    x1 is not equal to x2.
    (x3,y3) = (@,@) when P is equal to Q and y1 is equal to 0.

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    x3 = ((3*x1^2 + a)/(2*y1))^2 - 2*x1 and
    y3 = (x1-x3)*(3*x1^2 + a)/(2*y1) - y1 if P is equal to Q and y1 is
    not equal to 0.
 In the above equations, a, x1, x2, x3, y1, y2, and y3 are elements of
 the field Fp; thus, computation of x3 and y3 in practice must reduce
 the right-hand-side modulo p.  Pseudocode for the group operation is
 provided in Appendix F.1.
 The representation of elliptic curve points as a pair of integers in
 Zp is known as the affine coordinate representation.  This
 representation is suitable as an external data representation for
 communicating or storing group elements, though the point at infinity
 must be treated as a special case.
 Some pairs of integers are not valid elliptic curve points.  A valid
 pair will satisfy the curve equation, while an invalid pair will not.

3.1. Homogeneous Coordinates

 An alternative way to implement the group operation is to use
 homogeneous coordinates [K1987] (see also [KMOV1991]).  This method
 is typically more efficient because it does not require a modular
 inversion operation.
 An elliptic curve point (x,y) (other than the point at infinity
 (@,@)) is equivalent to a point (X,Y,Z) in homogeneous coordinates
 whenever x=X/Z mod p and y=Y/Z mod p.
 Let P1=(X1,Y1,Z1) and P2=(X2,Y2,Z2) be points on an elliptic curve,
 and suppose that the points P1 and P2 are not equal to (@,@), P1 is
 not equal to P2, and P1 is not equal to P2^-1.  Then the product
 P3=(X3,Y3,Z3) = P1 * P2 is given by
    X3 = v * (Z2 * (Z1 * u^2 - 2 * X1 * v^2) - v^3) mod p
    Y3 = Z2 * (3 * X1 * u * v^2 - Y1 * v^3 - Z1 * u^3) + u * v^3 mod p
    Z3 = v^3 * Z1 * Z2 mod p
 where u = Y2 * Z1 - Y1 * Z2 mod p and v = X2 * Z1 - X1 * Z2 mod p.
 When the points P1 and P2 are equal, then (X1/Z1, Y1/Z1) is equal to
 (X2/Z2, Y2/Z2), which is true if and only if u and v are both equal
 to zero.

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 The product P3=(X3,Y3,Z3) = P1 * P1 is given by
    X3 = 2 * Y1 * Z1 * (w^2 - 8 * X1 * Y1^2 * Z1) mod p
    Y3 = 4 * Y1^2 * Z1 * (3 * w * X1 - 2 * Y1^2 * Z1) - w^3 mod p
    Z3 = 8 * (Y1 * Z1)^3 mod p
 where w = 3 * X1^2 + a * Z1^2 mod p.  In the above equations, a, u,
 v, w, X1, X2, X3, Y1, Y2, Y3, Z1, Z2, and Z3 are integers in the set
 Fp.  Pseudocode for the group operation in homogeneous coordinates is
 provided in Appendix F.2.
 When converting from affine coordinates to homogeneous coordinates,
 it is convenient to set Z to 1.  When converting from homogeneous
 coordinates to affine coordinates, it is necessary to perform a
 modular inverse to find 1/Z mod p.

3.2. Other Coordinates

 Some other coordinate systems have been described; several are
 documented in [CC1986], including Jacobi coordinates.

3.3. ECC Parameters

 In cryptographic contexts, an elliptic curve parameter set consists
 of a cyclic subgroup of an elliptic curve together with a preferred
 generator of that subgroup.  When working over a prime order finite
 field with characteristic greater than three, an elliptic curve group
 is completely specified by the following parameters:
    The prime number p that indicates the order of the field Fp.
    The value a used in the curve equation.
    The value b used in the curve equation.
    The generator g of the subgroup.
    The order n of the subgroup generated by g.
 An example of an ECC parameter set is provided in Appendix D.
 Parameter generation is out of scope for this note.
 Each elliptic curve point is associated with a particular parameter
 set.  The elliptic curve group operation is only defined between two
 points in the same group.  It is an error to apply the group

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 operation to two elements that are from different groups, or to apply
 the group operation to a pair of coordinates that is not a valid
 point.  (A pair (x,y) of coordinates in Fp is a valid point only when
 it satisfies the curve equation.)  See Section 10.3 for further
 information.

3.3.1. Discriminant

 For each elliptic curve group, the discriminant -16*(4*a^3 + 27*b^2)
 must be nonzero modulo p [S1986]; this requires that
    4*a^3 + 27*b^2 != 0 mod p.

3.3.2. Security

 Security is highly dependent on the choice of these parameters.  This
 section gives normative guidance on acceptable choices.  See also
 Section 10 for informative guidance.
 The order of the group generated by g MUST be divisible by a large
 prime, in order to preclude easy solutions of the discrete logarithm
 problem [K1987].
 With some parameter choices, the discrete log problem is
 significantly easier to solve.  This includes parameter sets in which
 b = 0 and p = 3 (mod 4), and parameter sets in which a = 0 and
 p = 2 (mod 3) [MOV1993].  These parameter choices are inferior for
 cryptographic purposes and SHOULD NOT be used.

4. Elliptic Curve Diffie-Hellman (ECDH)

 The Diffie-Hellman (DH) key exchange protocol [DH1976] allows two
 parties communicating over an insecure channel to agree on a secret
 key.  It was originally defined in terms of operations in the
 multiplicative group of a field with a large prime characteristic.
 Massey [M1983] observed that it can be easily generalized so that it
 is defined in terms of an arbitrary cyclic group.  Miller [M1985] and
 Koblitz [K1987] analyzed the DH protocol over an elliptic curve
 group.  We describe DH following the former reference.
 Let G be a group, and g be a generator for that group, and let t
 denote the order of G.  The DH protocol runs as follows.  Party A
 chooses an exponent j between 1 and t-1, inclusive, uniformly at
 random, computes g^j, and sends that element to B.  Party B chooses
 an exponent k between 1 and t-1, inclusive, uniformly at random,
 computes g^k, and sends that element to A.  Each party can compute
 g^(j*k); party A computes (g^k)^j, and party B computes (g^j)^k.

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 See Appendix B regarding generation of random integers.

4.1. Data Types

 Each run of the ECDH protocol is associated with a particular
 parameter set (as defined in Section 3.3), and the public keys g^j
 and g^k and the shared secret g^(j*k) are elements of the cyclic
 subgroup associated with the parameter set.
 An ECDH private key z is an integer in Zt, where t is the order of
 the subgroup.

4.2. Compact Representation

 As described in the final paragraph of [M1985], the x-coordinate of
 the shared secret value g^(j*k) is a suitable representative for the
 entire point whenever exponentiation is used as a one-way function.
 In the ECDH key exchange protocol, after the element g^(j*k) has been
 computed, the x-coordinate of that value can be used as the shared
 secret.  We call this compact output.
 Following [M1985] again, when compact output is used in ECDH, only
 the x-coordinate of an elliptic curve point needs to be transmitted,
 instead of both coordinates as in the typical affine coordinate
 representation.  We call this the compact representation.  Its
 mathematical background is explained in Appendix C.
 ECDH can be used with or without compact output.  Both parties in a
 particular run of the ECDH protocol MUST use the same method.  ECDH
 can be used with or without compact representation.  If compact
 representation is used in a particular run of the ECDH protocol, then
 compact output MUST be used as well.

5. Elliptic Curve ElGamal Signatures

5.1. Background

 The ElGamal signature algorithm was introduced in 1984 [E1984a]
 [E1984b] [E1985].  It is based on the discrete logarithm problem, and
 was originally defined for the multiplicative group of the integers
 modulo a large prime number.  It is straightforward to extend it to
 use other finite groups, such as the multiplicative group of the
 finite field GF(2^w) [AMV1990] or an elliptic curve group [A1992].
 An ElGamal signature consists of a pair of components.  There are
 many possible generalizations of ElGamal signature methods that have
 been obtained by different rearrangements of the equation for the
 second component; see [HMP1994], [HP1994], [NR1994], [A1992], and

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 [AMV1990].  These generalizations are independent of the mathematical
 group used, and have been described for the multiplicative group
 modulo a prime number, the multiplicative group of GF(2^w), and
 elliptic curve groups [HMP1994] [NR1994] [AMV1990] [A1992].
 The Digital Signature Algorithm (DSA) [FIPS186] is an important
 ElGamal signature variant.

5.2. Hash Functions

 ElGamal signatures must use a collision-resistant hash function, so
 that it can sign messages of arbitrary length and can avoid
 existential forgery attacks; see Section 10.4.  (This is true for all
 ElGamal variants [HMP1994].)  We denote the hash function as h().
 Its input is a bit string of arbitrary length, and its output is a
 non-negative integer.
 Let H() denote a hash function whose output is a fixed-length bit
 string.  To use H in an ElGamal signature method, we define the
 mapping between that output and the non-negative integers; this
 realizes the function h() described above.  Given a bit string m, the
 function h(m) is computed as follows:
 1.  H(m) is evaluated; the result is a fixed-length bit string.
 2.  Convert the resulting bit string to an integer i by treating its
     leftmost (initial) bit as the most significant bit of i, and
     treating its rightmost (final) bit as the least significant bit
     of i.

5.3. KT-IV Signatures

 Koyama and Tsuruoka described a signature method based on Elliptic
 Curve ElGamal, in which the first signature component is the
 x-coordinate of an elliptic curve point reduced modulo q [KT1994].
 In this section, we recall that method, which we refer to as KT-IV.
 The algorithm uses an elliptic curve group, as described in
 Section 3.3, with prime field order p and curve equation parameters a
 and b.  We denote the generator as alpha, and the order of the
 generator as q.  We follow [FIPS186] in checking for exceptional
 cases.

5.3.1. Keypair Generation

 The private key z is an integer between 1 and q-1, inclusive,
 generated uniformly at random.  (See Appendix B regarding random
 integers.)  The public key is the group element Y = alpha^z.  Each

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 public key is associated with a particular parameter set as per
 Section 3.3.

5.3.2. Signature Creation

 To compute a KT-IV signature for a message m using the private key z:
 1.  Choose an integer k uniformly at random from the set of all
     integers between 1 and q-1, inclusive.  (See Appendix B regarding
     random integers.)
 2.  Calculate R = (r_x, r_y) = alpha^k.
 3.  Calculate s1 = r_x mod q.
 4.  Check if h(m) + z * s1 = 0 mod q; if so, a new value of k MUST be
     generated and the signature MUST be recalculated.  As an option,
     one MAY check if s1 = 0; if so, a new value of k SHOULD be
     generated and the signature SHOULD be recalculated.  (It is
     extremely unlikely that s1 = 0 or h(m) + z * s1 = 0 mod q if
     signatures are generated properly.)
 5.  Calculate s2 = k/(h(m) + z*s1) mod q.
 The signature is the ordered pair (s1, s2).  Both signature
 components are non-negative integers.

5.3.3. Signature Verification

 Given the message m, the generator g, the group order q, the public
 key Y, and the signature (s1, s2), verification is as follows:
 1.  Check to see that 0 < s1 < q and 0 < s2 < q; if either condition
     is violated, the signature SHALL be rejected.
 2.  Compute the non-negative integers u1 and u2, where
        u1 = h(m) * s2 mod q, and
        u2 = s1 * s2 mod q.
 3.  Compute the elliptic curve point R' = alpha^u1 * Y^u2.
 4.  If the x-coordinate of R' mod q is equal to s1, then the
     signature and message pass the verification; otherwise, they
     fail.

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5.4. KT-I Signatures

 Horster, Michels, and Petersen categorized many different ElGamal
 signature methods, demonstrated their equivalence, and showed how to
 convert signatures of one type to another type [HMP1994].  In their
 terminology, the signature method of Section 5.3 and [KT1994] is a
 Type IV method, which is why it is denoted as KT-IV.
 A Type I KT signature method has a second component that is computed
 in the same manner as that of the Digital Signature Algorithm.  In
 this section, we describe this method, which we refer to as KT-I.

5.4.1. Keypair Generation

 Keypairs and keypair generation are exactly as in Section 5.3.1.

5.4.2. Signature Creation

 To compute a KT-I signature for a message m using the private key z:
 1.  Choose an integer k uniformly at random from the set of all
     integers between 1 and q-1, inclusive.  (See Appendix B regarding
     random integers.)
 2.  Calculate R = (r_x, r_y) = alpha^k.
 3.  Calculate s1 = r_x mod q.
 4.  Calculate s2 = (h(m) + z*s1)/k mod q.
 5.  As an option, one MAY check if s1 = 0 or s2 = 0.  If either
     s1 = 0 or s2 = 0, a new value of k SHOULD be generated and the
     signature SHOULD be recalculated.  (It is extremely unlikely that
     s1 = 0 or s2 = 0 if signatures are generated properly.)
 The signature is the ordered pair (s1, s2).  Both signature
 components are non-negative integers.

5.4.3. Signature Verification

 Given the message m, the public key Y, and the signature (s1, s2),
 verification is as follows:
 1.  Check to see that 0 < s1 < q and 0 < s2 < q; if either condition
     is violated, the signature SHALL be rejected.
 2.  Compute s2_inv = 1/s2 mod q.

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 3.  Compute the non-negative integers u1 and u2, where
        u1 = h(m) * s2_inv mod q, and
        u2 = s1 * s2_inv mod q.
 4.  Compute the elliptic curve point R' = alpha^u1 * Y^u2.
 5.  If the x-coordinate of R' mod q is equal to s1, then the
     signature and message pass the verification; otherwise, they
     fail.

5.5. Converting KT-IV Signatures to KT-I Signatures

 A KT-IV signature for a message m and a public key Y can easily be
 converted into a KT-I signature for the same message and public key.
 If (s1, s2) is a KT-IV signature for a message m, then
 (s1, 1/s2 mod q) is a KT-I signature for the same message [HMP1994].
 The conversion operation uses only public information, and it can be
 performed by the creator of the pre-conversion KT-IV signature, the
 verifier of the post-conversion KT-I signature, or by any other
 entity.
 An implementation MAY use this method to compute KT-I signatures.

5.6. Rationale

 This subsection is not normative for this specification and is
 provided only as background information.
 [HMP1994] presents many generalizations of ElGamal signatures.
 Equation (5) of that reference shows the general signature equation
    A = x_A * B + k * C (mod q)
 where x_A is the private key, k is the secret value, and A, B, and C
 are determined by the Type of the equation, as shown in Table 1 of
 [HMP1994].  DSA [FIPS186] is an EG-I.1 signature method (as is KT-I),
 with A = m, B = -r, and C = s.  (Here we use the notation of
 [HMP1994] in which the first signature component is r and the second
 signature component is s; in KT-I and KT-IV these components are
 denoted as s1 and s2, respectively.  The private key x_A corresponds
 to the private key z.)  Its signature equation is
    m = -r * z + s * k (mod q).

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 The signature method of [KT1994] and Section 5.3 is an EG-IV.1
 method, with A = m * s, B = -r * s, C = 1.  Its signature equation is
    m * s = -r * s * z + k (mod q)
 The functions f and g mentioned in Table 1 of [HMP1994] are merely
 multiplication, as described under the heading "Fifth
 generalization".
 In the above equations, we rely on the implicit conversion of the
 message m from a bit string to an integer.  No hash function is shown
 in these equations, but, as described in Section 10.4, a hash
 function should be applied to the message prior to signing in order
 to prevent existential forgery attacks.
 Nyberg and Rueppel [NR1994] studied many different ElGamal signature
 methods and defined "strong equivalence" as follows:
    Two signature methods are called strongly equivalent if the
    signature of the first scheme can be transformed efficiently into
    signatures of the second scheme and vice versa, without knowledge
    of the private key.
 KT-I and KT-IV signatures are obviously strongly equivalent.
 A valid signature with s2=0 leaks the secret key, since in that case
 z = -h(m) / s1 mod q.  We follow [FIPS186] in checking for this
 exceptional case and the case that s1=0.  The s2=0 check was
 suggested by Rivest [R1992] and is discussed in [BS1992].
 [KT1994] uses "a positive integer q' that does not exceed q" when
 computing the signature component s1 from the x-coordinate r_x of the
 elliptic curve point R = (r_x, r_y).  The value q' is also used
 during signature validation when comparing the x-coordinate of a
 computed elliptic curve point to the value to s1.  In this note, we
 use the simplifying convention that q' = q.

6. Converting between Integers and Octet Strings

 A method for the conversion between integers and octet strings is
 specified in this section, following the established conventions of
 public key cryptography [R1993].  This method allows integers to be
 represented as octet strings that are suitable for transmission or
 storage.  This method SHOULD be used when representing an elliptic
 curve point or an elliptic curve coordinate as they are defined in
 this note.

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6.1. Octet-String-to-Integer Conversion

 The octet string S shall be converted to an integer x as follows.
 Let S1, ..., Sk be the octets of S from first to last.  Then the
 integer x shall satisfy
                        k
                  x =  SUM  2^(8(k-i)) Si .
                      i = 1
 In other words, the first octet of S has the most significance in the
 integer and the last octet of S has the least significance.
 Note: the integer x satisfies 0 <= x < 2^(8*k).

6.2. Integer-to-Octet-String Conversion

 The integer x shall be converted to an octet string S of length k as
 follows.  The string S shall satisfy
                        k
                  y =  SUM  2^(8(k-i)) Si .
                      i = 1
 where S1, ..., Sk are the octets of S from first to last.
 In other words, the first octet of S has the most significance in the
 integer, and the last octet of S has the least significance.

7. Interoperability

 The algorithms in this note can be used to interoperate with some
 other ECC specifications.  This section provides details for each
 algorithm.

7.1. ECDH

 Section 4 can be used with the Internet Key Exchange (IKE) versions
 one [RFC2409] or two [RFC5996].  These algorithms are compatible with
 the ECP groups defined by [RFC5903], [RFC5114], [RFC2409], and
 [RFC2412].  The group definition in this protocol uses an affine
 coordinate representation of the public key.  [RFC5903] uses the
 compact output of Section 4.2, while [RFC4753] (which was obsoleted
 by RFC 5903) does not.  Neither of those RFCs use compact
 representation.  Note that some groups indicate that the curve
 parameter "a" is negative; these values are to be interpreted modulo
 the order of the field.  For example, a parameter of a = -3 is equal
 to p - 3, where p is the order of the field.  The test cases in

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 Section 8 of [RFC5903] can be used to test an implementation; these
 cases use the multiplicative notation, as does this note.  The KEi
 and KEr payloads are equal to g^j and g^k, respectively, with 64 bits
 of encoding data prepended to them.
 The algorithms in Section 4 can be used to interoperate with the IEEE
 [P1363] and ANSI [X9.62] standards for ECDH based on fields of
 characteristic greater than three.  IEEE P1363 ECDH can be used in a
 manner that will interoperate with this note, with the following
 options and parameter choices from that specification:
    prime curves with a cofactor of 1,
    the ECSVDP-DH (Elliptic Curve Secret Value Derivation Primitive,
    Diffie-Hellman version),
    the Key Derivation Function (KDF) must be the "identity" function
    (equivalently, the KDF step should be omitted and the shared
    secret value should be output directly).

7.2. KT-I and ECDSA

 The Digital Signature Algorithm (DSA) is based on the discrete
 logarithm problem over the multiplicative subgroup of the finite
 field with large prime order [DSA1991] [FIPS186].  The Elliptic Curve
 Digital Signature Algorithm (ECDSA) [P1363] [X9.62] is an elliptic
 curve version of DSA.
 KT-I is mathematically and functionally equivalent to ECDSA, and can
 interoperate with the IEEE [P1363] and ANSI [X9.62] standards for
 Elliptic Curve DSA (ECDSA) based on fields of characteristic greater
 than three.  KT-I signatures can be verified using the ECDSA
 verification algorithm, and ECDSA signatures can be verified using
 the KT-I verification algorithm.

8. Validating an Implementation

 It is essential to validate the implementation of a cryptographic
 algorithm.  This section outlines tests that should be performed on
 the algorithms defined in this note.
 A known answer test, or KAT, uses a fixed set of inputs to test an
 algorithm; the output of the algorithm is compared with the expected
 output, which is also a fixed value.  KATs for ECDH and KT-I are set
 out in the following subsections.

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 A consistency test generates inputs for one algorithm being tested
 using a second algorithm that is also being tested, then checks the
 output of the first algorithm.  A signature creation algorithm can be
 tested for consistency against a signature verification algorithm.
 Implementations of KT-I should be tested in this way.  Their
 signature generation processes are non-deterministic, and thus cannot
 be tested using a KAT.  Signature verification algorithms, on the
 other hand, are deterministic and should be tested via a KAT.  This
 combination of tests provides coverage for all of the operations,
 including keypair generation.  Consistency testing should also be
 applied to ECDH.

8.1. ECDH

 An ECDH implementation can be validated using the known answer test
 cases from [RFC5903] or [RFC5114].  The correspondence between the
 notation in RFC 5903 and the notation in this note is summarized in
 the following table.  (Refer to Sections 3.3 and 4; the generator g
 is expressed in affine coordinate representation as (gx, gy)).
   +----------------------+---------------------------------------+
   | ECDH                 | RFC 5903                              |
   +----------------------+---------------------------------------+
   | order p of field Fp  | p                                     |
   | curve coefficient a  | -3                                    |
   | curve coefficient b  | b                                     |
   | generator g          | g=(gx, gy)                            |
   | private keys j and k | i and r                               |
   | public keys g^j, g^k | g^i = (gix, giy) and g^r = (grx, gry) |
   +----------------------+---------------------------------------+
 The correspondence between the notation in RFC 5114 and the notation
 in this note is summarized in the following table.
         +-----------------------+---------------------------+
         | ECDH                  | RFC 5114                  |
         +-----------------------+---------------------------+
         | order p of field Fp   | p                         |
         | curve coefficient a   | a                         |
         | curve coefficient b   | b                         |
         | generator g           | g=(gx, gy)                |
         | group order n         | n                         |
         | private keys j and k  | dA and dB                 |
         | public keys g^j, g^k  | g^(dA) = (x_qA, y_qA) and |
         |                       | g^(dB) = (x_qB, y_qB)     |
         | shared secret g^(j*k) | g^(dA*dB) = (x_Z, y_Z)    |
         +-----------------------+---------------------------+

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8.2. KT-I

 A KT-I implementation can be validated using the known answer test
 cases from [RFC4754].  The correspondence between the notation in
 that RFC and the notation in this note is summarized in the following
 table.
              +---------------------+------------------+
              | KT-I                | RFC 4754         |
              +---------------------+------------------+
              | order p of field Fp | p                |
              | curve coefficient a | -3               |
              | curve coefficient b | b                |
              | generator alpha     | g                |
              | group order q       | q                |
              | private key z       | w                |
              | public key Y        | g^w = (gwx,gwy)  |
              | random k            | ephem priv k     |
              | s1                  | r                |
              | s2                  | s                |
              | s2_inv              | sinv             |
              | u1                  | u = h*sinv mod q |
              | u2                  | v = r*sinv mod q |
              +---------------------+------------------+

9. Intellectual Property

 Concerns about intellectual property have slowed the adoption of ECC
 because a number of optimizations and specialized algorithms have
 been patented in recent years.
 All of the normative references for ECDH (as defined in Section 4)
 were published during or before 1989, and those for KT-I were
 published during or before May 1994.  All of the normative text for
 these algorithms is based solely on their respective references.

9.1. Disclaimer

 This document is not intended as legal advice.  Readers are advised
 to consult their own legal advisers if they would like a legal
 interpretation of their rights.
 The IETF policies and processes regarding intellectual property and
 patents are outlined in [RFC3979] and [RFC4879] and at
 https://datatracker.ietf.org/ipr/about/.

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10. Security Considerations

 The security level of an elliptic curve cryptosystem is determined by
 the cryptanalytic algorithm that is the least expensive for an
 attacker to implement.  There are several algorithms to consider.
 The Pohlig-Hellman method is a divide-and-conquer technique [PH1978].
 If the group order n can be factored as
    n = q1 * q2 * ... * qz,
 then the discrete log problem over the group can be solved by
 independently solving a discrete log problem in groups of order q1,
 q2, ..., qz, then combining the results using the Chinese remainder
 theorem.  The overall computational cost is dominated by that of the
 discrete log problem in the subgroup with the largest order.
 Shanks' algorithm [K1981v3] computes a discrete logarithm in a group
 of order n using O(sqrt(n)) operations and O(sqrt(n)) storage.  The
 Pollard rho algorithm [P1978] computes a discrete logarithm in a
 group of order n using O(sqrt(n)) operations, with a negligible
 amount of storage, and can be efficiently parallelized [VW1994].
 The Pollard lambda algorithm [P1978] can solve the discrete logarithm
 problem using O(sqrt(w)) operations and O(log(w)) storage, when the
 exponent is known to lie in an interval of width w.
 The algorithms described above work in any group.  There are
 specialized algorithms that specifically target elliptic curve
 groups.  There are no known subexponential algorithms against general
 elliptic curve groups, though there are methods that target certain
 special elliptic curve groups; see [MOV1993] and [FR1994].

10.1. Subgroups

 A group consisting of a nonempty set of elements S with associated
 group operation * is a subgroup of the group with the set of elements
 G, if the latter group uses the same group operation and S is a
 subset of G.  For each elliptic curve equation, there is an elliptic
 curve group whose group order is equal to the order of the elliptic
 curve; that is, there is a group that contains every point on the
 curve.
 The order m of the elliptic curve is divisible by the order n of the
 group associated with the generator; that is, for each elliptic curve
 group, m = n * c for some number c.  The number c is called the
 "cofactor" [P1363].  Each ECC parameter set as in Section 3.3 is
 associated with a particular cofactor.

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 It is possible and desirable to use a cofactor equal to 1.

10.2. Diffie-Hellman

 Note that the key exchange protocol as defined in Section 4 does not
 protect against active attacks; Party A must use some method to
 ensure that (g^k) originated with the intended communicant B, rather
 than an attacker, and Party B must do the same with (g^j).
 It is not sufficient to authenticate the shared secret g^(j*k), since
 this leaves the protocol open to attacks that manipulate the public
 keys.  Instead, the values of the public keys g^x and g^y that are
 exchanged should be directly authenticated.  This is the strategy
 used by protocols that build on Diffie-Hellman and that use end-
 entity authentication to protect against active attacks, such as
 OAKLEY [RFC2412] and the Internet Key Exchange [RFC2409] [RFC4306]
 [RFC5996].
 When the cofactor of a group is not equal to 1, there are a number of
 attacks that are possible against ECDH.  See [VW1996], [AV1996], and
 [LL1997].

10.3. Group Representation and Security

 The elliptic curve group operation does not explicitly incorporate
 the parameter b from the curve equation.  This opens the possibility
 that a malicious attacker could learn information about an ECDH
 private key by submitting a bogus public key [BMM2000].  An attacker
 can craft an elliptic curve group G' that has identical parameters to
 a group G that is being used in an ECDH protocol, except that b is
 different.  An attacker can submit a point on G' into a run of the
 ECDH protocol that is using group G, and gain information from the
 fact that the group operations using the private key of the device
 under attack are effectively taking place in G' instead of G.
 This attack can gain useful information about an ECDH private key
 that is associated with a static public key, i.e., a public key that
 is used in more than one run of the protocol.  However, it does not
 gain any useful information against ephemeral keys.
 This sort of attack is thwarted if an ECDH implementation does not
 assume that each pair of coordinates in Zp is actually a point on the
 appropriate elliptic curve.
 These considerations also apply when ECDH is used with compact
 representation (see Appendix C).

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10.4. Signatures

 Elliptic curve parameters should only be used if they come from a
 trusted source; otherwise, some attacks are possible [AV1996]
 [V1996].
 If no hash function is used in an ElGamal signature system, then the
 system is vulnerable to existential forgeries, in which an attacker
 who does not know a private key can generate valid signatures for the
 associated public key, but cannot generate a signature for a message
 of its own choosing.  (See [E1985] for instance.)  The use of a
 collision-resistant hash function eliminates this vulnerability.
 In principle, any collision-resistant hash function is suitable for
 use in KT signatures.  To facilitate interoperability, we recognize
 the following hashes as suitable for use as the function H defined in
 Section 5.2:
    SHA-256, which has a 256-bit output.
    SHA-384, which has a 384-bit output.
    SHA-512, which has a 512-bit output.
 All of these hash functions are defined in [FIPS180-2].
 The number of bits in the output of the hash used in KT signatures
 should be equal or close to the number of bits needed to represent
 the group order.

11. Acknowledgements

 The author expresses his thanks to the originators of elliptic curve
 cryptography, whose work made this note possible, and all of the
 reviewers, who provided valuable constructive feedback.  Thanks are
 especially due to Howard Pinder, Andrey Jivsov, Alfred Hoenes (who
 contributed the algorithms in Appendix F), Dan Harkins, and Tina
 Tsou.

12. References

12.1. Normative References

 [AMV1990]    Agnew, G., Mullin, R., and S. Vanstone, "Improved
              Digital Signature Scheme based on Discrete
              Exponentiation", Electronics Letters Vol. 26, No. 14,
              July, 1990.

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 [BC1989]     Bender, A. and G. Castagnoli, "On the Implementation of
              Elliptic Curve Cryptosystems", Advances in Cryptology -
              CRYPTO '89 Proceedings, Springer Lecture Notes in
              Computer Science (LNCS), volume 435, 1989.
 [CC1986]     Chudnovsky, D. and G. Chudnovsky, "Sequences of numbers
              generated by addition in formal groups and new primality
              and factorization tests", Advances in Applied
              Mathematics, Volume 7, Issue 4, December 1986.
 [D1966]      Deskins, W., "Abstract Algebra", MacMillan Company New
              York, 1966.
 [DH1976]     Diffie, W. and M. Hellman, "New Directions in
              Cryptography", IEEE Transactions in Information
              Theory IT-22, pp. 644-654, 1976.
 [FR1994]     Frey, G. and H. Ruck, "A remark concerning
              m-divisibility and the discrete logarithm in the divisor
              class group of curves.", Mathematics of Computation Vol.
              62, No. 206, pp. 865-874, 1994.
 [HMP1994]    Horster, P., Michels, M., and H. Petersen, "Meta-ElGamal
              signature schemes", University of Technology Chemnitz-
              Zwickau Department of Computer Science, Technical
              Report TR-94-5, May 1994.
 [K1981v2]    Knuth, D., "The Art of Computer Programming, Vol. 2:
              Seminumerical Algorithms", Addison Wesley , 1981.
 [K1987]      Koblitz, N., "Elliptic Curve Cryptosystems", Mathematics
              of Computation, Vol. 48, 1987, pp. 203-209, 1987.
 [KT1994]     Koyama, K. and Y. Tsuruoka, "Digital signature system
              based on elliptic curve and signer device and verifier
              device for said system", Japanese Unexamined Patent
              Application Publication H6-43809, February 18, 1994.
 [M1983]      Massey, J., "Logarithms in finite cyclic groups -
              cryptographic issues", Proceedings of the 4th Symposium
              on Information Theory, 1983.
 [M1985]      Miller, V., "Use of elliptic curves in cryptography",
              Advances in Cryptology - CRYPTO '85
              Proceedings, Springer Lecture Notes in Computer Science
              (LNCS), volume 218, 1985.

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 [MOV1993]    Menezes, A., Vanstone, S., and T. Okamoto, "Reducing
              Elliptic Curve Logarithms to Logarithms in a Finite
              Field", IEEE Transactions on Information Theory Vol. 39,
              No. 5, pp. 1639-1646, September, 1993.
 [R1993]      RSA Laboratories, "PKCS#1: RSA Encryption Standard",
              Technical Note version 1.5, 1993.
 [S1986]      Silverman, J., "The Arithmetic of Elliptic Curves",
              Springer-Verlag, New York, 1986.

12.2. Informative References

 [A1992]      Anderson, J., "Response to the proposed DSS",
              Communications of the ACM, v. 35, n. 7, p. 50-52,
              July 1992.
 [AV1996]     Anderson, R. and S. Vaudenay, "Minding Your P's and
              Q's", Advances in Cryptology - ASIACRYPT '96
              Proceedings, Springer Lecture Notes in Computer Science
              (LNCS), volume 1163, 1996.
 [BMM2000]    Biehl, I., Meyer, B., and V. Muller, "Differential fault
              analysis on elliptic curve cryptosystems", Advances in
              Cryptology - CRYPTO 2000 Proceedings, Springer Lecture
              Notes in Computer Science (LNCS), volume 1880, 2000.
 [BS1992]     Branstad, D. and M. Smid, "Response to Comments on the
              NIST Proposed Digital Signature Standard", Advances in
              Cryptology - CRYPTO '92 Proceedings, Springer Lecture
              Notes in Computer Science (LNCS), volume 740,
              August 1992.
 [DSA1991]    U.S. National Institute of Standards and Technology,
              "DIGITAL SIGNATURE STANDARD", Federal Register, Vol. 56,
              August 1991.
 [E1984a]     ElGamal, T., "Cryptography and logarithms over finite
              fields", Stanford University, UMI Order No. DA 8420519,
              1984.
 [E1984b]     ElGamal, T., "Cryptography and logarithms over finite
              fields", Advances in Cryptology - CRYPTO '84
              Proceedings, Springer Lecture Notes in Computer Science
              (LNCS), volume 196, 1984.

McGrew, et al. Informational [Page 25] RFC 6090 Fundamental ECC February 2011

 [E1985]      ElGamal, T., "A public key cryptosystem and a signature
              scheme based on discrete logarithms", IEEE Transactions
              on Information Theory, Vol. 30, No. 4, pp. 469-472,
              1985.
 [FIPS180-2]  U.S. National Institute of Standards and Technology,
              "SECURE HASH STANDARD", Federal Information Processing
              Standard (FIPS) 180-2, August 2002.
 [FIPS186]    U.S. National Institute of Standards and Technology,
              "DIGITAL SIGNATURE STANDARD", Federal Information
              Processing Standard FIPS-186, May 1994.
 [HP1994]     Horster, P. and H. Petersen, "Verallgemeinerte ElGamal-
              Signaturen", Proceedings der Fachtagung SIS '94, Verlag
              der Fachvereine, Zurich, 1994.
 [K1981v3]    Knuth, D., "The Art of Computer Programming, Vol. 3:
              Sorting and Searching", Addison Wesley, 1981.
 [KMOV1991]   Koyama, K., Maurer, U., Vanstone, S., and T. Okamoto,
              "New Public-Key Schemes Based on Elliptic Curves over
              the Ring Zn", Advances in Cryptology - CRYPTO '91
              Proceedings, Springer Lecture Notes in Computer Science
              (LNCS), volume 576, 1991.
 [L1969]      Lehmer, D., "Computer technology applied to the theory
              of numbers", M.A.A. Studies in Mathematics, 180-2, 1969.
 [LL1997]     Lim, C. and P. Lee, "A Key Recovery Attack on Discrete
              Log-based Schemes Using a Prime Order Subgroup",
              Advances in Cryptology - CRYPTO '97
              Proceedings, Springer Lecture Notes in Computer Science
              (LNCS), volume 1294, 1997.
 [NR1994]     Nyberg, K. and R. Rueppel, "Message Recovery for
              Signature Schemes Based on the Discrete Logarithm
              Problem", Advances in Cryptology - EUROCRYPT '94
              Proceedings, Springer Lecture Notes in Computer Science
              (LNCS), volume 950, May 1994.
 [P1363]      "Standard Specifications for Public Key Cryptography",
              Institute of Electric and Electronic Engineers
              (IEEE), P1363, 2000.
 [P1978]      Pollard, J., "Monte Carlo methods for index computation
              mod p", Mathematics of Computation, Vol. 32, 1978.

McGrew, et al. Informational [Page 26] RFC 6090 Fundamental ECC February 2011

 [PH1978]     Pohlig, S. and M. Hellman, "An Improved Algorithm for
              Computing Logarithms over GF(p) and its Cryptographic
              Significance", IEEE Transactions on Information
              Theory, Vol. 24, pp. 106-110, 1978.
 [R1988]      Rose, H., "A Course in Number Theory", Oxford
              University Press, 1988.
 [R1992]      Rivest, R., "Response to the proposed DSS",
              Communications of the ACM, v. 35, n. 7, p. 41-47,
              July 1992.
 [RFC2119]    Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119, March 1997.
 [RFC2409]    Harkins, D. and D. Carrel, "The Internet Key Exchange
              (IKE)", RFC 2409, November 1998.
 [RFC2412]    Orman, H., "The OAKLEY Key Determination Protocol",
              RFC 2412, November 1998.
 [RFC3979]    Bradner, S., "Intellectual Property Rights in IETF
              Technology", BCP 79, RFC 3979, March 2005.
 [RFC4086]    Eastlake, D., Schiller, J., and S. Crocker, "Randomness
              Requirements for Security", BCP 106, RFC 4086,
              June 2005.
 [RFC4306]    Kaufman, C., "Internet Key Exchange (IKEv2) Protocol",
              RFC 4306, December 2005.
 [RFC4753]    Fu, D. and J. Solinas, "ECP Groups For IKE and IKEv2",
              RFC 4753, January 2007.
 [RFC4754]    Fu, D. and J. Solinas, "IKE and IKEv2 Authentication
              Using the Elliptic Curve Digital Signature Algorithm
              (ECDSA)", RFC 4754, January 2007.
 [RFC4879]    Narten, T., "Clarification of the Third Party Disclosure
              Procedure in RFC 3979", BCP 79, RFC 4879, April 2007.
 [RFC5114]    Lepinski, M. and S. Kent, "Additional Diffie-Hellman
              Groups for Use with IETF Standards", RFC 5114,
              January 2008.
 [RFC5903]    Fu, D. and J. Solinas, "Elliptic Curve Groups modulo a
              Prime (ECP Groups) for IKE and IKEv2", RFC 5903,
              June 2010.

McGrew, et al. Informational [Page 27] RFC 6090 Fundamental ECC February 2011

 [RFC5996]    Kaufman, C., Hoffman, P., Nir, Y., and P. Eronen,
              "Internet Key Exchange Protocol Version 2 (IKEv2)",
              RFC 5996, September 2010.
 [SuiteB]     U. S. National Security Agency (NSA), "NSA Suite B
              Cryptography", <http://www.nsa.gov/ia/programs/
              suiteb_cryptography/index.shtml>.
 [V1996]      Vaudenay, S., "Hidden Collisions on DSS", Advances in
              Cryptology - CRYPTO '96 Proceedings, Springer Lecture
              Notes in Computer Science (LNCS), volume 1109, 1996.
 [VW1994]     van Oorschot, P. and M. Wiener, "Parallel Collision
              Search with Application to Hash Functions and Discrete
              Logarithms", Proceedings of the 2nd ACM Conference on
              Computer and communications security, pp. 210-218, 1994.
 [VW1996]     van Oorschot, P. and M. Wiener, "On Diffie-Hellman key
              agreement with short exponents", Advances in Cryptology
              - EUROCRYPT '96 Proceedings, Springer Lecture Notes in
              Computer Science (LNCS), volume 1070, 1996.
 [X9.62]      "Public Key Cryptography for the Financial Services
              Industry: The Elliptic Curve Digital Signature Algorithm
              (ECDSA)", American National Standards Institute (ANSI)
              X9.62.

McGrew, et al. Informational [Page 28] RFC 6090 Fundamental ECC February 2011

Appendix A. Key Words

 The definitions of these key words are quoted from [RFC2119] and are
 commonly used in Internet standards.  They are reproduced in this
 note in order to avoid a normative reference from after 1994.
 1.  MUST - This word, or the terms "REQUIRED" or "SHALL", means that
     the definition is an absolute requirement of the specification.
 2.  MUST NOT - This phrase, or the phrase "SHALL NOT", means that the
     definition is an absolute prohibition of the specification.
 3.  SHOULD - This word, or the adjective "RECOMMENDED", means that
     there may exist valid reasons in particular circumstances to
     ignore a particular item, but the full implications must be
     understood and carefully weighed before choosing a different
     course.
 4.  SHOULD NOT - This phrase, or the phrase "NOT RECOMMENDED", means
     that there may exist valid reasons in particular circumstances
     when the particular behavior is acceptable or even useful, but
     the full implications should be understood and the case carefully
     weighed before implementing any behavior described with this
     label.
 5.  MAY - This word, or the adjective "OPTIONAL", means that an item
     is truly optional.  One vendor may choose to include the item
     because a particular marketplace requires it or because the
     vendor feels that it enhances the product while another vendor
     may omit the same item.  An implementation which does not include
     a particular option MUST be prepared to interoperate with another
     implementation which does include the option, though perhaps with
     reduced functionality.  In the same vein an implementation which
     does include a particular option MUST be prepared to interoperate
     with another implementation which does not include the option
     (except, of course, for the feature the option provides.)

Appendix B. Random Integer Generation

 It is easy to generate an integer uniformly at random between zero
 and (2^t)-1, inclusive, for some positive integer t.  Generate a
 random bit string that contains exactly t bits, and then convert the
 bit string to a non-negative integer by treating the bits as the
 coefficients in a base-2 expansion of an integer.

McGrew, et al. Informational [Page 29] RFC 6090 Fundamental ECC February 2011

 It is sometimes necessary to generate an integer r uniformly at
 random so that r satisfies a certain property P, for example, lying
 within a certain interval.  A simple way to do this is with the
 rejection method:
 1.  Generate a candidate number c uniformly at random from a set that
     includes all numbers that satisfy property P (plus some other
     numbers, preferably not too many)
 2.  If c satisfies property P, then return c.  Otherwise, return to
     Step 1.
 For example, to generate a number between 1 and n-1, inclusive,
 repeatedly generate integers between zero and (2^t)-1, inclusive,
 stopping at the first integer that falls within that interval.
 Recommendations on how to generate random bit strings are provided in
 [RFC4086].

Appendix C. Why Compact Representation Works

 In the affine representation, the x-coordinate of the point P^i does
 not depend on the y-coordinate of the point P, for any non-negative
 exponent i and any point P.  This fact can be seen as follows.  When
 given only the x-coordinate of a point P, it is not possible to
 determine exactly what the y-coordinate is, but the y value will be a
 solution to the curve equation
    y^2 = x^3 + a*x + b (mod p).
 There are at most two distinct solutions y = w and y = -w mod p, and
 the point P must be either Q=(x,w) or Q^-1=(x,-w).  Thus P^n is equal
 to either Q^n or (Q^-1)^n = (Q^n)^-1.  These values have the same
 x-coordinate.  Thus, the x-coordinate of a point P^i can be computed
 from the x-coordinate of a point P by computing one of the possible
 values of the y coordinate of P, then computing the ith power of P,
 and then ignoring the y-coordinate of that result.
 In general, it is possible to compute a square root modulo p by using
 Shanks' method [K1981v2]; simple methods exist for some values of p.
 When p = 3 (mod 4), the square roots of z mod p are w and -w mod p,
 where
    w = z ^ ((p+1)/4) (mod p);

McGrew, et al. Informational [Page 30] RFC 6090 Fundamental ECC February 2011

 this observation is due to Lehmer [L1969].  When p satisfies this
 property, y can be computed from the curve equation, and either y = w
 or y = -w mod p, where
    w = (x^3 + a*x + b)^((p+1)/4) (mod p).
 Square roots modulo p only exist for a quadratic residue modulo p,
 [R1988]; if z is not a quadratic residue, then there is no number w
 such that w^2 = z (mod p).  A simple way to verify that z is a
 quadratic residue after computing w is to verify that
 w * w = z (mod p).  If this relation does not hold for the above
 equation, then the value x is not a valid x-coordinate for a valid
 elliptic curve point.  This is an important consideration when ECDH
 is used with compact output; see Section 10.3.
 The primes used in the P-256, P-384, and P-521 curves described in
 [RFC5903] all have the property that p = 3 (mod 4).

Appendix D. Example ECC Parameter Set

 For concreteness, we recall an elliptic curve defined by Solinas and
 Fu in [RFC5903] and referred to as P-256, which is believed to
 provide a 128-bit security level.  We use the notation of
 Section 3.3, and express the generator in the affine coordinate
 representation g=(gx,gy), where the values gx and gy are in Fp.
 p: FFFFFFFF00000001000000000000000000000000FFFFFFFFFFFFFFFFFFFFFFFF
 a: - 3
 b: 5AC635D8AA3A93E7B3EBBD55769886BC651D06B0CC53B0F63BCE3C3E27D2604B
 n: FFFFFFFF00000000FFFFFFFFFFFFFFFFBCE6FAADA7179E84F3B9CAC2FC632551
 gx: 6B17D1F2E12C4247F8BCE6E563A440F277037D812DEB33A0F4A13945D898C296
 gy: 4FE342E2FE1A7F9B8EE7EB4A7C0F9E162BCE33576B315ECECBB6406837BF51F5
 Note that p can also be expressed as
    p = 2^(256)-2^(224)+2^(192)+2^(96)-1.

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Appendix E. Additive and Multiplicative Notation

 The early publications on elliptic curve cryptography used
 multiplicative notation, but most modern publications use additive
 notation.  This section includes a table mapping between those two
 conventions.  In this section, a and b are elements of an elliptic
 curve group, and N is an integer.
          +-------------------------+-----------------------+
          | Multiplicative Notation | Additive Notation     |
          +-------------------------+-----------------------+
          | multiplication          | addition              |
          | a * b                   | a + b                 |
          | squaring                | doubling              |
          | a * a = a^2             | a + a = 2a            |
          | exponentiation          | scalar multiplication |
          | a^N = a * a * ... * a   | Na = a + a + ... + a  |
          | inverse                 | inverse               |
          | a^-1                    | -a                    |
          +-------------------------+-----------------------+

Appendix F. Algorithms

 This section contains a pseudocode description of the elliptic curve
 group operation.  Text that follows the symbol "//" is to be
 interpreted as comments rather than instructions.

F.1. Affine Coordinates

 To an arbitrary pair of elliptic curve points P and Q specified by
 their affine coordinates P=(x1,y1) and Q=(x2,y2), the group operation
 assigns a third point R = P*Q with the coordinates (x3,y3).  These
 coordinates are computed as follows:
   if P is (@,@),
      R = Q
   else if Q is (@,@),
      R = P
   else if P is not equal to Q and x1 is equal to x2,
      R = (@,@)
   else if P is not equal to Q and x1 is not equal to x2,
      x3 = ((y2-y1)/(x2-x1))^2 - x1 - x2 mod p and
      y3 = (x1-x3)*(y2-y1)/(x2-x1) - y1 mod p
   else if P is equal to Q and y1 is equal to 0,
      R = (@,@)
   else    // P is equal to Q and y1 is not equal to 0
      x3 = ((3*x1^2 + a)/(2*y1))^2 - 2*x1 mod p and
      y3 = (x1-x3)*(3*x1^2 + a)/(2*y1) - y mod p.

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 From the first and second case, it follows that the point at infinity
 is the neutral element of this operation, which is its own inverse.
 From the curve equation, it follows that for a given curve point P =
 (x,y) distinct from the point at infinity, (x,-y) also is a curve
 point, and from the third and the fifth case it follows that this is
 the inverse of P, P^-1.
 Note: The fifth and sixth case are known as "point squaring".

F.2. Homogeneous Coordinates

 An elliptic curve point (x,y) (other than the point at infinity
 (@,@)) is equivalent to a point (X,Y,Z) in homogeneous coordinates
 (with X, Y, and Z in Fp and not all three being zero at once)
 whenever x=X/Z and y=Y/Z. "Homogenous coordinates" means that two
 triples (X,Y,Z) and (X',Y',Z') are regarded as "equal" (i.e.,
 representing the same point) if there is some nonzero s in Fp such
 that X'=s*X, Y'=s*Y, and Z'=s*Z.  The point at infinity (@,@) is
 regarded as equivalent to the homogenous coordinates (0,1,0), i.e.,
 it can be represented by any triple (0,Y,0) with nonzero Y in Fp.
 Let P1=(X1,Y1,Z1) and P2=(X2,Y2,Z2) be points on the elliptic curve,
 and let u = Y2 * Z1 - Y1 * Z2 and v = X2 * Z1 - X1 * Z2.
 We observe that the points P1 and P2 are equal if and only if u and v
 are both equal to zero.  Otherwise, if either P1 or P2 are equal to
 the point at infinity, v is zero and u is nonzero (but the converse
 implication does not hold).
 Then, the product P3=(X3,Y3,Z3) = P1 * P2 is given by:
   if P1 is the point at infinity,
      P3 = P2
   else if P2 is the point at infinity,
      P3 = P1
   else if u is not equal to 0 but v is equal to 0,
      P3 = (0,1,0)
   else if both u and v are not equal to 0,
      X3 = v * (Z2 * (Z1 * u^2 - 2 * X1 * v^2) - v^3)
      Y3 = Z2 * (3 * X1 * u * v^2 - Y1 * v^3 - Z1 * u^3) + u * v^3
      Z3 = v^3 * Z1 * Z2
   else    // P2 equals P1, P3 = P1 * P1
       w = 3 * X1^2 + a * Z1^2
      X3 = 2 * Y1 * Z1 * (w^2 - 8 * X1 * Y1^2 * Z1)
      Y3 = 4 * Y1^2 * Z1 * (3 * w * X1 - 2 * Y1^2 * Z1) - w^3
      Z3 = 8 * (Y1 * Z1)^3

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 It thus turns out that the point at infinity is the identity element
 and for P1=(X,Y,Z) not equal to this point at infinity, P2=(X,-Y,Z)
 represents P1^-1.

Authors' Addresses

 David A. McGrew
 Cisco Systems
 510 McCarthy Blvd.
 Milpitas, CA  95035
 USA
 Phone: (408) 525 8651
 EMail: mcgrew@cisco.com
 URI:   http://www.mindspring.com/~dmcgrew/dam.htm
 Kevin M. Igoe
 National Security Agency
 Commercial Solutions Center
 United States of America
 EMail: kmigoe@nsa.gov
 Margaret Salter
 National Security Agency
 9800 Savage Rd.
 Fort Meade, MD  20755-6709
 USA
 EMail: msalter@restarea.ncsc.mil

McGrew, et al. Informational [Page 34]

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