





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.
















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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
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   publication of this document.  Please review these documents
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   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.






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   [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.




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   [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.



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   [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.

























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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.







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   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);







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   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



















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