Network Workign Group                                          A. Jivsov
Internet-Draft                                      Symantec Corporation
Intended status: Informational                              May 31, 2013
Expires: December 2, 2013


           Compact representation of an elliptic curve point
                      draft-jivsov-ecc-compact-01

Abstract

   This document defines a format for efficient storage representation
   of an elliptic curve point over prime fields, suitable for use with
   any IETF format or protocol.

Status of this Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

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   This Internet-Draft will expire on December 2, 2013.

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   document authors.  All rights reserved.

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   described in the Simplified BSD License.





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Table of Contents

   1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  3
   2.  Conventions used in this document  . . . . . . . . . . . . . .  3
   3.  Overview of the compact representation in IETF protocols . . .  3
   4.  The definition of the compact representation . . . . . . . . .  4
     4.1.  Encoding and decoding of an elliptic curve point . . . . .  5
     4.2.  The algorithms to generate a key pair  . . . . . . . . . .  5
       4.2.1.  The black box key generation algorithm . . . . . . . .  6
       4.2.2.  The deterministic key generation algorithm . . . . . .  6
     4.3.  The efficient square root algorithm for p=4*k+3  . . . . .  7
   5.  Interoperability considerations  . . . . . . . . . . . . . . .  7
   6.  IANA Considerations  . . . . . . . . . . . . . . . . . . . . .  8
   7.  Security Considerations  . . . . . . . . . . . . . . . . . . .  8
   8.  References . . . . . . . . . . . . . . . . . . . . . . . . . . 10
     8.1.  Normative References . . . . . . . . . . . . . . . . . . . 10
     8.2.  Informative References . . . . . . . . . . . . . . . . . . 10
   Appendix A.  Sample code change to add compliant key
                generation to libgcrypt . . . . . . . . . . . . . . . 11
   Author's Address . . . . . . . . . . . . . . . . . . . . . . . . . 12































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1.  Introduction

   The National Security Agency (NSA) of the United States specifies
   elliptic curve cryptography (ECC) for use in its [SuiteB] set of
   algorithms.  The NIST elliptic curves over the prime fields
   [FIPS-186], which include [SuiteB] curves, or the Brainpool curves
   [RFC5639] are the examples of curves over prime fields.

   This document provides an efficient format for compact representation
   of a point on an elliptic curve over a prime field.  It is intended
   as an open format that other IETF protocols can rely on to minimize
   space required to store an ECC point.  This document complements the
   [RFC6090] with the on-the-wire definition of an ECC point.

   One of the benefits of the ECC is the small size of field elements.
   The compact representation reduces the encoded size of an ECC element
   in half, which can be a substantial saving in cases such as
   encryption of a short message sent to multiple recipients.


2.  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 [RFC2119].


3.  Overview of the compact representation in IETF protocols

   IETF protocols often use the [SEC1] representation of a point on an
   elliptic curve, which is a sequence of the following fields:

   Field  Description
   ------ --------------------------------------------------------------
   B0     {02, 03, 04}, where 02 or 03 represent a compressed point (x
          only), while 04 represents a complete point (x,y)
   X      x coordinate of a point
   Y      y coordinate of a point, optional (present only for B0=04)

                         SEC1 point representation

   The [SEC1] is an example of a general-purpose elliptic curve point
   compression.  The idea behind these methods is the following:

   o  For the given point P=(x,y) the y coordinate can be derived from x
      by solving the corresponding equation of the ECC.





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   o  There are two possible y coordinates for any x of a given P

   o  The either of the two possibilities for y is encoded in some way
      in the compressed representation

   There are a few undesirable properties of the above representation:

   o  The requirement to store one bit to identify the 'y' means that
      the whole byte is required.

   o  For most well-known elliptic curves the extra byte removes the
      power of two alignment for the encoded point.

   o  The requirement for the balanced security calls for the ECC curve
      size to be equal the hash output size, yet the storage length of
      the ECC point is equal to the hash output size + 1.

   o  The encoded point is not a multi-precision integer, but a
      structured sequence of bytes.  For example, special wording is
      required to define the encoding of the [FIPS-186] P-521 to clarify
      how odd number of bits for x and y, or a bit representing y, are
      packed into bytes.

   o  Some protocols, such as ECDH, don't depend on the exact value of
      the y.  It is unnecessary to track the precise point P=(x,y) in
      such protocols.


4.  The definition of the compact representation

   This document is an improvement to the idea by [Miller] to not
   transmit the y coordinate of an ECC point in the elliptic curve
   Diffie-Hellman (ECDH) protocol.

   We will use the following notations for the ECC point Q and the
   features of the corresponding elliptic curve:

      Q = k*G, where

      Q = (x,y) is the point on an elliptic curve (the canonical
      represenation)

      k - the private key (a scalar)

      G - the elliptic curve generator point

      y^2 = x^3 + a*x + b is the standard Weierstrass equation linking x
      and y



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      p - the order of the underlying finite field to which x and y
      belong

      Ord - the order of the elliptic curve field, i.e. the number of
      points on the curve ( Ord*G = O, where O is the identity element )

   Q is a point that we need to represent in the compact form.  The
   integer operations considered in this document are performed modulo
   prime p and "(mod p)" is assumed in every formula with x and y.

   The steps to create and interpret the compact representation of a
   point are described next.  A special key generation algorithm is
   needed to make them possible, defined later in Section 4.2.

4.1.  Encoding and decoding of an elliptic curve point

   Encoding:  Given the canonical representation of Q=(x,y), return the
        x as the compact representation.

   Decoding:  Given the compact representation of Q, return canonical
        representation of Q=(x,y) as follows:

        1.   y' = sqrt( x^3 + a*x + b ), where y'>0

        2.   y = min(y',p-y')

        3.   Q=(x,y) is the canonical representation of the point

   Recall that the x is an element in the underlying finite field,
   represented by an integer.  Its precise encoding SHOULD be consistent
   with encoding of other multi-precision integers in the application,
   for example, it would be the same encoding as used for the r or s
   integer that is a part of the DSA signature and it is typically a
   sequence of big-endian bytes.

   The efficient algorithm to recover y for [SuiteB] or the Brainpool
   curves [RFC5639], among others, is given in Section 4.3.

   min(y,p-y) can be calculated with the help of the pre-calculated
   value p2=(p-1)/2. min(y,p-y) is y if y<p2 and p-y otherwise.

   The efficient encoding and decoding algorithms are possible with the
   special key generation algorithm, defined next.

4.2.  The algorithms to generate a key pair

   This document specifies two algorithms, called the "black box" and
   the "deterministic" key generation algorithms, to generate a key pair



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   {k, Q=k*G=(x,y)}, where k is the private key and Q=(x,y) is the
   public key.  A key pair generated according to the requirements in
   this section is called a compliant key pair, and the public key of
   such a key pair -- a compliant public key.  A compliant public key
   Q=(x,y) allows compact representation as x, as defined in
   Section 4.1.

   Both key generation algorithms can be built with any general purpose
   key generation algorithm which would be needed in any ECC
   implementation that generates keys, regardless of the support for any
   method defined in this document.  Such a general purpose key
   generation algorithm is referred in this section as "KG".

   The black box algorithm works in scenarios when the KG doesn't allow
   any adjustments to the private key.  The disadvantage of this
   algorithm is that multiple KGs may be needed to generate a single key
   pair {k, Q}.  The deterministic algorithm is similar, except that it
   is allowed to perform a simple and fast modification to the private
   key after the KG.  The advantage of the second algorithm is
   performance, in particular, the guarantee that only a single KG is
   needed.

4.2.1.  The black box key generation algorithm

   The following algorithm calculates a key pair {k, Q=k*G=(x,y)}, where
   k is the private key and Q=(x,y) is the public key.

   Black box generation:

        1.   Generate a key pair {k, Q=k*G=(x,y)} with KG

        2.   if( y != min(y,p-y) ) goto step 1

        3.   output {k, Q=(x,y)} as a key pair

   Note that the step 1 is a general purpose key generation algorithm,
   such as an algorithm compliant with [NIST-SP800-133].  Step 1 assumes
   neither changes to existing key generation methods nor access to the
   private key in clear.

   The expected number of iterations in the loop in the above algorithm
   is 2.  The step 2 is not needed for the ECDH keys.

4.2.2.  The deterministic key generation algorithm

   The following algorithm calculates a key pair {k, Q=k*G=(x,y)}, where
   k is the private key and Q=(x,y) is the public key.




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   Deterministic generation:

        1.   Generate a key pair {k, Q=k*G=(x,y)} with KG

        2.   if( y != min(y,p-y) ) k = Ord - k; y = p - y

        3.   output {k, Q=(x,y)} as a key pair

   The step 2 is not needed for the ECDH keys.

4.3.  The efficient square root algorithm for p=4*k+3

   When p mod 4 = 3, as is the case of [SuiteB] and the Brainpool curves
   [RFC5639], there is an efficient square root algorithm to recover the
   y, as follows:

      Given the compact representation of Q as x,

      y2 = x^3 + a*x + b

      y' = y2^((p+1)/4)

      y = min(y',p-y')

      Q=(x,y) is the canonical representation of the point

   See [Lehmer] for details.


5.  Interoperability considerations

   The compact representation described in this document allows two-
   phase introduction.

   First, key pairs must be generated as defined in Section 4.2 to allow
   compact representation.  No accompanied changes are needed elsewhere
   to use these keys.  This allows safe deployment of the new key
   generation, which, in turn, allows encoding and decoding of in
   compact representation, possibly at a later time.

   Finally, the encoding of public keys in the new compact
   representation format can be enabled after there is confidence in the
   universal support of new compact representation.  This event would
   not need to change any private key material, only public key
   representation.

   The above two phases can be implemented at once for new formats.




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   Most ECC cryptographic protocols, such as ECDSA [FIPS-186], are
   intended to work with persistently stored public keys that are
   generated as fresh key pairs, as opposed to some derivation function
   that transforms an ECC point.  The algorithm described in Section 4.2
   is possible in all these cases.  Furthermore, the typical
   instantiation of the ECDH protocol, as in [NIST-SP800-56A], makes
   |*any| ECC key compliant ( description Section 4.2 notes this
   simplification ).  The algorithm in Section 4.2 will even work for
   secure devices that never reveal the private key, such as smartcards
   or Hardware Security Modules.  A public key that is generated
   according to the Section 4.2 can be used without limitations in
   existing protocols that use ECC points encoded in other ways, such as
   [SEC1], with compression or not, with the added advantage that the
   keys generated according to the method in Section 4.2 will allow the
   Section 4.1 encoding.


6.  IANA Considerations

   This document defines the low-level format that may be suitable for a
   wide range of applications.  However, it is responsibility of the
   application that adopts this format to define the IDs that will
   enable the ECC compact point representation in that application.

   A new ID may not be always necessary.  For example, an application
   that currently allows the [SEC1] encoding may allow the compact
   representation defined in this document as an extension to the [SEC1]
   as follows.  Consider the encoding of a compressed [FIPS-186] P-256
   point, for example.  The [SEC1] compressed representation of a P-256
   point will always occupy exactly 33 bytes.  On the other hand, the
   compact representation defined in this document will never exceed 32
   bytes (it may occupy fewer that 32 bytes when the most significant
   byte has happened to be zero).  This size will allow reliable
   discrimination between two encoding formats.


7.  Security Considerations

   The key pair generation process in Section 4.2 excludes exactly half
   of the points on the elliptic curve.  What is left is the subset of
   points suitable for compact representation.  The filtering of points
   is based on a public criteria that are applied to the public output
   of the ECC one-way function.

   The set of Ord points on the elliptic curve can be subdivided as
   follows.  First, remove the point O, which leaves Ord-1 points.  Of
   these points there are exactly (Ord-1)/2 points that have unique x
   coordinate.  This document specifies a method to form the (Ord-1)/2



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   of points, each having a unique x coordinate.  These points are
   called compliant public keys in Section 4.2.

   For any two public keys P=(x,y) and P=(x,y') there is up to one bit
   of entropy in y' v.s. y and this information is public.  This bit of
   entropy doesn't contribute to the difficulty of the underlying hard
   problem of the ECC: the elliptic curve discrete logarithm problem
   (ECDLP).

   It will be shown next that breaking the ECDLP with a key generated
   according to Section 4.2 is not easier than breaking the ECDLP with a
   key obtained through a standard key generation algorithm, referred to
   as the KG algorithm in the Section 4.2.

   Let us assume that there is an algorithm A that solves the ECDLP for
   the KG.  The algorithm A can be transformed into the algorithm A' as
   follows.

   o  If P=(x,y) is a compliant public key, the ECDLP is solved with A
      for the point (x,y): the result is k, such that k*G=(x,y)

   o  If P=(x,y) is not a compliant public key, the ECDLP is solved with
      A for the point (x,p-y); assuming the result produced by A is k,
      the result produced by A' is set to (Ord-k).  Note that (Ord-k)*G
      = (x,p).

   A' is equivalent to A. The complexity of one additional substraction
   in the prime field is negligible even to the complexity of a single
   elliptic curve addition.  Observe that A' works on all public keys by
   performing the actual work only on compliant public keys.

   If we now consider only the compliant public keys, which cuts the
   number of points in half, we observe that the ECDLP solving algorithm
   A' doesn't get to break fewer public keys.  This concludes the proof.

   The same result can be observed based on the details of the current
   state of the art attacks on the ECDLP.  These attacks use Pollard's
   rho algorithm, which uses the collision search in the sequence(s) of
   generated points with the goal to produce the points P1=(x1,y1) and
   P2=(x2,y2), such that x1=x2 and y1=y2.  The match in the x coordinate
   is the sufficient event for the successful attack.  After this event
   has ocurred, the sequence(s) that led to x1=x2 collision can be
   adjusted in a constant number of steps to ensure that y1=y2, if this
   is not already the case.  Furthermore, collision search requires the
   storage of candidates for the collision.  It's wasteful to store
   (x,y) v.s. storing x and only calculating y when the collision in x
   is detected.  Thus, the ECDLP attack does not benefit from the
   unpredictability of the y.



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   Finally, note that a common design feature of an ECDH-based system is
   not to depend on the y coordinate, such as the one defined in the
   [NIST-SP800-56A].  Thus, the security of the system is unaffected if
   we fix either of the two possibilities for the point with the given x
   coordinate.


8.  References

8.1.  Normative References

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119, March 1997.

8.2.  Informative References

   [FIPS-186]
              National Institute of Standards and Technology, "Digital
              Signature Standard (DSS)", FIPS 186-3, June 2009,
              <http://csrc.nist.gov/publications/PubsSPs.html>.

   [Lehmer]   Lehmer, D., "Computer technology applied to the theory of
              numbers", 1969.

   [Miller]   Miller, V., "Use of elliptic curves in cryptography",
              Proceedings Lecture notes in computer sciences; 218 on
              Advances in cryptology -- CRYPTO 85, June 1986.

   [NIST-SP800-133]
              National Institute of Standards and Technology,
              "Recommendation for Cryptographic Key Generation", SP 800-
              133, November 2012,
              <http://csrc.nist.gov/publications/PubsSPs.html>.

   [NIST-SP800-56A]
              National Institute of Standards and Technology,
              "Recommendation for Pair-Wise Key Establishment Schemes
              Using Discrete Logarithm Cryptography", SP 800-56A
              Revision 1, March 2007,
              <http://csrc.nist.gov/publications/PubsSPs.html>.

   [RFC5639]  Lochter, M. and J. Merkle, "Elliptic Curve Cryptography
              (ECC) Brainpool Standard Curves and Curve Generation",
              RFC 5639, March 2010.

   [RFC6090]  McGrew, D., Igoe, K., and M. Salter, "Fundamental Elliptic
              Curve Cryptography Algorithms", RFC 6090, February 2011.




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   [SEC1]     STANDARDS FOR EFFICIENT CRYPTOGRAPHY, "SEC 1: Elliptic
              Curve Cryptography", September 2000, <www.secg.org/
              collateral/sec1_final.pdf>.

   [SuiteB]   National Security Agency, "NSA Suite B Cryptography",
              March 2010,
              <http://www.nsa.gov/ia/programs/suiteb_cryptography/>.


Appendix A.  Sample code change to add compliant key generation to
             libgcrypt

   This section shows complete changes that were needed to make
   libgcrypt library generate a compliant key.  Note that Q is the
   initial public key, G generator, and d is the corresponding private
   key. "-" prefix marks the two lines that were replaced with the lines
   starting with "+".  Lines starting with "+" represent the code that
   adds compliant key generation to libgcrypt.

   @@ generate_key (ECC_secret_key *sk,
        unsigned int nbits,
        const char *name,
      point_set (&sk->E.G, &E.G);
      sk->E.n = mpi_copy (E.n);
      point_init (&sk->Q);
   -  point_set (&sk->Q, &Q);
   -  sk->d    = mpi_copy (d);
   +
   +  /* We want the Q=(x,y) be a "compliant key" in terms of the
   +   * http://tools.ietf.org/html/draft-jivsov-ecc-compact,
   +   * which simply means that we choose either Q=(x,y) or -Q=(x,p-y)
   +   * such that we end up with the min(y,p-y) as the y coordinate.
   +   * Such a public key allows the most efficient compression: y can
   +   * simply be dropped because we know that it's a minimum of
   +   * the two possibilities without any loss of security.
   +   */
   +  {
   +    gcry_mpi_t x, p_y, y;
   +    const unsigned int nbits = mpi_get_nbits (E.p);
   +
   +    x = mpi_new (nbits);
   +    p_y = mpi_new (nbits);
   +    y = mpi_new (nbits);
   +
   +    if (_gcry_mpi_ec_get_affine (x, y, &Q, ctx))
   +      log_fatal ("ecgen: Failed to get affine coordinates for Q\n");
   +
   +    mpi_sub( p_y, E.p, y );  /* p_y = p-y */



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   +
   +    if( mpi_cmp( p_y /*p-y*/, y ) < 0 )  {  /* is p-y < p ? */
   +      gcry_mpi_t z = mpi_copy( mpi_const (MPI_C_ONE) );
   +      /* we need to end up with -Q; this assures that new Q's y
   +       * is the smallest one */
   +      sk->d = mpi_new (nbits);
   +      mpi_sub( sk->d, E.n, d );  /* d = order-d */
   +      /* log_mpidump ("ecgen d after ", sk->d); */
   +      gcry_mpi_point_set (&sk->Q, x, p_y/*p-y*/, z);  /* Q = -Q */
   +      if (DBG_CIPHER)
   +      {
   +        log_debug   ("ecgen converted Q to a compliant point\n");
   +      }
   +      mpi_free (z);
   +    }
   +    else
   +    {
   +      /* no change is needed exactly 50% of the time: just copy */
   +      sk->d = mpi_copy (d);
   +      point_set (&sk->Q, &Q);
   +      if (DBG_CIPHER)
   +      {
   +        log_debug   ("ecgen didn't need to convert Q to "
   +                     "a compliant point\n");
   +      }
   +    }
   +    mpi_free (x);
   +    mpi_free (p_y);
   +    mpi_free (y);
   +  }


Author's Address

   Andrey Jivsov
   Symantec Corporation

   Email: openpgp@brainhub.org













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