Network Workign Group A. Jivsov
Internet-Draft Symantec Corporation
Intended status: Informational March 15, 2014
Expires: September 16, 2014
Compact representation of an elliptic curve point
draft-jivsov-ecc-compact-05
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.
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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 . . . . . . . . . . 6
4.2.1. The black box key generation algorithm . . . . . . . . 6
4.2.2. The deterministic key generation algorithm . . . . . . 7
4.2.3. The key generation algorithm for the sum of points . . 7
4.2.4. The key generation algorithm for the multiples . . . . 8
4.3. The efficient square root algorithm for p=4*k+3 . . . . . 8
4.4. General applicability . . . . . . . . . . . . . . . . . . 9
5. Interoperability considerations . . . . . . . . . . . . . . . 10
6. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 10
7. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 11
8. Security Considerations . . . . . . . . . . . . . . . . . . . 11
9. References . . . . . . . . . . . . . . . . . . . . . . . . . . 12
9.1. Normative References . . . . . . . . . . . . . . . . . . . 12
9.2. Informative References . . . . . . . . . . . . . . . . . . 13
Appendix A. Sample code change to add compliant key
generation to libgcrypt and openssl . . . . . . . . . 14
A.1. libgcrypt changes . . . . . . . . . . . . . . . . . . . . 14
A.2. OpenSSL changes . . . . . . . . . . . . . . . . . . . . . 15
Author's Address . . . . . . . . . . . . . . . . . . . . . . . . . 15
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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. The
method described here can be applied to a various types of elliptic
curves, see Section 4.4.
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 elliptic curve equation.
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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 = C(x) is the appropriate Weierstrass equation linking x and
y; for example, C(x) = x^3 + a*x + b is used for the short
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Weierstrass form
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 )
This document specify how Q is represented 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.
Every elliptic curve for prime p (more generally, for any underlying
field that doesn't have characteristic 2) can be represented as y^2 =
C(x) with appropriate substitution, where C(x) is degree-3 polynomial
in x.
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 its compact representation.
Decoding: Given the compact representation of Q, return canonical
representation of Q=(x,y) as follows:
1. y' = sqrt( C(x) ), 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.
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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
{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
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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.
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.
4.2.3. The key generation algorithm for the sum of points
In some protocols a participant must produce a point Q, such that Q =
S + P. Q is to be shared with other participants. In this equation S
is some known point and P is a point for the key pair {k,
P=k*G=(x1,y1)} that the participant generates internally, but never
makes available to other participants of the protocol. The Q must be
compliant, while there is no such requirement for P. The following
algorithm is a generalization of the previous two algorithms to
produce a compliant Q.
It's easy to observe that the black box algorithm in section
Section 4.2.1 is easily adopted to this case. All that's needed is
to continue to generate a new P so that the Q (not P) is compliant.
A more efficient version of this algorithm that only requires one KG
is described next. This algorithm expects that the participant
generates a delta key pair {d, D=d*G} and caches it for a certain
period of time to ammortize the cost of generation of {d, D}. For
example, {d, D} might be generated once during the process start-up
and is kept in memory until the process is terminated. This allows
the same {d, D} to be used with many protocol runs. In this case the
formula is Q = S + (P + t*D), which means that the P is adjusted with
t additions of D until the point Q is a compliant point. The
expected value of t is 1.
Before the first protocol run:
1. Obtain the point S
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2. Generate a delta key pair {d, D=d*G} with KG; save it so it
can be used in subsequent protocol runs
Generation:
1. Generate an internal key pair {k, P=k*G=(x1,y1)} with KG
2. Calculate Q=(x,y)=P+S
3. Until ( y == min(y,p-y) ) do {Q += D; k += d;}
4. Output {k, P}, Q=(x,y), where Q is a compliant point and
{k, P} is a corresponding to Q internal key pair
4.2.4. The key generation algorithm for the multiples
In some protocols a participant must produce a point Q, such that Q =
k1*k2*...*kn*P. Q is to be shared with other participants. In this
equation P is a point for the key pair {k, P=k*G} that the
participant generates internally, but never makes available to other
participants of the protocol. The Q must be compliant, while there
is no such requirement for P.
It's easy to observe that the black box algorithm in section
Section 4.2.1 is easily adopted to this case. All that's needed is
to continue to generate a new P so that the Q (not P) is compliant.
However, there is a deterministic algorithm with a single KG, based
on the algorithm in section Section 4.2.2. Q can be rewritten as Q =
k1*k2*...*kn*k*G = k*(k1*k2*...*kn)G = k * S. The algorithm in
section Section 4.2.2 is used as is, except the G is replaced with S,
so that Q=k*G is interpreted as Q=k*S.
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 = C(x)
y' = y2^((p+1)/4)
y = min(y',p-y')
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Q=(x,y) is the canonical representation of the point
See [Lehmer] for details.
4.4. General applicability
The method described in this document should work for any elliptic
curve over the prime fields provided that the canonical repsentation
of the point is in affine coordinates. For the given x there are two
solutions to the equation y^2 = C(x); the method described here
selects the smallest y; this corresponds to the private keys k or
Ord-k.
The extension of the method described above is straightforward. Note
that the fundamental technique described above relies on the
relationship between {k,Ord-k} and {Q, -Q}. -Q refers to the explicit
affine negation formula. The transition between Q, -Q and k, Ord-k
is exceptionally fast. In general, the appropriate affine point
negation formula changes the coordinate that we will omit in the
compact representation (this is typically the y coordinate). The
code that recovers the omitted coordinate assumes that it is the
smallest possible coordinate y, such that y^2 = C(x). This
assumption is assured by the selection of the corresponding {k,
Ord-k}. This will be clear from the following examples.
Many definitions of elliptic curve are known and the list may keep
keep growing. [EFD] specifies the affine point negation formula for
various curves. Here are a few possibilities for -Q, where Q =
(x,y):
Formula Method Synopsis
--------- -------------------------------------------- --------------
(x,-y) Use step-by-step instructions in this y=min(y,p-y)
specification
(-x,y) Edwards curves. Given that the curve x=min(x,p-x);
definition is symmetric for x and y, rename y is compact
x and y in the description of the above representation
methods
(y,x) This specification, except change the y = min(x,y)
formula for negation to "exchange x with y"
(x/y,1/y) This specification, except change the y =
formula for negation to use division by y min(y,1/y),
x=x/y
Affine negation methods
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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 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.
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, a typical instantiation
of the ECDH protocol, such as ECDH specified in [NIST-SP800-56A],
makes any ECC key used in a DH key key agreement automatically
compliant for the purpose of this specification ( as noted in the
Section 4.2 ). 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. Acknowledgements
The methods described in this document eliminates one bit that is
tranditionally needed to identify either of the two y coordinates.
[FPE] is the earliest reference about such an approach to compact
representation in the prime underlying finite field that the author
is aware of, viewed in that work in the context of the format
preserving encryption.
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7. IANA Considerations
This document defines a 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.
8. 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
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.
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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 output produced by A is k,
the output produced by A' is set to (Ord-k). Note that (Ord-k)*G
= (x,y), which means that the output of A' is the correct
solution.
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 for 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 occurred, 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.
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.
9. References
9.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
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9.2. Informative References
[EFD] Bernstein, D. and T. Lange, "Genus-1 curves over large-
characteristic fields", 2014,
<http://www.hyperelliptic.org/EFD/index.html>.
[FIPS-186]
National Institute of Standards and Technology, "Digital
Signature Standard (DSS)", FIPS 186-3, June 2009,
<http://csrc.nist.gov/publications/PubsSPs.html>.
[FPE] Rogaway, P. and J. Black, "Ciphers with arbitrary finite
domains.", Proceedings Topics in Cryptology CT-RSA.
Springer Berlin Heidelberg, 2002,
<http://www.cs.ucdavis.edu/~rogaway/papers/subset.pdf>.
[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>.
[OpenSSL] Jivsov, A., "An enhancement to EC key generation to enable
compact point representation", June 2013,
<http://rt.openssl.org/Ticket/History.html?id=3069>.
[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.
[SEC1] STANDARDS FOR EFFICIENT CRYPTOGRAPHY, "SEC 1: Elliptic
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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 and openssl
The complete changes that were needed to make libgcrypt library
generate a compliant key are shown inline in this section, followed
by pending changes to OpenSSL.
A.1. libgcrypt changes
In the following changes, the 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);
+
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+ 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 */
+
+ 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);
+ }
A.2. OpenSSL changes
OpenSSL changes are even more compact than in Appendix A.1. They are
tracked at [OpenSSL].
Author's Address
Andrey Jivsov
Symantec Corporation
Email: crypto@brainhub.org
Jivsov Expires September 16, 2014 [Page 15]