Elliptic-curve cryptography

Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the mathematics of elliptic curves over finite fields. The use of elliptic curves in cryptography was suggested independently by Neal Koblitz ^[1] and Victor S. Miller ^[2] in 1985.

Elliptic curves are also used in several integer factorization algorithms that have applications in cryptography, such as, for instance, Lenstra elliptic curve factorization, but this use of elliptic curves is not usually referred to as "elliptic curve cryptography."

Introduction

Public key cryptography is based on the creation of mathematical puzzles that are difficult to solve without certain knowledge about how they were created. The creator keeps that knowledge secret (the private key) and publishes the puzzle (the public key). The puzzle can then be used to scramble a message in a way that only the creator can unscramble. Early public key systems used products of two large prime numbers as the puzzle, the RSA algorithm. Because of progress in factoring numbers, RSA public keys must be thousands of bits long to provide strong security. Another class of puzzle involves solving the equation a^b = c, where a and c are known. Such equations involving integers are easily solved using logarithms. In mathematical systems known as finite groups, a solution is much more difficult; this is known as the discrete logarithm problem.

An elliptic curve is a plane curve defined by an equation of the form

y² = x³ + a x + b.

Points on such a curve can be shown to form a group, and if the equation is taken over a finite field, the point group is finite as well. The discrete logarithm problem on elliptic curve groups is believed to be more difficult than over integer-based cyclic groups and therefor keys in elliptic curve cryptography can be shorter than integer-based systems. (see: cryptographic key length) As is the case for all public key cryptography, no mathematical proof of difficulty has been published as of 2006. While the RSA patent has expired, there are patents in force covering some types of ECC. The U.S. National Security Agency has incorporated ECC technology in its Suite B set of recommended allgorithms.

Mathematical introduction

Elliptic curves used in cryptography are typically defined over two types of finite fields: fields of odd characteristic ( $\mathbb {F} _{p}$ , where $p>3$ is a large prime number) and fields of characteristic two ( $\mathbb {F} _{2^{m}}$ ). When the distinction is not important we denote both of them as $\mathbb {F} _{q}$ , where $q=p$ or $q=2^{m}$ . In $\mathbb {F} _{p}$ the elements are integers ( $0\leq x<p$ ) which are combined using modular arithmetic. The case of $\mathbb {F} _{2^{m}}$ is slightly more complicated (see finite field arithmetic for details): there are several possible representations of the field elements as bitstrings and thus an irreducible binary polynomial $f(x)$ of degree $m$ shall be specified.

Pairs of affine coordinates $(x,y)$ , where $x\in \mathbb {F} _{q}$ and $y\in \mathbb {F} _{q}$ , form a plane $\mathbb {F} _{q}\times \mathbb {F} _{q}$ . Among all of them we consider only those which satisfy the elliptic curve equation and the point at infinity $O$ (note that contrary to common use of the word, a curve in the present context really only consists of individual, discrete and non-connected points since the underlying field is discrete). In the prime case the defining equation of $E(\mathbb {F} _{p})$ is as follows: $y^{2}=x^{3}+ax+b$ , where $a\in \mathbb {F} _{p}$ and $b\in \mathbb {F} _{p}$ are constants such that $4a^{3}+27b^{2}\neq 0$ . In the binary case the defining equation of $E(\mathbb {F} _{2^{m}})$ is $y^{2}+xy=x^{3}+ax^{2}+b$ , where $a\in \mathbb {F} _{2^{m}}$ and $b\in \mathbb {F} _{2^{m}}$ are constants and $b\neq 0$ . Although the point at infinity $O$ has no affine coordinates, it is convenient to use some pair of coordinates which does not satisfy the defining equation, for example, $O=(0,0)$ if $b\neq 0$ and $O=(0,1)$ otherwise. According to Hasse's theorem on elliptic curves the number of points on a curve is roughly the same as the size of the underlying field: $|E(\mathbb {F} _{q})|=q+1\pm 2{\sqrt {q}}$ .

For every two points on a curve ( $P\in E(\mathbb {F} _{q})$ and $Q\in E(\mathbb {F} _{q})$ ) it is possible to find a third point $S=P+Q\in E(\mathbb {F} _{q})$ such that certain relations hold for all points on the curve

$(P+Q)+R=P+(Q+R)$
$P+O=O+P=P$
there exists $(-P)$ such that $-P+P=P+(-P)=O$
P+Q = Q+P

and thus the set of all the points is an additive abelian group $(E(\mathbb {F} ),+)$ .

We already specified how $O$ is defined. The negative of the point $P=(x,y)$ is defined as $-P=(x,-y)$ for $P\in E(\mathbb {F} _{p})$ and $-P=(x,x+y)$ for $P\in E(\mathbb {F} _{2^{m}})$ . The exact addition rules are as follows:

if $Q=O$ then $P+Q=P$
if $Q=-P$ then $P+Q=O$
if $Q=P$ $Q=P$ then $P+Q=R$ $P+Q=R$ , where
- in the prime case $x_{R}=\lambda ^{2}-2x_{P}$ , $y_{R}=\lambda (x_{P}-x_{R})-y_{P}$ , and $\lambda ={\frac {3x_{P}^{2}+a}{2y_{P}}}$ , or
- in the binary case $x_{R}=\lambda ^{2}+\lambda +a$ , $y_{R}=x_{P}^{2}+(\lambda +1)x_{R}$ , and $\lambda =x_{P}+{\frac {y_{P}}{x_{P}}}$
if $Q\neq P$ $Q\neq P$ then $P+Q=R$ $P+Q=R$ , where
- in the prime case $x_{R}=\lambda ^{2}-x_{P}-x_{Q}$ , $y_{R}=\lambda (x_{P}-x_{R})-y_{P}$ , and $\lambda ={\frac {y_{Q}-y_{P}}{x_{Q}-x_{P}}}$ , or
- in the binary case $x_{R}=\lambda ^{2}+\lambda +x_{P}+x_{Q}+a$ , $\lambda (x_{P}+x_{R})+x_{R}+y_{P}$ , and $\lambda ={\frac {y_{P}+y_{Q}}{x_{P}+x_{Q}}}$

Certicom's Online ECC Tutorial contains a Java applet that can be used to experiment with addition in different EC groups.

We already described the underlying field $\mathbb {F} _{q}$ and the group of points of elliptic curve $E(\mathbb {F} _{q})$ but there is yet another mathematical structure commonly used in cryptography — a cyclic subgroup of $E(\mathbb {F} _{q})$ . For any point $G$ the set $(O,G,G+G,G+G+G,G+G+G+G,\ldots )$ is a cyclic group. It is convenient to use the following notation: $0G=O$ , $1G=G$ , $2G=G+G$ , $3G=G+G+G$ , et cetera. The calculation of $kG$ , where $k$ is an integer and $G$ is a point, is called scalar multiplication.

Cryptographic schemes

Since the the (additive) cyclic group described above can be considered similar to the (multiplicative) group of powers of an integer $g$ modulo prime $p$ : $(g^{0},g,g^{2},g^{3},g^{4},\ldots )$ , the problem of finding $k$ given points $kG$ and $G$ is called elliptic curve discrete logarithm problem (ECDLP). The assumed hardness of several problems related to the discrete logarithm in the subgroup of $E(\mathbb {F} _{q})$ allows cryptographic use of elliptic curves. Most of the elliptic curve cryptographic schemes are related to the discrete logarithm schemes which were originally formulated for usual modular arithmetic:

the Elliptic Curve Diffie-Hellman key agreement scheme is based on the Diffie-Hellman scheme,
the Elliptic Curve Digital Signature Algorithm is based on the Digital Signature Algorithm,
the ECMQV key agreement scheme is based on the MQV key agreement scheme.

Not all the DLP schemes should be ported to the elliptic curve domain. For example, the well known ElGamal encryption scheme was never standardized by official bodies and should not be directly used over an elliptic curve (the standard encryption scheme for ECC is called Elliptic Curve Integrated Encryption Scheme). The main reason is that although it is straightforward to convert an arbitrary message (of limited length) to an integer modulo $p$ , it is not that simple to convert a bitstring to a point of a curve (note for every $x_{0}$ there is an $y$ such that $(x,y)\in E(\mathbb {F} _{q})$ ). (Another factor is that ElGamal scheme is vulnerable to chosen-ciphertext attacks.)

There is a vision that ECDLP-based cryptography is going to replace cryptography based on integer factorization (e.g., RSA) and finite-field cryptography (e.g., DSA). At the RSA Conference 2005 the National Security Agency (NSA) announced Suite B which exclusively uses ECC for digital signature generation and key exchange. The suite is intended to protect both classified and unclassified national security systems and information.

Another major source of cryptographic applications of elliptic curves is bilinear operator (based on the Weil pairing or the Tate pairing) which allows, for example, to make efficient ID-based cryptography (see also The Pairing-Based Crypto Lounge).

Implementation considerations

Although the details of each particular elliptic curve scheme are described in the article referenced above some common implementation considerations are discussed here.

Domain parameters

To use ECC all parties must agree on all the elements defining the elliptic curve, that is domain parameters of the scheme. The field is defined by $p$ in the prime case and the pair of $m$ and $f$ in the binary case. The elliptic curve is defined by the constants $a$ and $b$ used in its defining equation. Finally, the cyclic subgroup is defined by its generator (aka. base point) $G$ . For cryptographic application the order of $G$ , that is the smallest non-negative number $n$ such that $nG=O$ , must be prime. Since $n$ is the size of a subgroup of $E(\mathbb {F} _{q})$ it follows from the Lagrange's theorem that the number $h={\frac {|E|}{n}}$ is integer. In cryptographic applications this number $h$ , called cofactor, at least must be small ( $h\leq 4$ ) and, preferably, $h=1$ . Let us summarize: in the prime case the domain parameters are $(p,a,b,G,n,h)$ and in the binary case they are $(m,f,a,b,G,n,h)$ .

Unless there is an assurance that domain parameters were generated by a party trusted with respect to their use, the domain parameters must be validated before use.

The generation of domain parameters is not usually done by each participant since this involves counting the number of points on a curve which is time-consuming and troublesome to implement. As a result several standard bodies published domain parameters of elliptic curves for several common field sizes:

Test vectors are also available [3].

If one (despite the said above) wants to build his own domain parameters he should select the underlying field and then use one of the following strategies to find a curve with appropriate (i.e., near prime) number of points using one of the following methods:

select a random curve and use a general point-counting algorithm, for example, Schoof's algorithm or Schoof-Elkies-Atkin algorithm,
select a random curve from a family which allows easy calculation of the number of points (e.g., Koblitz curves), or
select the number of points and generate a curve with this number of points using complex multiplication technique ^[4].

Several classes of curves are weak and shall be avoided:

curves over $\mathbb {F} _{2^{m}}$ with non-prime $m$ are vulnerable to Weil descent attacks ^[5], ^[6].
curves such that $n$ divides $q^{B}-1$ for small $B$ (say, $B<20$ ) are vulnerable to MOV attack ^[7] which applies usual DLP in a small degree extension field of $\mathbb {F} _{q}$ to solve ECDLP,
curves such that $|E(\mathbb {F} _{q})|=q$ are vulnerable to the attack that maps the points on the curve to the additive group of $\mathbb {F} _{q}$ ^[8], ^[9], ^[10].

Key sizes

Since all the fastest known algorithms that allow to solve the ECDLP (baby-step giant-step, Pollard's rho, etc.), need $O({\sqrt {n}})$ steps, it follows that the size of the underlying field shall be roughly twice the security parameter. For example, for 128-bit security one needs a curve over $\mathbb {F} _{q}$ , where $q\approx 2^{256}$ . This can be contrasted with finite-field cryptography (e.g., DSA) which requires^[11] 3072-bit public keys and 256-bit private keys, and integer factorization cryptography (e.g., RSA) which requires 3072-bit public and private keys. The hardest ECC scheme (publicly) broken to date had a 109-bit key (that is about 55 bits of security). It was broken near the beginning of 2003 using over 10,000 Pentium class PCs running continuously for over 540 days (see [12]).

Projective coordinates

A close examination of the addition rules shows that in order to add two points one needs not only several additions and multiplications in $\mathbb {F} _{q}$ but also an inversion operation. The inversion (for given $x\in \mathbb {F} _{q}$ find $y\in \mathbb {F} _{q}$ such that $xy=1$ ) is one to two orders of magnitude slower ^[13] than multiplication. Fortunately, points on a curve can be represented in different coordinate systems which do not require an inversion operation to add two points. Several such systems were proposed: in the projective system each point is represented by three coordinates $(X,Y,Z)$ using the following relation: $x={\frac {X}{Z}}$ , $y={\frac {y}{Z}}$ ; in the Jacobian system a point is also represented with three coordinates $(X,Y,Z)$ , but a different relation is used: $x={\frac {X}{Z^{2}}}$ , $y={\frac {y}{Z^{3}}}$ ; in the modified Jacobian system the same relations are used but four coordinates are stored and used for calculations $(X,Y,Z,aZ^{4})$ ; and in the Chudnovsky Jacobian system five coordinates are used $(X,Y,Z,Z^{2},Z^{3})$ . Note that there are may be different naming conventions, for example, IEEE P1363-2000 standard uses "projective coordinates" to refer to what is commonly called Jacobian coordinates. An additional speed-up is possible if mixed coordinates are used ^[14].

Fast reduction (NIST curves)

Reduction modulo $p$ (which is needed for addition and multiplication) can be executed much faster if the prime $p$ is a pseudo-Mersenne prime that is $p\approx 2^{d}$ , for example, $p=2^{521}-1$ or $p=2^{256}-2^{32}-2^{9}-2^{8}-2^{7}-2^{6}-2^{4}-1$ . Compared to Barrett reduction there can be an order of magnitude speed-up ^[15]. The curves over $\mathbb {F} _{p}$ with pseudo-Mersenne $p$ are recommended by NIST. Yet another advantage of the NIST curves is the fact that they use $a=-3$ which improves addition in Jacobian coordinates.

Side-channel attacks

Unlike DLP systems (where it is possible to use the same procedure for squaring and multiplication) the EC addition is significantly different for doubling ( $P=Q$ ) and general addition ( $P\neq Q$ ). Consequently, it is important to counteract side channel attacks (e.g., timing and simple power analysis attacks) using, for example, fixed pattern window (aka. comb) methods ^[16] (note that this does not increase the computation time).

Patents

Most of ECC (e.g., ECDH, ECIES, ECDSA) is not encumbered by patents whereas some other schemes (ECMQV) and some implementation techniques are covered. See ECC patents for details.

Open-source implementations

References

^ N. Koblitz, Elliptic curve cryptosystems, in Mathematics of Computation 48, 1987, pp. 203–209
^ V. Miller, Use of elliptic curves in cryptography, CRYPTO 85, 1985.
^ G. Lay and H. Zimmer, Constructing elliptic curves with given group order over large finite fields, Algorithmic Number Theory Symposium, 1994.
^ NIST, Recommendation for Key Management — Part 1: general, Special Publication 800-57, August 2005.
^ Y. Hitchcock, E. Dawson, A. Clark, and P. Montague, Implementing an efficient elliptic curve cryptosystem over GF(p) on a smart card, 2002.
^ S.D. Galbraith and N.P. Smart, A cryptographic application of the Weil descent, Cryptography and Coding, 1999.
^ P. Gaudry, F. Hess, and N.P. Smart, Constructive and destructive facets of Weil descent on elliptic curves, Hewlett Packard Laboratories Technical Report, 2000.
^ I. Semaev, Evaluation of discrete logarithm in a group of P-torsion points of an elliptic curve in characteristic P, Mathematics of Computation, number 67, 1998.
^ N. Smart, The discrete logarithm problem on elliptic curves of trace one, Journal of Cryptology, Volume 12, 1999.
^ T. Satoh and K. Araki, Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves, Commentarii Mathematici Universitatis Sancti Pauli, Volume 47, 1998.
^ A. Menezes, T. Okamoto, and S.A. Vanstone, Reducing elliptic curve logarithms to logarithms in a finite field, IEEE Transactions on Information Theory, Volume 39, 1993.
^ H. Cohen, A. Miyaji, T. Ono, Efficient Elliptic Curve Exponentiation Using Mixed Coordinates, ASIACRYPT 1998.
^ M. Brown, D. Hankerson, J. Lopez, and A. Menezes, Software Implementation of the NIST Elliptic Curves Over Prime Fields.
^ M. Hedabou, P. Pinel, and L. Beneteau, A comb method to render ECC resistant against Side Channel Attacks, 2004.