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elliptic curve cryptography

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Definition
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I thought I had correctly adapted the whole thing, but roughly the second half triggers an "unexpected error." I miss the simple LaTeX errors. But just in case a pair of younger eyes can detect what I did wrong, here's that second half:

$P + Q = Q+P$ (commutativity)

$(P + Q) + R = P + (Q + R)$ (associativity)
$P + O = O + P = P$ (existence of an identity element)

There exists $(-P)$ such that $-P + P = P + (-P) = O$ (existence of inverses)

We already specified how $O$ is defined. If we define the negative of a point $P = (x, y)$ to be $-P = (x, -y)$ for $P \in E(\mathbb{F}_p)$ and $-P = (x, x+y)$ for $P \in E(\mathbb{F}_{2^m})$, we can define the addition operation as follows:

if $Q = O$ then $P + Q = P$

if $Q = -P$ then $P + Q = O$

if $Q \ne P$ then $P + Q = R$, 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 $x_R = \lambda^2 + \lambda + x_P + x_Q + a$, $y_R = \lambda (x_P + x_R) + x_R + y_P$, and $\lambda = \frac{y_P + y_Q}{x_P + x_Q}$ (Geometrically, $P + Q$ is the inverse of the third point of intersection of the cubic with the line through $P$ and $Q$.)

If $Q = P$ then $P + Q = R$, where in the prime case $x_R = \lambda^2 - 2 x_P$, $y_R = \lambda(x_P - x_R) - y_P$, and $\lambda = \frac{3 x_P^2 + a}{2 y_P}$, or
in the binary case $x_R = \lambda^2 + \lambda + a$, $y_R = x_P^2 + (\lambda + 1) x_R$, and $\lambda = x_P + \frac{y_P}{x_P}$
(Geometrically, $2P$ is the inverse of the third point of intersection of the cubic with its tangent line at $P$.)

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: $0 G = O$, $1 G = G$, $2G = G+G$, $3G = G+G+G$, etc. The calculation of $k G$, where $k$ is an integer and $G$ is a point, is called ''scalar multiplication''.

Since 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 $k G$ and $G$ is called the 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 (not for every $x$ 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.)

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 or base point $G$. For cryptographic application the order of $G$, that is the smallest non-negative number $n$ such that $n G = O$, 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 ($h \le 4$) 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. % TODO: Validation procedure

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, NIST and SECG being the most important.

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.<ref>G. Lay and H. Zimmer, ''Constructing elliptic curves with given group order over large finite fields, '' Algorithmic Number Theory Symposium, 1994.</ref>

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.

Curves such that $n$ divides $p^B-1$ (where $p$ is the characteristic of the field – $q$ for a prime field, or $2$ for a binary field) for sufficiently small $B$ are vulnerable to MOV attack which applies usual DLP in a small degree extension field of $\mathbb{F}_p$ to solve ECDLP. The bound $B$ should be chosen so that discrete logarithms in the field $\mathbb{F}_{p^B}$ are at least as difficult to compute as discrete logs on the elliptic curve $E(\mathbb{F}_q)$.

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 $\mathbb{F}_q$

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 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). For the prime field case, it was broken near the beginning of 2003 using over 10, 000 Pentium class PCs running continuously for over 540 days . For the binary field case, it was broken in April 2004 using 2600 computers for 17 months.

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 $x y = 1$) is one to two orders of magnitude slower 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.

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. 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.

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 ($P \ne Q$). 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 (note that this does not increase the computation time).

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.

\begin{thebibliography}{7}
\bibitem{nk} N. Koblitz, ``Elliptic curve cryptosystems'', in {\it Mathematics of Computation} {\bf 48}, (1987): 203 - 209
\bibitem{vm} V. Miller, ``Use of elliptic curves in cryptography'', {\it CRYPTO 85} (1985)
\bibitem{sg} S.D. Galbraith \& N.P. Smart, ``A cryptographic application of the Weil descent'', {\it Cryptography and Coding} (1999)
\bibitem{am} A. Menezes, T. Okamoto, \& S.A. Vanstone, ``Reducing elliptic curve logarithms to logarithms in a finite field'', {\it IEEE Transactions on Information Theory} {\bf 39} (1993)
\bibitem{is} I. Semaev, ``Evaluation of discrete logarithm in a group of P-torsion points of an elliptic curve in characteristic $P$'', {\it Mathematics of Computation}, {\bf 67} (1998)
\end{thebibliography}

Hm. What you posted compiles if I paste into to emacs. Some times the "unexpected error" shows up in the document nowhere near where the error actually occurred, so maybe you should re-examine the first half.

> I thought I had correctly adapted the whole thing, but roughly the
> second half triggers an "unexpected error." I miss the simple LaTeX
> errors.

I made a temporary duplicate entry (including both
halves of the code) and couldn't reproduce the error.
I suspect it was a fluke. Perhaps you could try adding
in the second half again?

I've added the second half. I suspect the error was caused by an 'undigested' minus sign (or en dash). I was able to get a PM preview with no error messages, only that the dash in the paragraph "curves such that n divides p^B − 1 (where p is the characteristic of the field – q for a prime field ..." was being read as three letters A with assorted accents.

I got this same kind of error once with one of my biographical entries when I copied the birth/death dates from Wikipedia (which of course matched the ones in a book), the innocent dash somehow got garbled up.

Ah, that seems plausible. When I pasted it into emacs, I had to get fix some characters that got bungled in the translation. By doing that, I might have deleted the error I was searching for.

Cam

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