Schoof's algorithm: Difference between revisions

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{{Short description|Efficient algorithm to count points on elliptic curves}}
'''Schoof's algorithm''' is an efficient algorithm to count points on [[elliptic curvescurve]]s over [[finite fields]]. The algorithm has applications in [[elliptic curve cryptography]] where it is important to know the number of points to judge the difficulty of solving the [[discrete logarithm problem]] in the [[Group (mathematics)|group]] of points on an elliptic curve.

The algorithm was published by [[René Schoof]] in 1985 and it was a theoretical breakthrough, as it was the first deterministic polynomial time algorithm for [[counting points on elliptic curves]]. Before Schoof's algorithm, approaches to counting points on elliptic curves such as the naive and [[baby-step giant-step]] algorithms were, for the most part, tedious and had an exponential running time.

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with <math>A,B\in \mathbb{F}_{q}</math>. The set of points defined over <math>\mathbb{F}_{q}</math> consists of the solutions <math>(a,b)\in\mathbb{F}_{q}^2</math> satisfying the curve equation and a [[point at infinity]] <math>O</math>. Using the [[Elliptic curve#The group law|group law]] on elliptic curves restricted to this set one can see that this set <math>E(\mathbb{F}_{q})</math> forms an [[abelian group]], with <math>O</math> acting as the zero element.

In order to count points on an elliptic curve, we compute the cardinality of <math>E(\mathbb{F}_{q})</math>.

Schoof's approach to computing the cardinality <math>\sharp# E(\mathbb{F}_{q})</math> makes use of [[Hasse's theorem on elliptic curves]] along with the [[Chinese remainder theorem]] and [[division polynomials]].

==Hasse's theorem==

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</math>

This powerful result, given by Hasse in 1934, simplifies our problem by narrowing down <math>\# E(\mathbb{F}_{q})</math> to a finite (albeit large) set of possibilities. Defining <math>t</math> to be <math>q + 1 - \# E(\mathbb{F}_{q})</math>, and making use of this result, we now have that computing the cardinalityvalue of <math>t</math> modulo <math>N</math> where <math>N > 4\sqrt{q}</math>, is sufficient for determining <math>t</math>, and thus <math>\# E(\mathbb{F}_{q})</math>. While there is no efficient way to compute <math>t \pmod N</math> directly for general <math>N</math>, it is possible to compute <math>t \pmod l</math> for <math>l</math> a small prime, rather efficiently. We choose <math>S=\{l_1,l_2,...,l_r\}</math> to be a set of distinct primes such that <math>\prod l_i = N > 4\sqrt{q}</math>. Given <math>t \pmod {l_i}</math> for all <math>l_i\in S</math>, the [[Chinese remainder theorem]] allows us to compute <math>t \pmod N</math>.

In order to compute <math>t \pmod l</math> for a prime <math>l \neq p</math>, we make use of the theory of the Frobenius endomorphism <math>\phi</math> and [[division polynomials]]. Note that considering primes <math>l \neq p</math> is no loss since we can always pick a bigger prime to take its place to ensure the product is big enough. In any case Schoof's algorithm is most frequently used in addressing the case <math>q=p</math> since there are more efficient, so called <math>p</math> adic algorithms for small-characteristic fields.

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Given the elliptic curve <math>E</math> defined over <math>\mathbb{F}_{q}</math> we consider points on <math>E</math> over <math>\bar{\mathbb{F}}_{q}</math>, the [[algebraic closure]] of <math>\mathbb{F}_{q}</math>; i.e. we allow points with coordinates in <math>\bar{\mathbb{F}}_{q}</math>. The [[Frobenius endomorphism]] of <math>\bar{\mathbb{F}}_{q}</math> over <math>\mathbb{F}_q</math> extends to the elliptic curve by <math> \phi : (x, y) \mapsto (x^{q}, y^{q})</math>.

This map is the identity on <math>E(\mathbb{F}_{q})</math> and one can extend it to the point at infinity <math>O</math>, making it a [[group morphism]] from <math>E(\bar{\mathbb{F}}_{q}})</math> to itself.

The Frobenius endomorphism satisfies a quadratic polynomial which is linked to the cardinality of <math>E(\mathbb{F}_{q})</math> by the following theorem:

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We must split the problem into two cases: the case in which <math>(x^{q^{2}}, y^{q^{2}}) \neq \pm \bar{q}(x, y)</math>, and the case in which <math>(x^{q^{2}}, y^{q^{2}}) = \pm \bar{q}(x, y)</math>. Note that these equalities are checked modulo <math>\psi_l</math>.

===Case 1: <math>(x^{q^{2===}}, y^{q^{2}}) \neq \pm \bar{q}(x, y)</math>}}===

By using the [[Elliptic curves#The group law|addition formula]] for the group <math>E(\mathbb{F}_{q})</math> we obtain:

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: <math>

(x^3+Ax+B)((x^3+Ax+B)^{\frac{q^{2}-1}{2}}-\theta(x))^2

</math>

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: <math>

X(x)\equiv (x^3+Ax+B)\left(\frac{(x^3+Ax+B)^{\frac{q^{2}-1}{2}}-\theta(x)}{x^{q^2}-x_{\bar{q}}}\right)^2\bmod \psi_l(x).

</math>

Here, it seems not right, we throw away <math>x^{q^{2}}-x_{\bar{q}}</math>?

Now if <math>X \equiv x^{q} _ {\bar{t}}\bmod \psi_l(x)</math> for onesome <math>\bar{t}\in [0,(l-1)/2]</math>, then <math>\bar{t}</math> satisfies

: <math>

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As mentioned earlier, using {{mvar|Y}} and <math>y_{\bar{t}}^{q}</math> we are now able to determine which of the two values of <math>\bar{t}</math> (<math>\bar{t}</math> or <math>-\bar{t}</math>) works. This gives the value of <math>t\equiv \bar{t}\pmod l</math>. Schoof's algorithm stores the values of <math>\bar{t}\pmod l</math> in a variable <math>t_l</math> for each prime {{mvar|l}} considered.

===Case 2: <math>(x^{q^{2===}}, y^{q^{2}}) = \pm \bar{q}(x, y)</math>}}===

We begin with the assumption that <math>(x^{q^{2}}, y^{q^{2}}) = \bar{q}(x, y)</math>. Since {{mvar|l}} is an odd prime it cannot be that <math>\bar{q}(x, y)=-\bar{q}(x, y)</math> and thus <math>\bar{t}\neq 0</math>. The characteristic equation yields that <math>\bar{t} \phi(P) = 2\bar{q} P</math>. And consequently that <math>\bar{t}^{2}\bar{q} \equiv (2q)^{2} \pmod l</math>.

This implies that {{mvar|q}} is a square modulo {{mvar|l}}. Let <math>q \equiv w^{2} \pmod l</math>. Compute <math>w\phi(x,y)</math> in <math>\mathbb{F}_{q}[x,y]/(y^{2}-x^{3}-Ax-B, \psi_{l})</math> and check whether <math>

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==Implementations==

Several algorithms were implemented in [[C++]] by Mike Scott and are available with [ftp://ftp.computing.dcu.ie/pub/crypto/ source code]. The implementations are free (no terms, no conditions), and make use of the [https://web.archive.org/web/20121114015633/http://certivoxgithub.com/solutions/miracl-crypto-sdk/MIRACL MIRACL] library which is distributed under the [[AGPLv3]].

* Schoof's algorithm [ftphttps://ftpgithub.computing.dcu.iecom/miracl/MIRACL/blob/master/pubsource/cryptocurve/schoof.cpp implementation] for <math>E(\mathbb{F}_p)</math> with prime <math>p</math>.

* Schoof's algorithm [ftphttps://ftpgithub.computing.dcu.iecom/miracl/MIRACL/blob/master/pubsource/cryptocurve/schoof2.cpp implementation] for <math>E(\mathbb{F}_{2^m})</math>.

==See also==

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{{Number-theoretic algorithms}}

{{Algebraic curves navbox}}

[[Category:Asymmetric-key algorithms]]