Schoof's algorithm: Difference between revisions

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{{Short description|Efficient algorithm to count points on elliptic curves}}

{{Underconstruction}}

'''Schoof's algorithm''' is an efficient algorithm to count points on [[elliptic curve]]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.

'''Schoof's Algorithm''' is an important tool of [[Elliptic Curve Cryptography]] that provides us with an efficient way of counting points on [[elliptic curves]]. Published as a paper by René Schoof in 1985, the algorithm was a theoretical break through, as it was the first deterministic polynomial time algorithm.

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.

This article explains Schoof's approach, laying emphasis on the mathematical ideas underlying the structure of the algorithm.

==Introduction==

InLet order<math>E</math> tobe count points onan [[elliptic curve,]] wedefined computeover the cardinalityfinite offield <math>E(\mathbb{F}_{q})</math>, thewhere group<math>q=p^n</math> offor an<math>p</math> elliptica curveprime and <math>E(\mathbb{F}_{q})n</math> overan a [[finite field]]integer <math>\mathbb{F}_{q}geq 1</math>. Over a field of [[characteristic]] <math>p\neq 2, 3</math>. an elliptic curve can be given by a (short) Weierstrass equation

: <math>

Schoof's approach to compute the cardinality <math>\sharp E(\mathbb{F}_{q})</math> makes use of [[Hasse's theorem]] along with the [[Chinese Remainder Theorem]] and [[Division Polynomials]].

y^2 = x^3 + Ax + B

We treat an elliptic curve <math>E(\mathbb{F}_{q})</math> as the zero locus of an algebraic equation of a special form, commonly known as the Weierstrass form. Over a field of characteristic <math>\neq 2, 3</math>, this equation can be written as

<math>

y^{2} = x^{3} + Ax + B.

</math>

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>\# 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==

The set of points defined over <math>\mathbb{F}_{q}</math> satisfying this equation, together with an additional `point at infinity' <math>O</math>, form the [[abelian group]] <math>E(\mathbb{F}_{q})</math>, with <math>O</math> acting as the zero element.

{{main|Hasse's theorem on elliptic curves}}

Hasse's theorem states that if <math>E/\mathbb{F}_{q}</math> is an elliptic curve over the finite field <math>\mathbb{F}_{q}</math>, then <math>\# E(\mathbb{F}_q)</math> satisfies

==Hasse's Theorem==

[[Hasse's theorem]] states that if <math>E/\mathbb{F}_{q}</math> is an elliptic curve over the finite field <math>\mathbb{F}_{q}</math>, then <math>\sharp E(\mathbb{F}_{q})</math> satisfies

: <math>

\mid q + 1 - \sharp# E(\mathbb{F}_{q}) \mid \leq 2\sqrt{q}.

</math>

This powerful result, given by Hasse in 1934, simplifies our problem by narrowing down <math>\sharp# E(\mathbb{F}_{q})</math> to a finite (albeit large) set of possibilities. Defining <math>t</math> to be <math>q + 1 - \sharp# 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>\sharp# 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 ll_i = N > 4\sqrt{q}</math>. Given <math>t \pmod l{l_i}</math> for all <math>l l_i\in S</math>, the [[Chinese Remainderremainder Theoremtheorem]] 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[[division Polynomialspolynomials]]. 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.

==The Frobenius Endomorphismendomorphism==

Given the elliptic curve <math>E(</math> defined over <math>\mathbb{F}_{q})</math> we consider points on <math>E</math> toover be<math>\bar{\mathbb{F}}_{q}</math>, the curve[[algebraic withclosure]] same defining equation asof <math>E(\mathbb{F}_{q})</math>,; buti.e. nowwe allowingallow points with coordinates in <math>\overlinebar{\mathbb{F}}_{q}}</math>,. theThe [[algebraicFrobenius closureendomorphism]] of <math>\bar{\mathbb{F}}_{q}</math>. Given curveover <math>E\mathbb{F}_q</math>, thereextends existsto athe mapelliptic (thecurve [[Frobeniusby endomorphism]]<math> \phi : (x, y) given\mapsto by(x^{q}, y^{q})</math>.

<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:

'''Theorem:''' The Frobenius endomorphism given by <math>\phi</math> satisfies the characteristic equation

<math> \phi ^{2} - t\phi + q = 0 </math> where <math> t = q + 1 - \sharp E(\mathbb{F}_{q}) </math>

: <math> \phi ^2 - t\phi + q = 0,</math> where <math> t = q + 1 - \# E(\mathbb{F}_q) </math>

Thus we have for all <math>P=(x, y) \in E</math>, that <math>(x^{q^{2}}, y^{q^{2}} ) + q(x, y) = t(x^{q}, y^{q})</math>, where + denotes addition on the elliptic curve and <math>q(x,y)</math> and <math>t(x^{q},y^{q})</math>

denote scalar multiplication of <math>(x,y)</math> by <math>q</math> and of <math>(x^{q},y^{q})</math> by <math>t</math>.

Fixing <math>l</math>, we now move on to solving the problem of determining <math>t_{l}</math>, defined as <math>t \pmod l</math>, for a given prime <math>l \neq 2, p</math>.

If a point <math>(x, y)</math> is in the [[torsion subgroup]] <math>E[l]=\{P\in E(\bar{\mathbb{F}_{q}}) \mid lP=O \}</math>, then <math>qP = \bar{q}P</math> where <math>\bar{q}</math> is the unique integer such that <math>q \equiv \bar{q} \pmod l</math> and <math>\mid \bar{q} \mid< l/2</math>.

Being a group morphism, we have that <math>\phi(O) = O</math> and if <math>r</math> is an integer, then <math>r\phi (P) = \phi (rP)</math>. Thus <math>\phi (P)</math> will have the same order as <math>P</math>. Thus for <math>(x, y)</math> belonging to <math>E[l]</math>, we also have <math>t(x^{q}, y^{q})= \bar{t}(x^{q}, y^{q})</math> if <math>t \equiv \bar{t} \pmod l</math>. Hence we have reduced our problem to solving the equation

One could try to symbolically compute these points <math> (x^{q^{2}}, y^{q^{2}})</math>, +<math>(x^{q}, \bary^{q})</math> and <math>q(x, y)</math> \equivas functions in the [[Imaginary hyperelliptic curve#Coordinate ring|coordinate ring]] <math>\barmathbb{tF}(x^_{q}[x, y]/(y^{q2}-x^{3}-Ax-B)</math> of \pmod l <math>E</math>

and then search for a value of <math>t</math> which satisfies the equation. However, the degrees get very large and this approach is impractical.

Schoof's idea was to carry out this computation restricted to points of order <math>l</math> for various small primes <math>l</math>.

Fixing an odd prime <math>l</math>, we now move on to solving the problem of determining <math>t_{l}</math>, defined as <math>t \pmod l</math>, for a given prime <math>l \neq 2, p</math>.

If a point <math>(x, y)</math> is in the <math>l</math>-[[torsion subgroup]] <math>E[l]=\{P\in E(\bar{\mathbb{F}_{q}}) \mid lP=O \}</math>, then <math>qP = \bar{q}P</math> where <math>\bar{q}</math> is the unique integer such that <math>q \equiv \bar{q} \pmod l</math> and <math>\mid \bar{q} \mid< l/2</math>.

Note that <math>\phi(O) = O</math> and that for any integer <math>r</math> we have <math>r\phi (P) = \phi (rP)</math>. Thus <math>\phi (P)</math> will have the same order as <math>P</math>. Thus for <math>(x, y)</math> belonging to <math>E[l]</math>, we also have <math>t(x^{q}, y^{q})= \bar{t}(x^{q}, y^{q})</math> if <math>t \equiv \bar{t} \pmod l</math>. Hence we have reduced our problem to solving the equation

: <math> (x^{q^{2}}, y^{q^{2}}) + \bar{q}(x, y) \equiv \bar{t}(x^{q}, y^{q}),</math>

where <math>\bar{t}</math> and <math>\bar{q}</math> have integer values in <math>[-(l-1)/2,(l-1)/2]</math>.

==Computation modulo primes==

The {{mvar|l}}th [[division polynomial]] is such that its roots are precisely the {{mvar|x}} coordinates of points of order {{mvar|l}}. Thus, to restrict the computation of <math>(x^{q^{2}}, y^{q^{2}}) + \bar{q}(x, y)</math> to the {{mvar|l}}-torsion points means computing these expressions as functions in the coordinate ring of {{mvar|E}} ''and'' modulo the {{mvar|l}}th division polynomial. I.e. we are working in <math>\mathbb{F}_{q}[x,y]/(y^{2}-x^{3}-Ax-B, \psi_{l})</math>. This means in particular that the degree of {{mvar|X}} and {{mvar|Y}} defined via <math>(X(x,y),Y(x,y)):=(x^{q^{2}}, y^{q^{2}}) + \bar{q}(x, y)</math> is at most 1 in {{mvar|y}} and at most <math>(l^2-3)/2</math>

We must now compute <math>(x^{q^{2}}, y^{q^{2}}) + \bar{q}(x, y)</math> as a pair of rational functions <math>(x', y'):=(x^{q^{2}}, y^{q^{2}}) + \bar{q}(x, y)</math> in terms of <math>x</math> and <math>y</math> (remember that we are working modulo <math>l</math>). We do so by using [[Division Polynomials]], which lie at the heart of the remaining part of Schoof's algorithm.

in {{mvar|x}}.

The scalar multiplication <math>\bar{q}(x, y)</math> can be done either by [[exponentiation by squaring|double-and-add]] methods or by using the <math>\bar{q}</math>th division polynomial. The latter approach gives:

We obtain:

: <math>

\bar{q} (x,y) = (x_{\bar{q}},y_{\bar{q}}) = \left( x - \frac {\psi_{\bar{q}-1} \psi_{\bar{q}+1}}{\psi^{2}_{\bar{q}}}, \frac{\psi_{2\bar{q}}}{2\psi^{4}_{\bar{q}}} \right)

</math>

where <math>\psi_{n}</math>, a function of <math>x</math> and <math>y</math>, is the <math>{{mvar|n</math>}}th division polynomial. Note that

By replacing <math>y^{2}</math> by <math>x^3+Ax+B</math>, we have that <math>\theta(x)=y_{\bar{q}}/y</math> is ana expressionfunction in {{mvar|x}} only and denote it by <math>\theta(x)</math>.

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:

: <math>

X(x',y) = \left( \frac{y^{q^{2}} - y_{\bar{q}}}{x^{q^{2}} - x_{\bar{q}}} \right) ^{2} - x^{q^{2}} - x_{\bar{q}}.

</math>

Note that this computation fails in case the assumption of inequality was wrong.

We are now able to use the <math>{{mvar|x</math>}}-coordinate to narrow down the choice of <math>\bar{t}</math> to two possibilities, namely the positive and negative case. Using the <math>{{mvar|y</math>}}-coordinate one later determines which of the two cases holds.

We first show that <math>x'</math>{{mvar|X}} is a function in <math>{{mvar|x</math>}} alone. Consider <math>(y^{q^{2}} - y_{\bar{q}})^{2}=y^{2}(y^{q^{2}-1}-y_{\bar{q}}/y)^{2}</math>.

Since <math>q^{2}-1</math> is even, by replacing <math>y^{2}</math> by <math>x^3+Ax+B</math>, we rewrite the expression as

<math>

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

</math>

Note that such a substitution would mean that we are working in the [[quotient ring]] <math>\mathbb{F}_{q}[x,y]/(y^{2}-x^{3}-Ax-B)</math>.

: <math>

Now if <math>x' = x^{q} _ {\bar{t}}</math> for one point <math>P=(x,y)</math> in <math>E[l] \setminus \{O\} </math> then <math>\bar{t}</math> satisfies

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

<math>

\phi ^{2}(P) - \bar{t} \phi(P) + qP = O

</math>

implying that <math>x' = x^{q} _ {\bar{t}}</math> for all points <math>P</math> in <math>E[l] \setminus \{O\} </math> and confirming that <math>\bar{t}=t_{l}</math>.

and have that

But since the roots of the <math>l</math>-th Division Polynomial <math>\psi_{l}</math> are exactly the <math>x</math>-coordinates of the points of <math>E[l] \setminus \{O\} </math> and are simple, we can reduce our problem to solving the equation

: <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).

x' \equiv x^{q}_{\bar{t}} \pmod { \psi _{l}} </math> (where <math>\psi_{l}</math> is the <math>l</math>-th division polynomial)

</math>

ThatNow is,if we<math>X are\equiv nowx^{q} working_ in{\bar{t}}\bmod \psi_l(x)</math> thefor ringsome <math>\mathbbbar{F}_{qt}\in [x0,y]/(y^{2}l-x^{3}-Ax-B1)/2]</math>, then <math>\psi_bar{lt})</math>. satisfies

: <math>

As mentioned earlier, using the <math>y'</math> 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.

\phi ^{2}(P) \mp \bar{t} \phi(P) + \bar{q}P = O

This computation fails in case the assumption of inequality was wrong.

</math>

for all {{mvar|l}}-torsion points {{mvar|P}}.

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

We begin with the assumption that <math>(x^{q^{2}}, y^{q^{2}}) = +q(x, y)</math>.

For a point <math>P</math> in <math>E(\mathbb{F}_{q})</math> satisfying this, plugging it into the characteristic equation yields that <math>\bar{t} \phi(P) = 2\bar{q} P</math>. And consequently that <math>\bar{t}^{2}\bar{q} P \equiv (2q)^{2}P \pmod l</math>.

This implies that <math>q</math> is a square modulo <math>l</math> unless <math>\bar{t}</math> is <math>0 mod \, l</math>. We discount the latter case as it yields a contradiction. We find <math>q</math>'s square-roots over <math>\mathbb{F}_{l}</math>. If <math>q \equiv w^{2} \pmod l</math>, we find that <math>t_{l}</math> will be <math>\pm2w</math> mod <math>l</math> depending on the y-coordinate.

If <math>q</math> turns out not to be a square modulo <math>l</math>, our assumption that <math>(x^{q^{2}}, y^{q^{2}}) = +q(x, y)</math> is rendered false, forcing us to consider the case where <math>(x^{q^{2}}, y^{q^{2}}) = - q(x, y)</math>, in which case our argument asserts that <math>\bar{t}</math> is <math>0 \, ( mod \, l)</math>.

As youmentioned might noticeearlier, itusing is{{mvar|Y}} possible for <math>q</math> to be a squareand <math>y_{\pmod l</math> but <math>(x^bar{q^{2t}}, y^{q^{2}}) = -q(x, y)</math>. Inwe thisare casenow however, it sufficesable to checkdetermine whetherwhich orof notthe eithertwo squarevalues rootof <math>\pm wbar{t}</math> satisfies (<math>\phi(P) = \pm wPbar{t}</math>, foror <math>P \in E[l] \setminus -\bar{O\t}</math>,) andworks. this amountsThis togives checkingthe whethervalue or notof <math>t\gcd(numerator(x^equiv \bar{qt}-x_{w}),\phi_{pmod l})=1</math>. IfSchoof's italgorithm is <math>1</math> then we are instores the minusvalues case andof <math>t_\bar{lt}=0\pmod l</math>; ifin a variable <math>\neq 1t_l</math> wefor proceedeach asprime in the plus case with <math>t_{{mvar|l}=\pm} 2w</math>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>

\bar{q}(x, y)=w\phi(x,y)</math>. If so, <math>t_{l}</math> is <math>\pm 2w \pmod l</math> depending on the y-coordinate.

If {{mvar|q}} turns out not to be a square modulo {{mvar|l}} or if the equation does not hold for any of {{mvar|w}} and <math>-w</math>, our assumption that <math>(x^{q^{2}}, y^{q^{2}}) = +\bar{q}(x, y)</math> is false, thus <math>(x^{q^{2}}, y^{q^{2}}) = - \bar{q}(x, y)</math>. The characteristic equation gives <math>t_l=0</math>.

===Additional Case:case <math>l = 2</math>===

If you recall, our initial considerations omit the case of <math>l = 2</math>.

Since we assume <math>{{mvar|q</math>}} to be odd, <math>q + 1 - t \equiv t \pmod 2</math> and in particular, <math>t_{2} \equiv 0 \pmod 2</math> if and only if <math>E(\mathbb{F}_{q})</math> has an element of order 2. By definition of addition in the group, any element of order 2 must be of the form <math>(x_{0}, 0)</math>. Thus <math>t_{2} \equiv 0 \pmod 2</math> if and only if the polynomial <math>x^{3} + Ax + B</math> has a root in <math>\mathbb{F}_{q}</math>, if and only if <math>\gcd(x^{q}-x, x^{3} + Ax + B)\neq 1</math>.

==The algorithm==

Input:

1. An elliptic curve <math>E = y^{2}-x^{3}-Ax-B</math>.

2. An integer {{mvar|q}} for a finite field <math>F_q</math> with <math>q=p^{b}, b \ge 1</math>.

Output:

The number of points of {{mvar|E}} over <math>F_q</math>.

Choose a set of odd primes {{mvar|S}} not containing {{mvar|p}} {{nowrap|such that <math>N=\prod_{l\in S} l > 4\sqrt{q}.</math>}}

{{nowrap|'''Put''' <math>t_2=0</math> '''if''' <math>\gcd(x^{q}-x, x^{3} + Ax + B)\neq 1</math>, '''else''' <math>t_2=1</math>.}}

{{nowrap|Compute the division polynomial <math>\psi_l</math>}}.

All computations in the loop below are performed {{nowrap|in the ring <math>\mathbb{F}_{q}[x,y]/(y^{2}-x^{3}-Ax-B, \psi_{l}).</math>}}

{{nowrap|'''For''' <math>l \in S</math> '''do''':}}

{{nowrap|'''Let''' <math>\bar{q}</math> be the unique integer}} {{nowrap|such that <math>q \equiv \bar{q} \pmod l</math> and <math>\mid \bar{q} \mid< l/2</math>.}}

{{nowrap|Compute <math>(x^{q}, y^{q})</math>, <math>(x^{q^{2}}, y^{q^{2}})</math> and <math>(x_{\bar{q}},y_{\bar{q}})</math>.}}

{{nowrap|'''if''' <math>x^{q^{2}}\neq x_{\bar{q}}</math> '''then'''}}

{{nowrap|Compute <math>(X,Y)</math>.}}

{{nowrap|'''for''' <math>1\leq \bar{t} \leq (l-1)/2</math> '''do''':}}

{{nowrap|'''if''' <math>X = x^{q} _ {\bar{t}}</math> '''then'''}}

{{nowrap|'''if''' <math>Y = y^{q} _ {\bar{t}}</math> '''then'''}}

{{nowrap|<math>t_{l}=\bar{t}</math>;}}

'''else'''

{{nowrap|<math>t_{l}=-\bar{t}</math>.}}

'''else if''' {{mvar|q}} is a square modulo {{mvar|l}} '''then'''

{{nowrap|compute {{mvar|w}} with <math>q\equiv w^{2} \pmod l</math>}}

{{nowrap|compute <math>w(x^{q}, y^{q})</math>}}

{{nowrap|'''if''' <math>w(x^{q}, y^{q})=(x^{q^{2}}, y^{q^{2}})</math> '''then'''}}

{{nowrap|<math>t_l=2w</math>}}

{{nowrap|'''else if''' <math>w(x^{q}, y^{q})=(x^{q^{2}}, -y^{q^{2}})</math> '''then'''}}

{{nowrap|<math>t_l=-2w</math>}}

'''else'''

{{nowrap|<math>t_{l}=0</math>}}

'''else'''

{{nowrap|<math>t_{l}=0</math>}}

Use the [[Chinese Remainder Theorem]] to compute {{mvar|t}} modulo {{mvar|N}}

from the equations <math>t \equiv t_{l} \pmod l</math>, where <math>l \in S</math>.

Output <math>q+1-t</math>.

==The AlgorithmComplexity==

Most of the computation is taken by the evaluation of <math>\phi(P)</math> and <math>\phi^{2}(P)</math>, for each prime <math>l</math>, that is computing <math>x^q</math>, <math>y^q</math>, <math>x^{q^2}</math>, <math>y^{q^2}</math> for each prime <math>l</math>. This involves exponentiation in the ring <math>R = \mathbb{F}_{q}[x, y]/ (y^2-x^3-Ax-B, \psi_l)</math> and requires <math>O(\log q)</math> multiplications. Since the degree of <math>\psi_l</math> is <math>\frac{l^2-1}{2}</math>, each element in the ring is a polynomial of degree <math>O(l^2)</math>. By the [[prime number theorem]], there are around <math>O(\log q)</math> primes of size <math>O(\log q)</math>, giving that <math>l</math> is <math>O(\log q)</math> and we obtain that <math>O(l^2) = O(\log^2q)</math>. Thus each multiplication in the ring <math>R</math> requires <math>O(\log^4 q)</math> multiplications in <math>\mathbb{F}_{q}</math> which in turn requires <math>O(\log^2 q)</math> bit operations. In total, the number of bit operations for each prime <math>l</math> is <math>O(\log^7 q)</math>. Given that this computation needs to be carried out for each of the <math>O(\log q)</math> primes, the total complexity of Schoof's algorithm turns out to be <math>O(\log^8 q)</math>. Using fast polynomial and integer arithmetic reduces this to <math>\tilde{O}(\log^5 q)</math>.

==Improvements to Schoof's algorithm==

(1) Choose a set of primes <math>S</math>, <math>p \notin S</math> such that <math>N=\prod_{l\in S} l > 4\sqrt{q}</math>

(2) For <math>l=2</math>, <math>t_{l}=0</math> if and only if <math>\gcd(x^{q}-x, x^{3} + Ax + B)\neq 1</math>

(3) For <math>l \in S\setminus \{2\}</math> do

(a) let <math>\bar{q}</math> be the unique integer such that <math>q \equiv \bar{q} \pmod l</math> and <math>\mid \bar{q} \mid< l/2</math>

(b) compute <math>x'</math> as above.

(c) for <math>1\leq \bar{t} \leq (l-1)/2</math> do

(i) if <math>x' = x^{q} _ {\bar{t}} \pmod {\psi_{l}}</math> go to ii. . Otherwise try next value of <math>\bar{t}</math>. If you have unsuccesfully tried <math>(l-1)/2</math>, go to (d).

(ii) if <math>(y' - y^{q} _ {\bar{t}})/y \equiv 0 \pmod {\psi_{l}}</math> then <math>t_{l}=\bar{t}</math>; otherwise <math>t_{l}=-\bar{t}</math>

(d) Find <math>q</math>'s square roots modulo <math>l</math>,if they exist. If they don't exist then <math>t_{l}=0</math>. Otherwise let <math>q\equiv w^{2} \pmod l</math>

(e) If <math>\gcd(numerator(x^{q}-x_{w}), \phi_{l})=1</math> <math>t_{l}-2w</math>. Otherwise <math>t_{l}\equiv 2w</math>

(4) Use Chinese Remainder Theorem to compute <math>\sharp E(\mathbb{F}_{q}) \pmod{N}</math>.

{{main|Schoof–Elkies–Atkin algorithm}}

Note that since the set <math>S</math> was chosen so that <math>N>4\sqrt{q}</math>, by Hasse's theorem, we in fact know <math> \sharp E(\mathbb{F}_{q})</math> precisely.

==Complexity of Schoof's Algorithm==

Most of the computation is taken by the evaluation of <math>\phi(P)</math> and <math>\phi^{2}(P)</math>, for each prime <math>l</math>, that is computing <math>x^q</math>, <math>y^q</math>, <math>x^{q^2}</math>, <math>y^{q^2}</math> for each prime <math>l</math>. This involves exponentiation in the ring <math>R = \mathbb{F}_{q}[x, y]/ (y^2-x^3-Ax-B, \psi_l)</math> and requires <math>O(\log q)</math> multiplications. Since the degree of <math>\psi_l</math> is <math>\frac{l^2-1}{2}</math>, each element in the ring is a polynomial of degree <math>O(l^2)</math>, and we obtain that <math>O(l^2) = O(\log^2q)</math>. Thus each multiplication in the ring <math>R</math> requires <math>O(\log^4 q)</math> multiplications in <math>\mathbb{F}_{q}</math> which in turn requires <math>O(\log^2 q)</math> bit operations. In total, the number of bit operations for each prime <math>l</math> is <math>O(\log^7 q)</math>. By the [[prime number theorem]], we have around <math>O(\log q)</math> primes, and thus the total complexity of Schoof's algorithm turns out to be <math>O(\log^7 q) \cdot O(\log q) = O(\log^8 q)</math>.

==Improvements to Schoof's Algorithm==

In the 1990s, Elkies, followed by Atkin devised improvements to Schoof's basic algorithm by restricting the set of primes <math>S = \{l_1, \ldots, l_s\}</math> considered before to primes of a certain kind. These came to be called Elkies primes and Atkin primes respectively. A prime <math>l</math> is called an Elkies prime if the characteristic equation: <math>\phi^2-t\phi+ q = 0</math> splits over <math>\mathbb{F}_l</math>, while an Atkin prime is a prime that is not an Elkies prime. Atkin showed how to combine information obtained from the Atkin primes with the information obtained from Elkies primes to produce an efficient algorithm, which came to be known as the [[Schoof-Elkies-Atkin algorithm]]. The first problem to address is to determine whether a given prime is Elkies or Atkin. In order to do so, we make use of modular polynomials, which come from the study of [[modular forms]] and an interpretation of elliptic curves over the complex numbers as lattices. Once we have determined which case we are in, instead of using Division Polynomials, we proceed by working modulo the modular polynomials <math>f_l</math> which have a lower degree than the corresponding division polynomial <math>\psi_l</math> (degree <math>O(l)</math> rather than <math>O(l^2)</math>). This results in a further reduction in the running time, giving us an algorithm more efficient than Schoof's, with complexity <math>O(\log^6 q)</math>.

In the 1990s, [[Noam Elkies]], followed by [[A. O. L. Atkin]], devised improvements to Schoof's basic algorithm by restricting the set of primes <math>S = \{l_1, \ldots, l_s\}</math> considered before to primes of a certain kind. These came to be called Elkies primes and Atkin primes respectively. A prime <math>l</math> is called an Elkies prime if the characteristic equation: <math>\phi^2-t\phi+ q = 0</math> splits over <math>\mathbb{F}_l</math>, while an Atkin prime is a prime that is not an Elkies prime. Atkin showed how to combine information obtained from the Atkin primes with the information obtained from Elkies primes to produce an efficient algorithm, which came to be known as the [[Schoof–Elkies–Atkin algorithm]]. The first problem to address is to determine whether a given prime is Elkies or Atkin. In order to do so, we make use of modular polynomials, which come from the study of [[modular forms]] and an interpretation of [[Elliptic curve#Elliptic curves over the complex numbers|elliptic curves over the complex numbers]] as lattices. Once we have determined which case we are in, instead of using [[division polynomials]], we are able to work with a polynomial that has lower degree than the corresponding division polynomial: <math>O(l)</math> rather than <math>O(l^2)</math>. For efficient implementation, probabilistic root-finding algorithms are used, which makes this a [[Las Vegas algorithm]] rather than a deterministic algorithm.

Under the heuristic assumption that approximately half of the primes up to an <math>O(\log q)</math> bound are Elkies primes, this yields an algorithm that is more efficient than Schoof's, with an expected running time of <math>O(\log^6 q)</math> using naive arithmetic, and <math>\tilde{O}(\log^4 q)</math> using fast arithmetic. Although this heuristic assumption is known to hold for most elliptic curves, it is not known to hold in every case, even under the [[Generalized Riemann Hypothesis|GRH]].

==Implementations==

Several algorithms were implemented in [[C++]] by Mike Scott and are available with [ftp://ftp.compapp.dcu.ie/pub/crypto/ source code]. The implementations are free (no terms, no conditions), butand theymake use of the [httphttps://indigogithub.iecom/~mscottmiracl/MIRACL MIRACL] library which is onlydistributed freeunder forthe non-commercial use[[AGPLv3]]. Note that (unmodified) programs may be used to generate curves for commercial use. There are

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

==See Alsoalso==

* [[Elliptic Curve Cryptographycurve cryptography]]

* [[Counting points on elliptic curves]]

* [[Hasse's theorem]]

* [[Chinese Remainder Theorem]]

* [[Division Polynomials]]

* [[Frobenius endomorphism]]

==References==

* R. Schoof: Elliptic Curves over Finite Fields and the Computation of Square Roots mod p. Math. Comp., 44(170):483-494483–494, 1985. AvaliableAvailable at http://www.mat.uniroma2.it/~schoof/ctpts.pdf

* R. Schoof: Counting Points on Elliptic Curves over Finite Fields. J. Theor. Nombres Bordeaux 7:219-254219–254, 1995. AvaliableAvailable at http://www.mat.uniroma2.it/~schoof/ctg.pdf

* G. Musiker: Schoof's Algorithm for Counting Points on <math>E(\mathbb{F}_{q})</math>. AvaliableAvailable at http://www-.math.mitumn.edu/~musiker/schoof.pdf

* V. Müller : Die Berechnung der Punktanzahl von elliptischen kurven über endlichen Primkörpern. Master's Thesis. Universität des Saarlandes, Saarbrücken, 1991. Available at http://lecturer.ukdw.ac.id/vmueller/publications.php

* A. Brown: Algorithms for Elliptic Curves over Finite Fields, EPFL - LMA. Avaliable at http://algo.epfl.ch/handouts/en/andrew.pdf

* V. Müller : Die Berechnung der Punktanzahl von elliptischen kurven über endlichen Primkörpern. Master's Thesis. Universität des Saarlandes, Saarbrücken, 1991. Avaliable at http://lecturer.ukdw.ac.id/vmueller/publications.php

* A. Enge: Elliptic Curves and their Applications to Cryptography: An Introduction. Kluwer Academic Publishers, Dordrecht, 1999.

* L. C. Washington: Elliptic Curves: Number Theory and Cryptography. Chapman & Hall/CRC, New York, 2003.

* N. Koblitz: A Course in Number Theory and Cryptography, Graduate Texts in Math. No. 114, Springer-Verlag, 1987. Second edition, 1994

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