Schoof's algorithm
Encyclopedia
Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields. The algorithm has applications in elliptic curve cryptography
Elliptic curve cryptography
Elliptic curve cryptography is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor S...

 where it is important to know the number of points to judge the difficulty of solving the discrete logarithm problem in the group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 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
Counting points on elliptic curves
An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve. There have been several approaches to do so, and the algorithms devised have proved to be useful tools in the study of various fields such as number theory, and more recently in...

. Before Schoof's algorithm, approaches to counting points on elliptic curves such as the naive and baby-step giant-step
Baby-step giant-step
In group theory, a branch of mathematics, the baby-step giant-step is a meet-in-the-middle algorithm computing the discrete logarithm. The discrete log problem is of fundamental importance to the area of public key cryptography...

 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

Let be an elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

 defined over the finite field , where for a prime and an integer . Over a field of characteristic an elliptic curve can be given by a (short) Weierstrass equation

with . The set of points defined over consists of the solutions satisfying the curve equation and a point at infinity . Using the group law on elliptic curves restricted to this set one can see that this set forms an abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

, with acting as the zero element.
In order to count points on an elliptic curve, we compute the cardinality of .
Schoof's approach to computing the cardinality makes use of Hasse's theorem on elliptic curves along with the Chinese remainder theorem
Chinese remainder theorem
The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.In its most basic form it concerned with determining n, given the remainders generated by division of n by several numbers...

 and division polynomials
Division polynomials
In mathematics the division polynomials provide a way to calculate multiples of points on elliptic curves over Finite fields. They play a central role in the study of counting points on elliptic curves in Schoof's algorithm.- Definition :...

.

Hasse's theorem

Hasse's theorem states that if is an elliptic curve over the finite field , then satisfies


This powerful result, given by Hasse in 1934, simplifies our problem by narrowing down to a finite (albeit large) set of possibilities. Defining to be , and making use of this result, we now have that computing the cardinality of modulo where , is sufficient for determining , and thus . While there is no efficient way to compute directly for general , it is possible to compute for a small prime, rather efficiently. We choose to be a set of distinct primes such that . Given for all , the Chinese remainder theorem
Chinese remainder theorem
The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.In its most basic form it concerned with determining n, given the remainders generated by division of n by several numbers...

 allows us to compute .

In order to compute for a prime , we make use of the theory of the Frobenius endomorphism and division polynomials
Division polynomials
In mathematics the division polynomials provide a way to calculate multiples of points on elliptic curves over Finite fields. They play a central role in the study of counting points on elliptic curves in Schoof's algorithm.- Definition :...

. Note that considering primes 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 since there are more efficient, so called adic algorithms for small characteristic fields.

The Frobenius endomorphism

Given the elliptic curve defined over we consider points on over , the algebraic closure
Algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....

 of ; i.e. we allow points with coordinates in . The Frobenius endomorphism
Frobenius endomorphism
In commutative algebra and field theory, the Frobenius endomorphism is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields. The endomorphism maps every element to its pth power...

 of over extends to the elliptic curve by .

This map is the identity on and one can extend it to the point at infinity , making it a group morphism from to itself.

The Frobenius endomorphism satisfies a quadratic polynomial which is linked to the cardinality of by the following theorem:

Theorem: The Frobenius endomorphism given by satisfies the characteristic equation
where


Thus we have for all that , where + denotes addition on the elliptic curve and and
denote scalar multiplication of by and of by .

One could try to symbolically compute these points , and as functions in the coordinate ring  of
and the search for a value of 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 for various small primes
Fixing an odd prime , we now move on to solving the problem of determining , defined as , for a given prime .
If a point is in the -torsion subgroup
Torsion subgroup
In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order...

 , then where is the unique integer such that and .
Note that and that for any integer we have . Thus will have the same order as . Thus for belonging to , we also have if . Hence we have reduced our problem to solving the equation


where and have integer values in .

Computation modulo primes

The th division polynomial is such that its roots are precisely the coordinates of points of order . Thus, to restrict the computation of to the -torsion points means computing these expressions as functions in the coordinate ring of and modulo the th division polynomial. I.e. we are working in . This means in particular that the degree of and defined via is at most 1 in and at most
in .

The scalar multiplication can be done either by double-and-add
Exponentiation by squaring
Exponentiating by squaring is a general method for fast computation of large integer powers of a number. Some variants are commonly referred to as square-and-multiply algorithms or binary exponentiation. In additive notation the appropriate term is double-and-add...

 methods or by using the th division polynomial. The latter approach gives:


where is the th division polynomial. Note that
is a function in only and denote it by .

We must split the problem into two cases: the case in which , and the case in which . Note that these equalities are checked modulo .

Case 1:

By using the addition formula for the group we obtain:

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

We are now able to use the -coordinate to narrow down the choice of to two possibilities, namely the positive and negative case. Using the -coordinate one later determines which of the two cases holds.

We first show that is a function in alone. Consider .
Since is even, by replacing by , we rewrite the expression as


and have that


Now if for one then satisfies


for all -torsion points .

As mentioned earlier, using and we are now able to determine which of the two values of ( or ) works. This gives the value of . Schoof's algorithm stores the values of in a variable for each prime considered.

Case 2:

We begin with the assumption that . Since is an odd prime it cannot be that and thus . The characteristic equation yields that . And consequently that .
This implies that is a square modulo . Let . Compute in and check whether . If so, is depending on the y-coordinate.

If turns out not to be a square modulo or if the equation does not hold for any of and , our assumption that is false, thus . The characteristic equation gives .

Additional case:

If you recall, our initial considerations omit the case of .
Since we assume to be odd, and in particular, if and only if has an element of order 2. By definition of addition in the group, any element of order 2 must be of the form . Thus if and only if the polynomial has a root in , if and only if .

The algorithm

Choose a set of odd primes , such that put if , else . for do:
(a) Let be the unique integer such that and . Compute . All following computations in this loop are in the following ring:
(b) Compute , and .
(c) if then
(i) compute .
(ii)for do
(iii)if then
(iv)if then ; else .
(d)else if is a square modulo then
(i)compute with
(ii)compute
(iii)if then
(iv)else if then
(v)else
(e)else Use the Chinese Remainder Theorem to compute .


Note that since the set was chosen so that , by Hasse's theorem, we in fact know precisely.

Complexity of Schoof's algorithm

Most of the computation is taken by the evaluation of and , for each prime , that is computing , , , for each prime . This involves exponentiation in the ring and requires multiplications. Since the degree of is , each element in the ring is a polynomial of degree . By the prime number theorem
Prime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....

, we have around primes of size giving that is and we obtain that . Thus each multiplication in the ring requires multiplications in which in turn requires bit operations. In total, the number of bit operations for each prime is . Given that this computation needs to be carried out for each of the primes, the total complexity of Schoof's algorithm turns out to be . Note that fast polynomial arithmetic reduces the costs down to .

Improvements to Schoof's algorithm

In the 1990s, Noam Elkies
Noam Elkies
Noam David Elkies is an American mathematician and chess master.At age 14, Elkies received a gold medal with a perfect score at the International Mathematical Olympiad, the youngest ever to do so...

, followed by A. O. L. Atkin
A. O. L. Atkin
Arthur Oliver Lonsdale Atkin , who published under the name A. O. L. Atkin, was a Professor Emeritus of mathematics at the University of Illinois at Chicago. As an undergraduate during World War II, he worked at Bletchley Park cracking German codes. He received his Ph.D...

, devised improvements to Schoof's basic algorithm by restricting the set of primes considered before to primes of a certain kind. These came to be called Elkies primes and Atkin primes respectively. A prime is called an Elkies prime if the characteristic equation: splits over , 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
Division polynomials
In mathematics the division polynomials provide a way to calculate multiples of points on elliptic curves over Finite fields. They play a central role in the study of counting points on elliptic curves in Schoof's algorithm.- Definition :...

, we proceed by working modulo the modular polynomials which have a lower degree than the corresponding division polynomial (degree rather than ). This results in a further reduction in the running time, giving us an algorithm more efficient than Schoof's, with complexity for standard arithmetic and .

Implementations

Several algorithms were implemented in C++
C++
C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell...

 by Mike Scott and are available with [ftp://ftp.computing.dcu.ie/pub/crypto/ source code]. The implementations are free (no terms, no conditions), but they use MIRACL library which is only free for non-commercial use. Note that (unmodified) programs may be used to generate curves for commercial use. There are
  • Schoof's algorithm [ftp://ftp.computing.dcu.ie/pub/crypto/schoof.cpp implementation] for with prime .
  • Schoof's algorithm [ftp://ftp.computing.dcu.ie/pub/crypto/schoof2.cpp implementation] for .

See also

  • Elliptic curve cryptography
    Elliptic curve cryptography
    Elliptic curve cryptography is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor S...

  • Counting points on elliptic curves
    Counting points on elliptic curves
    An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve. There have been several approaches to do so, and the algorithms devised have proved to be useful tools in the study of various fields such as number theory, and more recently in...

  • Division Polynomials
    Division polynomials
    In mathematics the division polynomials provide a way to calculate multiples of points on elliptic curves over Finite fields. They play a central role in the study of counting points on elliptic curves in Schoof's algorithm.- Definition :...

  • Frobenius endomorphism
    Frobenius endomorphism
    In commutative algebra and field theory, the Frobenius endomorphism is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields. The endomorphism maps every element to its pth power...

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