Schoof–Elkies–Atkin algorithm

Key takeaways

Schoof–Elkies–Atkin algorithm belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Schoof–Elkies–Atkin algorithm to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Schoof–Elkies–Atkin algorithm from memory before moving on to harder problems.

Reference excerpt

The Schoof–Elkies–Atkin algorithm (SEA) is an algorithm used for finding the order of or calculating the number of points on an elliptic curve over a finite field. Its primary application is in elliptic curve cryptography. The algorithm is an extension of Schoof's algorithm by Noam Elkies and A. O. L. Atkin to significantly improve its efficiency (under heuristic assumptions).

Details The Elkies-Atkin extension to Schoof's algorithm works by restricting the set of primes S = { l 1 , … , l s } {\displaystyle S=\{l_{1},\ldots ,l_{s}\}} considered to primes of a certain kind. These came to be called Elkies primes and Atkin primes respectively. A prime l {\displaystyle l} is called an Elkies prime if the characteristic equation: ϕ 2 − t ϕ + q = 0 {\displaystyle \phi ^{2}-t\phi +q=0} splits over F l {\displaystyle \mathbb {F} _{l}} , 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 Φ l ( X , Y ) {\displaystyle \Phi _{l}(X,Y)} that parametrize pairs of l {\displaystyle l} -isogenous elliptic curves in terms of their j-invariants (in practice alternative modular polynomials may also be used but for the same purpose). If the instantiated polynomial Φ l ( X , j ( E ) ) {\displaystyle \Phi _{l}(X,j(E))} has a root j ( E ′ ) {\displaystyle j(E')} in F q {\displaystyle \mathbb {F} _{q}} then l {\displaystyle l} is an Elkies prime, and we may compute a polynomial f l ( X ) {\displaystyle f_{l}(X)} whose roots correspond to points in the kernel of the l {\displaystyle l} -isogeny from E {\displaystyle E} to E ′ {\displaystyle E'} . The polynomial f l {\displaystyle f_{l}} is a divisor of the corresponding division polynomial used in Schoof's algorithm, and it has significantly lower degree, O ( l ) {\displaystyle O(l)} versus O ( l 2 ) {\displaystyle O(l^{2})} . For Elkies primes, this allows one to compute the number of points on E {\displaystyle E} modulo l {\displaystyle l} more efficiently than in Schoof's algorithm. In the case of an Atkin prime, we can gain some information from the factorization pattern of Φ l ( X , j ( E ) ) {\displaystyle \Phi _{l}(X,j(E))} in F l [ X ] {\displaystyle \mathbb {F} _{l}[X]} , which constrains the possibilities for the number of points modulo l {\displaystyle l} , but the asymptotic complexity of the algorithm depends entirely on the Elkies primes. Provided there are sufficiently many small Elkies primes (on average, we expect half the primes l {\displaystyle l} to be Elkies primes), this results in a reduction in the running time. The resulting algorithm is probabilistic (of Las Vegas type), and its expected running time is, heuristically, O ~ ( log 4 ⁡ q ) {\displaystyle {\tilde {O}}(\log ^{4}q)} , making it more efficient in practice than Schoof's algorithm. Here the O ~ {\displaystyle {\tilde {O}}} notation is a variant of big O notation that suppresses terms that are logarithmic in the main term of an expression.

Implementations The Schoof–Elkies–Atkin algorithm is implemented in the PARI/GP computer algebra system in the GP function ellap.

External links "Schoof: Counting points on elliptic curves over finite fields" article on Mathworld "Remarks on the Schoof-Elkies-Atkin algorithm" "The SEA Algorithm in Characteristic 2"

Worked examples

Example 1 — a first encounter with Schoof–Elkies–Atkin algorithm

Start with the simplest possible case. Write down what Schoof–Elkies–Atkin algorithm claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Schoof–Elkies–Atkin algorithm before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Schoof–Elkies–Atkin algorithm ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Schoof–Elkies–Atkin algorithm

In research: Schoof–Elkies–Atkin algorithm appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Schoof–Elkies–Atkin algorithm in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Schoof–Elkies–Atkin algorithm is common in secondary-school and first-year university syllabi. It links to neighbouring topics Asymmetric-key algorithms, Elliptic curve cryptography, Finite fields, so understanding it makes those chapters shorter.
In everyday life: Look for Schoof–Elkies–Atkin algorithm outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Schoof–Elkies–Atkin algorithm in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Schoof–Elkies–Atkin algorithm means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Schoof–Elkies–Atkin algorithm out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Schoof–Elkies–Atkin algorithm in simple terms?

The Schoof–Elkies–Atkin algorithm (SEA) is an algorithm used for finding the order of or calculating the number of points on an elliptic curve over a finite field. Its primary application is in elliptic curve cryptography.

Why does Schoof–Elkies–Atkin algorithm matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Schoof–Elkies–Atkin algorithm?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Schoof–Elkies–Atkin algorithm.