Schoof–Elkies–Atkin (SEA) point-counting algorithm

E753155

algorithm computational number theory method elliptic curve algorithm point-counting algorithm

The Schoof–Elkies–Atkin (SEA) point-counting algorithm is an efficient method in computational number theory and elliptic curve cryptography for determining the number of points on an elliptic curve over a finite field.

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Statements (47)

Predicate	Object
instanceOf	algorithm ⓘ computational number theory method ⓘ elliptic curve algorithm ⓘ point-counting algorithm ⓘ
advantageOver	exponential-time point-counting methods ⓘ
appliesTo	elliptic curves over finite fields of large characteristic ⓘ elliptic curves used in cryptography ⓘ
basedOn	Schoof algorithm NERFINISHED ⓘ
comparedTo	brute-force point counting ⓘ
computes	number of rational points on an elliptic curve over a finite field ⓘ trace of Frobenius of an elliptic curve over a finite field ⓘ
developedInContextOf	elliptic curve cryptography ⓘ
extends	Schoof algorithm NERFINISHED ⓘ
field	computational number theory ⓘ elliptic curve cryptography NERFINISHED ⓘ
goal	efficient point counting on large elliptic curves ⓘ
hasPart	Atkin prime stage ⓘ Chinese remainder reconstruction stage ⓘ Elkies prime stage NERFINISHED ⓘ
improvesOn	Schoof algorithm NERFINISHED ⓘ
input	elliptic curve over a finite field ⓘ
namedAfter	A. O. L. Atkin NERFINISHED ⓘ Noam Elkies NERFINISHED ⓘ René Schoof NERFINISHED ⓘ
output	cardinality of the elliptic curve group over the finite field ⓘ
property	deterministic ⓘ uses Chinese remainder theorem NERFINISHED ⓘ uses classification of primes into Elkies and Atkin primes ⓘ uses isogenies of elliptic curves ⓘ uses modular polynomials ⓘ
relatedTo	Frobenius endomorphism NERFINISHED ⓘ Hasse bound for elliptic curves NERFINISHED ⓘ isogeny volcanoes ⓘ modular polynomials Φ_l ⓘ
requires	arithmetic in finite fields ⓘ computation of isogenies of small prime degree ⓘ computation of modular polynomials ⓘ
timeComplexity	polynomial in log(q) where q is the size of the finite field ⓘ
typicalUseCase	selection of secure elliptic curves for public-key cryptography ⓘ verification of claimed curve orders in standards ⓘ
usedFor	computing the order of an elliptic curve over a finite field ⓘ constructing elliptic curves with prescribed order ⓘ counting points on elliptic curves over finite fields ⓘ elliptic curve cryptosystem parameter generation ⓘ
usedIn	CM (complex multiplication) method for generating elliptic curves NERFINISHED ⓘ construction of cryptographically secure elliptic curves ⓘ elliptic curve cryptography parameter validation ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Hasse bound for elliptic curves → usedIn → Schoof–Elkies–Atkin (SEA) point-counting algorithm ⓘ