An Algorithmic Construction of Finite Elliptic Curves of Order Divisible by a Large Prime

Abstract

Given a square-free integer ∆ < 0, we present an algorithm constructing a pair of primes p and q such that q|p + 1 − t and 4p − t 2 = ∆f 2 , where |t| ≤ 2 √ p for some integers f, t. Together with a CM method presented in the paper, such primes p and q are used for a construction of an elliptic curve E over a finite field F p such that the order of E is divisible by a large prime. It is shown that our algorithm works in polynomial time.

Explicit Zero-Counting Theorem for Hecke–landau Zeta-Functions

“…Our proof is similar in spirit to that of Trudgian [12], although the explicit expressions for the numerical constants do not contain the parameter 0 < η ≤ 1 2 which appears in [12]. Explicit results of this kind are useful for estimating the computational complexity of an algorithm which generates special primes [4]. Such primes are needed to construct an elliptic curve over a prime field using complex multiplication.…”

A variant of the large sieve inequality with explicit constants

“…It is central to the proof of various version of the Bombieri-Vinogradov theorem [6], [7], [8]. As an example of the application of the large sieve inequality to computational number theory and cryptography we refer the reader to [9], [10], [11]. In [9] author proposes the polynomial time algorithm that generates primes satisfying the following definition.…”

“…In [9] a polynomial time algorithm for constructing primes of the form (7) is given. The main idea of the algorithm is as follows.…”

“…Given α, the second procedure generates β ∈ O K such that β ≡ 1 (mod (α)) and N K/Q (β) = p is a prime. For sufficiently large p, q and 0 < ε < 2 5 the algorithm finds primes (7) satisfying (log p)/(log q) ≪ 5/(2 − 5ε) [9]. However, it is interesting to known whether the order of magnitude of the generated primes coincide with practical expectations.…”

Explicit Bound for the Prime Ideal Theorem in Residue Classes

“…Let φ be the function which occur in(6).For T ≥ 1 we have ∞ T φ(t, r 2 , η, ∆, f)t −2 dt ≤ c 9 T −1 log(e(T + 4)),where c 9 = 821.212r2 2 log (|∆|N f). Proof.…”