Implementing the asymptotically fast version of the elliptic curve primality proving algorithm

A LLT-like test for proving the primality of Fermat numbers

This paper provides a proof of a LLT-like test for Fermat numbers, based on the properties of Lucas Sequences and on the method of Lehmer.

Dual Elliptic Primes and Applications to Cyclotomy Primality Proving

Two rational primes p, q are called dual elliptic if there is an elliptic curve E mod p with q points. They were introduced as an interesting means for combining the strengths of the elliptic curve and cyclotomy primality proving algorithms. By extending to elliptic curves some notions of galois theory of rings used in the cyclotomy primality tests, one obtains a new algorithm which has heuristic cubic run time and generates certificates that can be verified in quadratic time. After the break through of Agrawal, Kayal and Saxena has settled the complexity theoretical problem of primality testi…

Deterministic elliptic curve primality proving for a special sequence of numbers

We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers N in a certain sequence, using an elliptic curve E/Q with complex multiplication by the ring of integers of Q(sqrt(-7)). The algorithm uses O(log N) arithmetic operations in the ring Z/NZ, implying a bit complexity that is quasi-quadratic in log N. Notably, neither of the classical "N-1" or "N+1" primality tests apply to the integers in our sequence. We discuss how this algorithm may be applied, in combination with sieving techniques, to efficiently search for very large primes. This has a…

Four primality testing algorithms

In this expository paper we describe four primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or 'very probably' prime. The second test is a deterministic polynomial time algorithm to prove that a given numer is either prime or composite. The third and fourth primality tests are at present most widely used in practice. Both tests are capable of proving that a given number is prime or composite, but neither algorithm is deterministic. The third algorithm exploits the arithmetic of cyclotomic fields. Its running time is almost…

$Kp^n+1$数的质数测试及安全质数和索菲·热尔曼质数的推广

A primality test for $Kp^n+1$ numbers and a generalization of Safe Primes and Sophie Germain Primes

本文对形式为$Kp^n+1$的整数推广了普罗特定理，提出了一种仅需一次模幂运算的质数测试方法，无需计算任何GCD。同时提供了两种测试以提高验证$Kp^n+1$数质数的可能性。作为这些测试的推论，我们给出了三个整数家族$N$，其质数性只能通过证明$a^{N-1} \equiv 1 \pmod N$（费马测试）来确认。还推广了安全质数，定义为$a$-安全质数，因其$N-1$具有与安全质数相同的大型质因数。我们讨论了这些数的分布问题，并提出了关于相关数$a$-索菲·热尔曼质数分布的猜想。

On the Probability of Relative Primality in the Gaussian Integers

This paper studies the interplay between probability, number theory, and geometry in the context of relatively prime integers in the ring of integers of a number field. In particular, probabilistic ideas are coupled together with integer lattices and the theory of zeta functions over number fields in order to show that $$P(\gcd(z_{1},z_{2})=1) = \frac{1}{ζ_{\Q(i)}(2)}$$ where $z_{1},z_{2} \in \mathbb{Z}[i]$ are randomly chosen and $ζ_{\Q(i)}(s)$ is the Dedekind zeta function over the Gaussian integers. Our proof outlines a lattice-theoretic approach to proving the generalization of this theore…

Kummer surfaces for primality testing

We use the arithmetic of the Kummer surface associated to the Jacobian of a hyperelliptic curve to study the primality of integers of the form $4m^2 5^n-1$. We provide an algorithm capable of proving the primality or compositeness of most of the integers in these families and discuss in detail the necessary steps to implement this algorithm in a computer. Although an indetermination is possible, in which case another choice of initial parameters should be used, we prove that the probability of reaching this situation is exceedingly low and decreases exponentially with $n$.

Universal elliptic Gauß sums and applications

We present new ideas for computing elliptic Gauß sums, which constitute an analogue of the classical cyclotomic Gauß sums and whose use has been proposed in the context of counting points on elliptic curves and primality tests. By means of certain well-known modular functions we define the universal elliptic Gauß sums and prove they admit an efficiently computable representation in terms of the $j$-invariant and another modular function. After that, we show how this representation can be used for obtaining the elliptic Gauß sum associated to an elliptic curve over a finite field $\mathbb{F}_p$…

三个Brillhart-Lehmer-Selfridge素性证明用于Wagstaff数

Three Brillhart-Lehmer-Selfridge primality proofs for Wagstaff numbers

本文基于Brillhart-Lehmer-Selfridge (BLS) N-1准则，对Wagstaff数W_2617、W_10501和W_12391进行了严格验证的素性证明，不依赖ECPP，仅使用经典N-1方法，并利用Cunningham项目表和FactorDB进行因子分解。

Generalizing and Implementing Michael Hirschhorn's Amazing Algorithm for Proving Ramanujan-Type Congruences

When Mike Hirschhorn showed us his lovely gem, that gives the simplest-to-date proof of Ramanujan's famous result that p(11n+6) is divisible by 11, we realized that his amazing method can be extended, and taught to a computer, and can prove even deeper identities. We would have done much more if not for the existence of Silviu Radu's powerful algorithm that handle any Ramanujan type congruence for any modular form (of a very general type), but it is still nice to know that in order to prove such simply stated results, that can be explained to a seven-year-old, one does not need the intimidatin…

Computing the cardinality of CM elliptic curves using torsion points

Let E be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field K. The field of definition of E is the ring class field Omega of the order. If the prime p splits completely in Omega, then we can reduce E modulo one the factors of p and get a curve Ep defined over GF(p). The trace of the Frobenius of Ep is known up to sign and we need a fast way to find this sign. For this, we propose to use the action of the Frobenius on torsion points of small order built with class invariants a la Weber, in a manner reminiscent of the Schoof-Elkies-Atkin al…

Constructing elliptic curves with a known number of points over a prime field

Elliptic curves with a known number of points over a given prime field with n elements are often needed for use in cryptography. In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication to construct such curves. One of the steps in this method is the calculation of a root modulo n of the Hilbert class polynomial H(X) for a fundamental discriminant D. The usual way is to compute H(X) over the integers and then to find the root modulo n. We present a modified version of the Chinese remainder theorem (CRT) to compute H(X) modulo n directly fr…