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**11 - 18**of**18**### ELLIPTIC PERIODS AND PRIMALITY PROVING (EXTENTED VERSION)

, 810

"... Abstract. We construct extension rings with fast arithmetic using isogenies between elliptic curves. As an application, we give an elliptic version of the AKS primality criterion. ..."

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Abstract. We construct extension rings with fast arithmetic using isogenies between elliptic curves. As an application, we give an elliptic version of the AKS primality criterion.

### unknown title

, 2008

"... Solutions by radicals at singular values kN from new class invariants for N ≡ 3 mod 8 ..."

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Solutions by radicals at singular values kN from new class invariants for N ≡ 3 mod 8

### unknown title

, 2008

"... Solutions by radicals at singular values kN from new class invariants for N ≡ 3 mod 8 ..."

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Solutions by radicals at singular values kN from new class invariants for N ≡ 3 mod 8

### Journal de Théorie des Nombres

"... Computing the cardinality of CM elliptic curves using torsion points par François MORAIN ∗ Résumé. Soit E/Q une courbe elliptique avec multiplications complexes par un ordre d’un corps quadratique imaginaire K. Le corps de définition de E est le corps de classe de rayon Ω associé à l’ordre. Si le no ..."

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Computing the cardinality of CM elliptic curves using torsion points par François MORAIN ∗ Résumé. Soit E/Q une courbe elliptique avec multiplications complexes par un ordre d’un corps quadratique imaginaire K. Le corps de définition de E est le corps de classe de rayon Ω associé à l’ordre. Si le nombre premier p est scindé dans Ω, on peut réduire E modulo un des facteurs de p et obtenir une courbe E définie sur Fp. La trace du Frobenius de E est connue au signe près et nous cherchons à déterminer ce signe de la manière la plus rapide possible, avec comme application l’algorithme de primalité ECPP. Dans ce but, nous expliquons comment utiliser l’action du Frobenius sur des points de torsion d’ordre petit obtenus à partir d’invariants de classes qui généralisent les fonctions de Weber. Abstract. Let E/Q 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 Ω of the order. If the prime p splits completely in Ω, then we can reduce E modulo one the factors of p and get a curve E defined over Fp. The trace of the Frobenius of E is known up to sign and we need a fast way to find this sign, in the context of the Elliptic Curve Primality Proving algorithm (ECPP). For this purpose, we propose to use the action of the Frobenius on torsion points of small order built with class invariants generalizing the classical Weber functions. 1.

### DETERMINISTIC ELLIPTIC CURVE PRIMALITY PROVING FOR A SPECIAL SEQUENCE OF NUMBERS

"... Abstract. 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 ( √ −7). The algorithm uses O(log N) arithmetic operations in the ring Z/ ..."

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Abstract. 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 ( √ −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 allowed us to prove the primality of several integers with more than 100,000 decimal digits, the largest of which has more than a million bits in its binary representation. We believe that this is the largest proven prime N for which no significant partial factorization of N − 1 or N + 1 is known. 1.

### 18.783 Elliptic Curves Spring 2013 Lecture #13 03/21/2013

"... In this lecture, we consider the following problem: given a positive integer N, how can we efficiently determine whether N is prime or not? This question is intimately related to the problem of factoring N. Without a method for determining primality, we have no way of knowing when we have completely ..."

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In this lecture, we consider the following problem: given a positive integer N, how can we efficiently determine whether N is prime or not? This question is intimately related to the problem of factoring N. Without a method for determining primality, we have no way of knowing when we have completely factored N. This is a serious issue for probabilistic factorization algorithms such as ECM: if we attempt to factor a prime number N with the ECM algorithm, the algorithm will never terminate. This problem is in not unique to ECM; currently every known factorization algorithm that achieves a subexponential running time (even heuristically) is a randomized algorithm; in the absence of an explicit primality test most of these algorithms will simply fail to terminate on prime inputs. Even if we are able to ensure termination, there is still the issue of correctness. If a Monte Carlo algorithm outputs the factorization N = pq, it is easy to check whether the product of p and q is in fact equal to N. But how do we know that this is the complete factorization of N? We need a way to unequivocally prove that p and q are both prime. 13.1 Classical primality tests The most elementary approach to the problem is trial division: attempt to divide N by every integer p ≤ √ N. If no such p divides N, then N must be prime. This takes time O ( √ NM(log N)), which is exponential in log N. Remark 13.1. This complexity bound can be slightly improved. Using fast sieving techniques [6, Alg. 3.2.2], we can enumerate the primes p up to √ N in O ( √ N log N / log log N) time and then perform trial divisions by just the primes p ≤ √ N. Applying the prime number theorem and the Schönhage-Strassen bound, the sieving time dominates the cost of the divisions and the overall complexity of trial division becomes O ( √ N log N / log log N). Many early primality tests were based on Fermat’s little theorem. Theorem 13.2 (Fermat). If N is prime, then for all a ∈ Z/NZ: a N = a. This implies that if a N = a for some a ∈ Z/NZ, then N cannot be prime. This gives us a way to efficiently prove that certain integers are composite. For example, N = 91 is not prime, since: 2 91 ≡ 37 mod 91. But this does not always work. For example, 341 = 11 · 31 is not clearly not prime, but