Primality Proving Using Elliptic Curves: An Update
نویسنده
چکیده
In 1986, following the work of Schoof on counting points on elliptic curves over finite fields, new algorithms for primality proving emerged, due to Goldwasser and Kilian on the one hand, and Atkin on the other. The latter algorithm uses the theory of complex multiplication. The algorithm, now called ECPP, has been used for nearly ten years. The purpose of this paper is to give an account of the recent theoretical and practical improvements of ECPP, as well as new benchmarks for integers of various sizes and a new primality record.
منابع مشابه
Elliptic Curves, Primality Proving and Some Titanic Primes
We describe how to generate large primes using the primality proving algorithm of Atkin. Figure 1: The Titanic .
متن کاملCyclotomy 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 heurist...
متن کاملPrimality Proving with Elliptic Curves
Elliptic curves are fascinating mathematical objects. In this paper, we present the way they have been represented inside the COQ system, and how we have proved that the classical composition law on the points is internal and gives them a group structure. We then describe how having elliptic curves inside a prover makes it possible to derive a checker for proving the primality of natural number...
متن کاملSome remarks on primality proving and elliptic curves
We give an overview of a method for using elliptic curves with complex multiplication to give efficient deterministic polynomial time primality tests for the integers in sequences of a special form. This technique has been used to find the largest proven primes N for which there was no known significant partial factorization of N − 1 or N + 1.
متن کامل18 . 783 Elliptic Curves Spring 2013
Andrew V. Sutherland A key ingredient to improving the efficiency of elliptic curve primality proving (and many other algorithms) is the ability to directly construct an elliptic curve E/Fq with a specified number of rational points, rather than generating curves at random until a suitable curve is found. To do this we need to develop the theory of complex multiplication. Recall from Lecture 7 ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1998