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Implementing the asymptotically fast version of the elliptic curve primality proving algorithm
 Math. Comp
"... Abstract. The elliptic curve primality proving (ECPP) algorithm is one of the current fastest practical algorithms for proving the primality of large numbers. Its running time cannot be proven rigorously, but heuristic arguments show that it should run in time Õ((log N)5) to prove the primality of N ..."
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Cited by 27 (1 self)
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Abstract. The elliptic curve primality proving (ECPP) algorithm is one of the current fastest practical algorithms for proving the primality of large numbers. Its running time cannot be proven rigorously, but heuristic arguments show that it should run in time Õ((log N)5) to prove the primality of N. An asymptotically fast version of it, attributed to J. O. Shallit, runs in time Õ((log N)4). The aim of this article is to describe this version in more details, leading to actual implementations able to handle numbers with several thousands of decimal digits. 1.
On the bounded sumofdigits discrete logarithm problem in finite fields
 In Proc. of the 24th Annual International Cryptology Conference (CRYPTO
, 2004
"... Abstract. In this paper, we study the bounded sumofdigits discrete logarithm problem in finite fields. Our results concern primarily with fields Fqn where nq − 1. The fields are called Kummer extensions of Fq. It is known that we can efficiently construct an element g with order greater than 2 n ..."
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Cited by 4 (1 self)
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Abstract. In this paper, we study the bounded sumofdigits discrete logarithm problem in finite fields. Our results concern primarily with fields Fqn where nq − 1. The fields are called Kummer extensions of Fq. It is known that we can efficiently construct an element g with order greater than 2 n in the fields. Let Sq(•) be the function from integers to the sum of digits in their qary expansions. We first present an algorithm that given g e (0 ≤ e < q n) finds e in random polynomial time, provided that Sq(e) < n. We then show that the problem is solvable in random polynomial time for most of the exponent e with Sq(e) < 1.32n, by exploring an interesting connection between the discrete logarithm problem and the problem of list decoding of ReedSolomon codes, and applying the GuruswamiSudan algorithm. As a side result, we obtain a sharper lower bound on the number of congruent polynomials generated by linear factors than the one based on StothersMason ABCtheorem. We also prove that in the field Fqq−1, the bounded sumofdigits discrete logarithm with respect to g can be computed in random time O(f(w) log 4 (q q−1)), where f is a subexponential function and w is the bound on the qary sumofdigits of the exponent, hence the problem is fixed parameter tractable. These results are shown to be generalized to ArtinSchreier extension Fpp where p is a prime. Since every finite field has an extension of reasonable degree which is a Kummer extension, our result reveals an unexpected property of the discrete logarithm problem, namely, the bounded sumofdigits discrete logarithm problem in any given finite field becomes polynomial time solvable in certain low degree extensions. 1