Results 1  10
of
12
Primality testing using elliptic curves
 Journal of the ACM
, 1999
"... Abstract. We present a primality proving algorithm—a probabilistic primality test that produces short certificates of primality on prime inputs. We prove that the test runs in expected polynomial time for all but a vanishingly small fraction of the primes. As a corollary, we obtain an algorithm for ..."
Abstract

Cited by 22 (0 self)
 Add to MetaCart
Abstract. We present a primality proving algorithm—a probabilistic primality test that produces short certificates of primality on prime inputs. We prove that the test runs in expected polynomial time for all but a vanishingly small fraction of the primes. As a corollary, we obtain an algorithm for generating large certified primes with distribution statistically close to uniform. Under the conjecture that the gap between consecutive primes is bounded by some polynomial in their size, the test is shown to run in expected polynomial time for all primes, yielding a Las Vegas primality test. Our test is based on a new methodology for applying group theory to the problem of prime certification, and the application of this methodology using groups generated by elliptic curves over finite fields. We note that our methodology and methods have been subsequently used and improved upon, most notably in the primality proving algorithm of Adleman and Huang using hyperelliptic curves and
Proving primality in essentially quartic random time
 Math. Comp
, 2003
"... Abstract. This paper presents an algorithm that, given a prime n, finds and verifies a proof of the primality of n in random time (lg n) 4+o(1). Several practical speedups are incorporated into the algorithm and discussed in detail. 1. ..."
Abstract

Cited by 18 (0 self)
 Add to MetaCart
Abstract. This paper presents an algorithm that, given a prime n, finds and verifies a proof of the primality of n in random time (lg n) 4+o(1). Several practical speedups are incorporated into the algorithm and discussed in detail. 1.
List decoding for binary Goppa codes
, 2008
"... This paper presents a listdecoding algorithm for classical irreducible binary Goppa codes. The algorithm corrects, in polynomial time, approximately n − p n(n − 2t − 2) errors in a lengthn classical irreducible degreet binary Goppa code. Compared to the best previous polynomialtime listdecoding ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
This paper presents a listdecoding algorithm for classical irreducible binary Goppa codes. The algorithm corrects, in polynomial time, approximately n − p n(n − 2t − 2) errors in a lengthn classical irreducible degreet binary Goppa code. Compared to the best previous polynomialtime listdecoding algorithms for the same codes, the new algorithm corrects approximately t 2 /2n extra errors. 1.
Sharpening PRIMES is in P for a large family of numbers
 Math. Comp
, 2005
"... We present algorithms that are deterministic primality tests for a large family of integers, namely, integers n ≡ 1 (mod 4) for which an integer a is given such that the Jacobi symbol ( a) = −1, and n integers n ≡ −1 (mod 4) for which an integer a is given such that ( a 1−a) = ( ) = −1. The algo ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
We present algorithms that are deterministic primality tests for a large family of integers, namely, integers n ≡ 1 (mod 4) for which an integer a is given such that the Jacobi symbol ( a) = −1, and n integers n ≡ −1 (mod 4) for which an integer a is given such that ( a 1−a) = ( ) = −1. The algorithms n n we present run in 2 − min(k,[2 log log n]) Õ(log n) 6 time, where k = ν2(n − 1) is the exact power of 2 dividing n − 1 when n ≡ 1 (mod 4) and k = ν2(n + 1) if n ≡ −1 (mod 4). The complexity of our algorithms improves up to Õ(log n)4 when k ≥ [2 log log n]. We also give tests for more general family of numbers and study their complexity.
Reducing lattice bases to find smallheight values of univariate polynomials
 in [13] (2007). URL: http://cr.yp.to/papers.html#smallheight. Citations in this document: §A
, 2004
"... Abstract. This paper generalizes several previous results on finding divisors in residue classes (Lenstra, Konyagin, Pomerance, Coppersmith, HowgraveGraham, Nagaraj), finding divisors in intervals (Rivest, Shamir, Coppersmith, HowgraveGraham), finding modular roots (Hastad, Vallée, Girault, Toffin ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
Abstract. This paper generalizes several previous results on finding divisors in residue classes (Lenstra, Konyagin, Pomerance, Coppersmith, HowgraveGraham, Nagaraj), finding divisors in intervals (Rivest, Shamir, Coppersmith, HowgraveGraham), finding modular roots (Hastad, Vallée, Girault, Toffin, Coppersmith, HowgraveGraham), finding highpower divisors (Boneh, Durfee, HowgraveGraham), and finding codeword errors beyond half distance (Sudan, Guruswami, Goldreich, Ron, Boneh) into a unified algorithm that, given f and g, finds all rational numbers r such that f(r) and g(r) both have small height. 1.
Divisors in Residue Classes, Constructively
 URL: http://eprint.iacr.org/2004/339. Citations in this paper
, 2004
"... Let r, s, n be integers satisfying 0 , # > 1/4, and gcd(r, s) = 1. Lenstra showed that the number of integer divisors of n equivalent to r (mod s) is upper bounded by O((# 1/4) 2 ). ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Let r, s, n be integers satisfying 0 , # > 1/4, and gcd(r, s) = 1. Lenstra showed that the number of integer divisors of n equivalent to r (mod s) is upper bounded by O((# 1/4) 2 ).
Approximate Constructions In Finite Fields
"... this paper are new, we do not give complete detailed proofs but indicate the underlying ideas. Here we present a list of possible applications (which is certainly incomplete). We start from pointing out some general purpose applications: ffl Coding Theory : AP1, AP3, AP6 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
this paper are new, we do not give complete detailed proofs but indicate the underlying ideas. Here we present a list of possible applications (which is certainly incomplete). We start from pointing out some general purpose applications: ffl Coding Theory : AP1, AP3, AP6
On the Divisibility of Fermat Quotients
"... We show that for a prime p the smallest a with a p−1 ̸ ≡ 1 (mod p 2) does not exceed (log p) 463/252+o(1) which improves the previous bound O((log p) 2) obtained by H. W. Lenstra in 1979. We also show that for almost all primes p the bound can be improved as (log p) 5/3+o(1). Keywords: sieve. Fermat ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We show that for a prime p the smallest a with a p−1 ̸ ≡ 1 (mod p 2) does not exceed (log p) 463/252+o(1) which improves the previous bound O((log p) 2) obtained by H. W. Lenstra in 1979. We also show that for almost all primes p the bound can be improved as (log p) 5/3+o(1). Keywords: sieve. Fermat quotients, smooth numbers, Heilbronn sums, large AMS Mathematics Subject Classification: 11A07, 11L40, 11N25 1