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29
Factorization of a 768bit RSA modulus
, 2010
"... This paper reports on the factorization of the 768bit number RSA768 by the number field sieve factoring method and discusses some implications for RSA. ..."
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Cited by 23 (6 self)
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This paper reports on the factorization of the 768bit number RSA768 by the number field sieve factoring method and discusses some implications for RSA.
Factorization of the tenth and eleventh Fermat numbers
, 1996
"... . We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a ..."
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Cited by 17 (8 self)
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. We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a new 27decimal digit factor of the thirteenth Fermat number. This number has four known prime factors and a 2391decimal digit composite factor. All the new factors reported here were found by the elliptic curve method (ECM). The 40digit factor of the tenth Fermat number was found after about 140 Mflopyears of computation. We discuss aspects of the practical implementation of ECM, including the use of specialpurpose hardware, and note several other large factors found recently by ECM. 1. Introduction For a nonnegative integer n, the nth Fermat number is F n = 2 2 n + 1. It is known that F n is prime for 0 n 4, and composite for 5 n 23. Also, for n 2, the factors of F n are of th...
The role of smooth numbers in number theoretic algorithms
 In International Congress of Mathematicians
, 1994
"... A smooth number is a number with only small prime factors. In particular, a positive integer is ysmooth if it has no prime factor exceeding y. Smooth numbers are a useful tool in number theory because they not only have a simple multiplicative structure, but are also fairly numerous. These twin pr ..."
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Cited by 9 (0 self)
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A smooth number is a number with only small prime factors. In particular, a positive integer is ysmooth if it has no prime factor exceeding y. Smooth numbers are a useful tool in number theory because they not only have a simple multiplicative structure, but are also fairly numerous. These twin properties of smooth numbers
Cryptanalysis of RSA using the ratio of the primes
 In: B. Preneel (Ed.) Africacrypt 2009, LNCS 5580
, 2009
"... Abstract. Let N = pq be an RSA modulus, i.e. the product of two large unknown primes of equal bitsize. In the X9.311997 standard for public key cryptography, Section 4.1.2, there are a number of recommendations for the generation of the primes of an RSA modulus. Among them, the ratio of the primes ..."
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Cited by 4 (4 self)
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Abstract. Let N = pq be an RSA modulus, i.e. the product of two large unknown primes of equal bitsize. In the X9.311997 standard for public key cryptography, Section 4.1.2, there are a number of recommendations for the generation of the primes of an RSA modulus. Among them, the ratio of the primes shall not be close to the ratio of small integers. In this paper, we show that if the public exponent e satisfies an equation eX − (N − (ap + bq))Y = Z with suitably small integers X, Y, Z, where a q is an unknown convergent of the continued fraction expansion of b p, then N can be factored efficiently. In addition, we show that the number of such exponents is at least N 3 4 −ε where ε is arbitrarily small for large N.
SQUARE FORM FACTORIZATION
, 2007
"... We present a detailed analysis of SQUFOF, Daniel Shanks’ Square Form Factorization algorithm. We give the average time and space requirements for SQUFOF. We analyze the effect of multipliers, either used for a single factorization or when racing the algorithm in parallel. ..."
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Cited by 4 (0 self)
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We present a detailed analysis of SQUFOF, Daniel Shanks’ Square Form Factorization algorithm. We give the average time and space requirements for SQUFOF. We analyze the effect of multipliers, either used for a single factorization or when racing the algorithm in parallel.
Efficient SIMD arithmetic modulo a Mersenne number
 20TH IEEE SYMPOSIUM ON COMPUTER ARITHMETIC
, 2011
"... This paper describes carryless arithmetic operations modulo an integer 2 M −1 in the thousandbit range, targeted at single instruction multiple data platforms and applications where overall throughput is the main performance criterion. Using an implementation on a cluster of PlayStation 3 game con ..."
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Cited by 2 (2 self)
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This paper describes carryless arithmetic operations modulo an integer 2 M −1 in the thousandbit range, targeted at single instruction multiple data platforms and applications where overall throughput is the main performance criterion. Using an implementation on a cluster of PlayStation 3 game consoles a new record was set for the elliptic curve method for integer factorization.
Integer Factoring
, 2000
"... Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization. ..."
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Cited by 2 (0 self)
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Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization.
Computational Methods in Public Key Cryptology
, 2002
"... These notes informally review the most common methods from computational number theory that have applications in public key cryptology. ..."
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Cited by 1 (1 self)
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These notes informally review the most common methods from computational number theory that have applications in public key cryptology.