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34
Some integer factorization algorithms using elliptic curves
 Australian Computer Science Communications
, 1986
"... Lenstra’s integer factorization algorithm is asymptotically one of the fastest known algorithms, and is also ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase. Under some plausible assumptions, the speedup is of order ..."
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Cited by 52 (13 self)
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Lenstra’s integer factorization algorithm is asymptotically one of the fastest known algorithms, and is also ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase. Under some plausible assumptions, the speedup is of order log(p), where p is the factor which is found. In practice the speedup is significant. We mention some refinements which give greater speedup, an alternative way of implementing a second phase, and the connection with Pollard’s “p − 1” factorization algorithm. 1
Efficient Generation of Prime Numbers
, 2000
"... The generation of prime numbers underlies the use of most publickey schemes, essentially as a major primitive needed for the creation of key pairs or as a computation stage appearing during various cryptographic setups. Surprisingly, despite decades of intense mathematical studies on primality test ..."
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Cited by 13 (4 self)
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The generation of prime numbers underlies the use of most publickey schemes, essentially as a major primitive needed for the creation of key pairs or as a computation stage appearing during various cryptographic setups. Surprisingly, despite decades of intense mathematical studies on primality testing and an observed progressive intensification of cryptographic usages, prime number generation algorithms remain scarcely investigated and most reallife implementations are of rather poor performance. Common generators typically output a nbit prime in heuristic average complexity O(n^4) or O(n^4/log n) and these figures, according to experience, seem impossible to improve significantly: this paper rather shows a simple way to substantially reduce the value of hidden constants to provide much more efficient prime generation algorithms. We apply our...
Mijajlović: On Kurepa problems in number theory
 Publ. Inst. Math. (N.S
, 1995
"... Dedicated to the memory of Prof.Duro Kurepa ..."
Efficient Algorithms for Implementing Elliptic Curve PublicKey Schemes
 Master's thesis, ECE Dept., Worcester Polytechnic Institute
, 1996
"... The recent developments in the study of elliptic curve publickey algorithms have shown that they could play a major factor in the design of cryptosystems of the future. This thesis describes efficient algorithms for two important aspects of such systems. The first part describes a structured approa ..."
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Cited by 5 (0 self)
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The recent developments in the study of elliptic curve publickey algorithms have shown that they could play a major factor in the design of cryptosystems of the future. This thesis describes efficient algorithms for two important aspects of such systems. The first part describes a structured approach for finding cryptographically secure curves. A comprehensive lists of elliptic curves over subfields GF (2 n ), n = 8; 9; : : : 18, was generated, which are cryptographically secure over GF ((2 n ) m ), n \Delta m = 150; : : : ; 200. The second part describes efficient algorithms for fast software implementations of elliptic curve computations which can be used in a variety of publickey protocols. These algorithms, which perform group operations over nonsupersingular elliptic curves, are optimized through the use of composite Galois fields of the form GF ((2 n ) m ). An elliptic curve keyexchange protocol over the composite field GF ((2 16 ) 11 ) was implemented using op...
GCDFree Algorithms for Computing Modular Inverses
 Cryptographic Hardware and Embedded Systems CHES 2003, Springer LNCS
, 1997
"... Abstract. This paper describes new algorithms for computing a modular inverse e−1 mod f given coprime integers e and f. Contrary to previously reported methods, we neither rely on the extended Euclidean algorithm, nor impose conditions on e or f. The main application of our gcdfree technique is the ..."
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Cited by 4 (0 self)
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Abstract. This paper describes new algorithms for computing a modular inverse e−1 mod f given coprime integers e and f. Contrary to previously reported methods, we neither rely on the extended Euclidean algorithm, nor impose conditions on e or f. The main application of our gcdfree technique is the computation of an RSA private key in both standard and CRT modes based on simple modular arithmetic operations, thus boosting reallife implementations on cryptoaccelerated devices.
Large Sophie Germain primes
 Math. Comp
, 1996
"... Abstract. If P is a prime and 2P+1 is also prime, then P is a Sophie Germain prime. In this article several new Sophie Germain primes are reported, which are the largest known at this time. The search method and the expected search times are discussed. 1. ..."
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Cited by 4 (1 self)
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Abstract. If P is a prime and 2P+1 is also prime, then P is a Sophie Germain prime. In this article several new Sophie Germain primes are reported, which are the largest known at this time. The search method and the expected search times are discussed. 1.
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.
Fast Generation of Prime Numbers of Portable Devices: An Update
 Proceedings of CHES 2006, LNCS 4249
, 2006
"... Abstract. The generation of prime numbers underlies the use of most publickey cryptosystems, essentially as a primitive needed for the creation of RSA key pairs. Surprisingly enough, despite decades of intense mathematical studies on primality testing and an observed progressive intensification of ..."
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Cited by 3 (1 self)
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Abstract. The generation of prime numbers underlies the use of most publickey cryptosystems, essentially as a primitive needed for the creation of RSA key pairs. Surprisingly enough, despite decades of intense mathematical studies on primality testing and an observed progressive intensification of cryptography, prime number generation algorithms remain scarcely investigated and most reallife implementations are of dramatically poor performance. We show simple techniques that substantially improve all algorithms previously suggested or extend their capabilities. We derive fast implementations on appropriately equipped portable devices like smartcards embedding a cryptographic coprocessor. This allows onboard generation of RSA keys featuring a very attractive (average) processing time. Our motivation here is to help transferring this task from terminals where this operation usually took place so far, to portable devices themselves in near future for more confidence, security, and compliance with networkscaled distributed protocols such as electronic cash or mobile commerce.
Landau’s function for one million billions, in "Journal de Théorie des Nombres de Bordeaux", 2009, à paraître, http://hal.archivesouvertes.fr/hal00264057/en/. CACAO
"... À Henri Cohen pour son soixantième anniversaire. Let Sn denote the symmetric group with n letters, and g(n) the maximal order of an element of Sn. If the standard factorization of M into primes is M = q α1 1 qα2 2... q αk k, we define ℓ(M) to be qα1 1 + qα2 2 +... + qα k k; one century ago, E. Landa ..."
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Cited by 2 (0 self)
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À Henri Cohen pour son soixantième anniversaire. Let Sn denote the symmetric group with n letters, and g(n) the maximal order of an element of Sn. If the standard factorization of M into primes is M = q α1 1 qα2 2... q αk k, we define ℓ(M) to be qα1 1 + qα2 2 +... + qα k k; one century ago, E. Landau proved that g(n) = maxℓ(M)≤n M and that, when n goes to infinity, log g(n) ∼ p nlog(n). There exists a basic algorithm to compute g(n) for 1 ≤ n ≤ N; its running time is O N 3/2 / √ ” log N and the needed memory is O(N); it allows computing g(n) up to, say, one million. We describe an algorithm to calculate g(n) for n up to 10 15. The main idea is to use the socalled ℓsuperchampion numbers. Similar numbers, the superior highly composite numbers, were introduced by S. Ramanujan to study large values of the divisor function τ(n) = P d  n 1. Key words: arithmetical function, symmetric group, maximal order, highly
Cryptology
"... Cryptology has advanced tremendously since 1976; this chapter provides a brief overview of the current stateoftheart in the field. Several major themes predominate in the development. One such theme is the careful elaboration of the definition of security for a cryptosystem. A second theme has be ..."
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Cited by 1 (0 self)
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Cryptology has advanced tremendously since 1976; this chapter provides a brief overview of the current stateoftheart in the field. Several major themes predominate in the development. One such theme is the careful elaboration of the definition of security for a cryptosystem. A second theme has been the search for provably secure cryptosystems, based on plausible assumptions about the difficulty of specific numbertheoretic problems or on the existence of certain kinds of functions (such as oneway functions). A third theme is the invention of many novel and surprising cryptographic capabilities, such as publickey cryptography, digital signatures, secretsharing, oblivious transfers, and zeroknowledge proofs. These themes have been developed and interwoven so that today theorems of breathtaking generality and power assert the existence of cryptographic techniques capable of solving almost any imaginable cryptographic problem.