H. Cohen and H. W. Lenstra, Jr., Primality testing and Jacobi sums, Math. Comp. Vol. 42 (1984), 297-330.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....theorem [8] hold, we have (6.1) d # 1 (mod F 1 ) Similarly, by Morrison s theorem [2, Theorem 16] 6.2) d # 1 (mod F 2 ) where F 2 = 43 2 73 = 134977. Next, we confirm that the conditions for the APRCL test (see, for example, Cohen and A. K. Lenstra [4] or Cohen and H. W. Lenstra [5]) are satisfied with the prime powers p k : 2 5 , 3 3 , 5 2 , 7, 11, 13 , and primes q: 11, 17, 19, 23, 29, 31, 37, 41, 53, 61, 67, 71, 79, 89, 97, 101, 109, 113, 127, 131, 151, 157, 181, 199, 211, 241, 271, 281, 313, 331, 337, 353, 379, 397, 401, 421, 433, 463, 521, 541, 547, 601, ....

H. Cohen and H. W. Lenstra, Primality testing and Jacobi sums, Math. Comp., 42 (1984), 297-330.

....with a proof, for any given number N . Of course, it should at the same time be as fast as possible. Examples of such tests include: cyclotomy tests: O( log N) c log log log N ) Gauss sums test: Adleman, Pomerance, Rumely (1979) 2, 15] Jacobi sums test: Cohen, Lenstra, Lenstra (1980) [8]. cyclotomy tests: Bosma van der Hulst (1990) Mihailescu (1997) 17] elliptic curves tests: O( log N) c ) Bosma, Chudnovsy Chudnovsky (1985) 7] Goldwasser Kilian [10] Atkin Morain (1986) 5, 4] genus 2 curves: O( log N) Adleman Huang (1986) 1] the interest of ....

Cohen (H.) and Lenstra, Jr. (H. W.). -- Primality testing and Jacobi sums. Mathematics of Computation, vol. 42, n 165, 1984, pp. 297--330.

....prime factors of N Gamma 1 (or N 1, and became impractical for numbers having more than 100 digits. A breakthrough came when Adleman, Pomerance and Rumely [1] found a method to test primality of much larger numbers. This test was simplified and improved by H. Cohen and H.W. Lenstra, Jr. [29]. The resulting test was implemented by A.K. Lenstra and H. Cohen with the help of Dik Winter [30] and made it possible to prove primality of numbers up to 300 decimal digits in a few minutes CPU time. At present, one is able to prove primality of numbers with 1000 and more digits [4, 16] For an ....

H. Cohen and H.W. Lenstra, Jr. Primality testing and Jacobi sums. Mathematics of Computation, 42:297--330, 1984.

.... 1) 3, the first ordinary Titanic prime. 1 Introduction For cryptographical purposes [7] it is desirable to generate large primes as fast as possible. This can be done via ad hoc techniques [30, 12, 14, 4] or by means of a general purpose primality testing algorithm such as that described in [1, 11, 10, 6] or the Elliptic Curve Primality Proving (ECPP) algorithm due to Atkin [2, 25, 28] For a survey of primality testing, see [18] Another point is to certify large primes, such as the Cunningham numbers [8] which sometimes have more than 400 digits. The purpose of this paper is to explain how ....

....the new Mersenne s conjecture [3] Bateman, Selfridge and Wagstaff tested some numbers of the form N p = 2 p 1) 3 for primality. They found that N p was a probable prime for p 2 f1709; 2617; 3539g. During EUROCRYPT 89 (April 10 13, 1989) it appeared that both ECPP and the Jacobi Sums test [11, 10, 6] were able to attack numbers as large as 1000 digits. This was the very start of a stimulating competition with W. Bosma and M. P. van der Hulst. Indeed, the first of these numbers (p = 1709, N p with 514 digits) was the first number proven prime using ECPP in its distributed version. This was ....

H. Cohen and H. W. Lenstra, Jr. Primality testing and Jacobi sums. Math. Comp. 42, 165 (1984), 297--330.

....the authors describe a primality testing algorithm using some ideas which are implicit in [117] They are able to prove that the running time of this algorithm is O( log n) c log log log n ) for some effectively computable constant c 0. This test, improved by H. Cohen and H. W. Lenstra ([33]) has been implemented by the first author and A. K. Lenstra ( 34] The results are very impressive. We give below some typical times for their algorithm, on a CDC Cyber 170 750. The time used for elementary operations on multiples, 16 words of 47 bits, and doubles, 32 words of 47 bits, are also ....

H. Cohen, H. W. Lenstra, Jr. Primality testing and Jacobi sums. Math. of Comp., 42, 165, 1984, pp. 297-330.

....proving algorithms of the Lucas Lehmer style. The further research started and inspired by H. W. Lenstra Jr. Le1] Le2] BovdH] Mi1] showed that the contrary was the case. The Adleman et al. algorithm was first implemented in a slightly generalized shape by H. Cohen and A. Lenstra [CoLe]. The deeper connection to Lucas Lehmer was later exploited in the implementations of W. Bosma and M. van der Hulst [BovdH] and myself [Mi1] While being the most efficient general primality test in use, cyclotomy is not a polynomial algorithm. We shall consider this algorithm in x4. Elliptic ....

....a high power of a constant s; for s = 2, elements ff of high order can be found using quadratic reciprocity. We show that Gauss sums can be used successfully also in this context. Gauss sums together with the main properties used in the context of primality proving are explained, for instance, in [CoLe]. Theorem 4. Let N(m) a Delta p m ffl with ffl 2 f Gamma1; 1g be such that p m p n a with p a prime. Let f(x) and R be defined as above and 2 R be a p Gammath primitive root of unity. Let q = hp 1 be a prime such that N(m) h 6= 1 mod q, ZZ= q Delta ZZ) Gamma be a ....

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H.Cohen, H.W.Lenstra Jr.: Primality Testing and Jacobi sums, Math. Comp. 1984, pp 297-330.

....a selection of values of n between 127 and 521 for which the complete factorization of 2 n 1 is known and includes a very large prime factor. This table is drawn from [14] except that the primality of the 105 digit factor of 2 373 1 was proved by the author using the CohenLenstra [17] version of the Adleman Pomerance Rumely primality test [2] Also included are the two values n = 881 and n = 1063, for which the cofactors have not been shown to be prime, although they almost definitely are, since they pass pseudoprime tests. Any one of these values of n will give a ....

H. Cohen and H. W. Lenstra, Jr., Primality testing and Jacobi sums, Math. Comp., 42 (1984), 297330.

....the set of possible divisors of n. This set of divisors is then tested, and if all members of the set fail to divide n, then n is declared prime. This test does not yield a certificate for its input, and has not proved to be practical. The test was modified and made practical by Cohen and Lenstra [CL1], CL2] In place of Gauss sums, they use Jacobi sums, which permit them to work in the simpler ring (Z=nZ) i p ] An improved Jacobi sum test, designed and implemented by Bosma and van der Hulst [BV] compares favorably with today s most successful general purpose primality tests. 10 P Q P Q ....

H. Cohen, H. W. Lenstra Jr., Primality Testing and Jacobi Sums, Math. Comp., 42, 1984, pp. 297-330.

.... Le premier algorithme de primalit e tous usages est bas e sur des lois de r eciprocit e de haut degr e et utilise les propri et es des corps cyclotomiques (corps K = Q(i n ) o u i n = exp(2i =n) Il a et e introduit par Adleman, Pomerance et Rumely [APR83] et perfectionn e par Cohen et Lenstra [CL84] et implant e par Cohen et Lenstra [CL87, BvdH90] Cet al..gorithme permet de prouver la primalit e d entiers de 200 chiffres en moins de quelques minutes de Cray. Son temps de calcul est O( log N) c log log log N ) 4.3 Courbes elliptiques Dans un corps k quelconque (de caract eristique ....

Cohen (H.) et Lenstra, Jr. (H. W.). -- Primality testing and Jacobi sums. Math. Comp., vol. 42, n 165, 1984, pp. 297--330.

....this test, see [4] Another way is to use deterministic primality testing algorithms which yield a proof for a number to be a prime. The first general purpose deterministic algorithm was introduced by Adleman, Rumely and Pomerance [1] and refined by H. Cohen, H. W. Lenstra (Jr. and A. K. Lenstra [11, 10] (and more recently by Bosma and van der Hulst [6] It gives good running times (on a huge computer) However, the proof given by their program is yes or no and the only way for someone else to verify the results is to rewrite and rerun the entire program. One of the most recent primality testing ....

H. Cohen and H. W. Lenstra, Jr. Primality testing and Jacobi sums. Math. Comp., 42(165):297--330, 1984.

....in the affirmative by Adleman and Huang [AH92] As a result of the work of H. Maier on gaps between consecutive primes, the exact formulation of Cram er s conjecture has now been called into question, however the conjecture required for [GK86] is unaffected. Ref1 [Guy77] Knu81] Len81] [CL84], Pom81] Rab80a] Rie85b] Rie85a] Wil78] 2 Testing an infinite set of primes Let S ae N. C2 Input n 2 N. Output 1 if n 2 S, 0 otherwise. O2 Does there exist an infinite set S ae Primes such that C2 is in P Rem2 86 In light of Rem1 86 it is remarkable that O2 remains unsettled. The ....

H. Cohen and H. W. Lenstra, Jr. Primality testing and Jacobi sums. Mathematics of Computation, 42:297--330, 1984.

....and the 105 digit prime 163117845256502920687558543656697020247411364462120385893160945804553250187 472604743326476435522680378897: Primality of these numbers was proved with help of the Jacobi sum test of Adleman, Pomerance, Rumely, H. Cohen and H.W. Lenstra, Jr. [1, 3], as implemented by H. Cohen and A.K. Lenstra [4] with the help of D.T. Winter at CWI. Acknowledgements We thank the Dutch National Computing Facilities Foundation (NCF) for providing access to the Cray C90, our colleague Walter Lioen for technical assistance in this project, and all those CWI ....

H. Cohen and H.W. Lenstra, Jr. (1984). Primality testing and Jacobi sums. Mathematics of Computation, 42, 297--330.

....For quite a long time, it has been known that one could quickly recognize most composite numbers using Fermat s little theorem. For cryptographical purposes, this idea was extended and it has yielded some fast probabilistic compositeness algorithms (for this, we refer to [52] the introduction of [27] and [8] On the contrary, testing an arbitrary number for primality depended on integer factorization. For this era, see [17, 98, 101] The reader interested in large or curious primes is referred to [83] as well as [66] The year 1979 saw the appearance of the first general purpose primality ....

....of the first general purpose primality testing algorithm, designed by Adleman, Pomerance and Rumely [3] The running time of the algorithm was proved to be O( log N ) c log log log N ) for some effective c 0. This algorithm was simplified and made practical by H. W. Lenstra and H. Cohen [27] and then successfully implemented by H. Cohen and A. K. Lenstra [26] Motivated by our results with elliptic curves (see below) the algorithm was recently optimized by Bosma and Van der Hulst [14] see also [63] However, it is not possible to check the results of this algorithm independently ....

H. Cohen and H. W. Lenstra, Jr. Primality testing and Jacobi sums. Math. Comp. 42, 165 (1984), 297--330.

....For quite a long time, it has been known that one could quickly recognize most composite numbers using Fermat s little theorem. For cryptographical purposes, this idea was extended and it has yielded some fast probabilistic compositeness algorithms (for this, we refer to [52] the introduction of [28] and [9] On the contrary, testing an arbitrary number for primality depended on integer factorization. For this era, see [18, 92, 95] The reader interested in large or curious primes is referred to [80] as well as [68] The year 1979 saw the appearance of the first general purpose primality ....

....of the first general purpose primality testing algorithm, designed by Adleman, Pomerance and Rumely [3] The running time of the algorithm was proved to be O( log N) c log log log N ) for some effective c 0. This algorithm was simplified and made practical by H. W. Lenstra and H. Cohen [28] and then successfully implemented by H. Cohen and A. K. Lenstra [27] Motivated by our results with elliptic curves (see below) the algorithm was recently optimized by Bosma and Van der Hulst [15] see also [60] However, it is not possible to check the results of this algorithm independently ....

H. Cohen and H. W. Lenstra, Jr. Primality testing and Jacobi sums. Math. Comp. 42, 165 (1984), 297--330.

..... 1. Introduction. During the last ten years, primality testing evolved at great speed. Motivated by the RSA cryptosystem [3] the first deterministic primality proving algorithm was designed by Adleman, Pomerance and Rumely [2] and made practical by Cohen, H. W. Lenstra and A. K. Lenstra (see [9, 10] and more recently [5] It was then proved that the time needed to test an arbitrary integer N for primality is O( log N) c log log log N ) for some positive constant c 0. When implemented on a huge computer, the algorithm was able to test 200 digit numbers in about 10 minutes of CPU time. A ....

H. Cohen, H. W. Lenstra, Jr. Primality testing and Jacobi sums. Math. of Comp., 42, 165, 1984, pp. 297-330.

....can be proved to be prime if they are indeed prime using a general purpose primality test. Compared to the probabilistic compositeness test from 3.5.2, these tests are rather involved. They are hardly relevant for cryptology, and are therefore beyond the scope of these notes. See [8, 64] or [3, 14, 23, 24, 42, 82] for more details. It is also possible to generate primes uniformly in such a way that very simple primality proofs (based on Pocklington s theorem and generalizations thereof [8, 14, 64] can be applied to them. See [73] for details. 3.5.6 Prime generation with trial division. Most random odd ....

H. Cohen, H.W. Lenstra, Jr., Primality testing and Jacobi sums, Math. Comp. 42 (1984) 297-330.

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H. Cohen and H. W. Lenstra, Jr., Primality testing and Jacobi sums, Math. Comp. Vol. 42 (1984), 297-330.

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H. Cohen and H. W. Lenstra, Jr., Primality testing and Jacobi sums, Math. Comp. 42 (1984), 297--330.

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H. Cohen and H. W. Lenstra, Jr., Primality testing and Jacobi sums, Math. Comp. 42 (1984), 297--330.

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Cohen, H., Lenstra, H. W., Jr.: Primality testing and Jacobi sums. Math. Comp. 42, 297--330 (1984).

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H.Cohen, H.W.Lenstra Jr.: "Primality Testing and Jacobi sums", Math. Comp. vol 48 (1984), pp 297-330.

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H. Cohen, H.W. Lenstra, Jr., "Primality testing and Jacobi sums", Math. Comp. 42 (1984), 297--330.

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