Showing papers on "Prime number published in 2002"
TL;DR: A polynomial-time algorithm that provably recovers the signer's secret DSA key when a few consecutive bits of the random nonces k are known for a number of DSA signatures at most linear in log q, under a reasonable assumption on the hash function used in DSA.
Abstract: We present a polynomial-time algorithm that provably recovers the signer's secret DSA key when a few consecutive bits of the random nonces k (used at each signature generation) are known for a number of DSA signatures at most linear in log q (q denoting as usual the small prime of DSA), under a reasonable assumption on the hash function used in DSA. For most significant or least significant bits, the number of required bits is about log1/2 q , but can be decreased to log log q with a running time qO(1/log log q) subexponential in log q , and even further to two in polynomial time if one assumes access to ideal lattice basis reduction, namely an oracle for the lattice closest vector problem for the infinity norm. For arbitrary consecutive bits, the attack requires twice as many bits. All previously known results were only heuristic, including those of Howgrave-Graham and Smart who recently introduced that topic. Our attack is based on a connection with the hidden number problem (HNP) introduced at Crypto '96 by Boneh and Venkatesan in order to study the bit-security of the Diffie--Hellman key exchange. The HNP consists, given a prime number q , of recovering a number ? ? Fq such that for many known random t ? Fq a certain approximation of t ? is known. To handle the DSA case, we extend Boneh and Venkatesan's results on the HNP to the case where t has not necessarily perfectly uniform distribution, and establish uniformity statements on the DSA signatures, using exponential sum techniques. The efficiency of our attack has been validated experimentally, and illustrates once again the fact that one should be very cautious with the pseudo-random generation of the nonce within DSA.
271 citations
Book•
1 Jan 2002TL;DR: The third edition of the book as discussed by the authors is available in a completely revised third edition, which reflects the exciting developments in number theory during the past two decades that culminated in the proof of Fermat's Last Theorem.
Abstract: First published in 1979 and written by two distinguished mathematicians with a special gift for exposition, this book is now available in a completely revised third edition. It reflects the exciting developments in number theory during the past two decades that culminated in the proof of Fermat's Last Theorem. Intended as a upper level textbook, it is also eminently suited as a text for self-study.
169 citations
TL;DR: Algorithms are developed to construct rank-1 lattice rules in weighted Korobov spaces of periodic functions and shifted rank- 1 latticerules in weighted Sobolev spaces of non-periodic functions and which show that the rules so constructed achieve strong QMC tractability error bounds.
51 citations
TL;DR: In this article, Liu et al. showed that the ideas underlying this progress permit estimates for exceptional sets in a variety of additive problems to be significantly slimmed whenever sufficiently many excess variables are available.
Abstract: Given that available technology permits one to establish that almost all natural numbers satisfying appropriate congruence conditions are represented as the sum of three squares of prime numbers, one expects strong estimates to be attainable for exceptional sets in the analogous problem involving sums of four squares of primes. Let E(N)$ due to J. Liu and M.-C. Liu. It transpires that the ideas underlying this progress permit estimates for exceptional sets in a variety of additive problems to be significantly slimmed whenever sufficiently many excess variables are available. Such ideas are illustrated for several additional problems involving sums of four squares. 2000 Mathematical Subject Classification: 11P32, 11P05, 11P55.
49 citations
TL;DR: In this paper, the authors extended the range for which the primality of n! ± 1 and p# ± 1 are known and found two new primes of the first form (6380! + 1, 6917! - 1) and one of the second (42209# + 1).
Abstract: For each prime p, let p# be the product of the primes less than or equal to p. We have greatly extended the range for which the primality of n! ± 1 and p# ± 1 are known and have found two new primes of the first form (6380! + 1, 6917! - 1) and one of the second (42209# + 1). We supply heuristic estimates on the expected number of such primes and compare these estimates to the number actually found.
30 citations
TL;DR: In this paper, the effects of certain hypothetical configurations of zeros of Dirichlet L-functions lying off the critical line on the distribution of primes in arithmetic progressions are examined.
Abstract: We examine the effects of certain hypothetical configurations of zeros of Dirichlet L-functions lying off the critical line on the distribution of primes in arithmetic progressions.
27 citations
25 citations
Posted Content•
TL;DR: The authors survey results about prime number races, that is, results about the relative sizes of prime counting functions with respect to fixed and varying values of the number of entries in a prime counting function.
Abstract: We survey results about prime number races, that is, results about the relative sizes of prime counting functions $\pi_{q,a}(x)$, with $q$ fixed and $a$ varying. In particular, we describe recent work by the authors on these problems.
24 citations
Journal Article•
TL;DR: In this paper, it was shown that 1 c 237/214 is a symptotic formula in prime numbers p1, p2, p3 for sufficiently N and e≥ N -1/c(237/214-c)+v for some v 0.1.
Abstract: It this paper, we proved that if 1 c 237/214. then the quantity D(N):= Σ|pc1+pc2+pc3-N| elogp1logp2logp3 has an symptotic formula in prime numbers p1, p2, p3 for sufficiently N and e≥ N -1/c(237/214-c)+v for some v 0. which improves Kumchev and Nedeva's result 1 c 1.1 in [1].
22 citations
TL;DR: In this paper, it was shown that the triple product of twisted L-functions does not vanish for a positive proportion of weight 2 primitive forms for Γ 0(q) when q goes to infinity through the set of prime numbers.
Abstract: Given three distinct primitive complex characters χ1,χ2,χ3 satisfying some technical conditions, we prove that the triple product of twisted L-functions L(f·χ1,1/2) L(f·χ2,1/2) L(f·χ3,1/2) does not vanish for a positive proportion of weight 2 primitive forms for Γ0(q), when q goes to infinity through the set of prime numbers. This result, together with some variants, implies the existence of quotients of J0(q) of large dimension satisfying the Birch–Swinnerton-Dyer conjecture over cyclic number fields of degree less than 5.
TL;DR: In this paper, it was shown that the sum of class numbers of orders in complex cubic fields obeys an asymptotic law similar to the prime numbers as the bound on the regulators tends to infinity.
TL;DR: In this article, the distribution on the torus of the images through h of the -points of the points of a rational map defined on the irreducible curve is investigated.
Abstract: Let p be a prime number, let be the algebraic closure of , let be an irreducible curve in and a rational map defined on the curve We investigate the distribution on the torus of the images through h of the -points of
TL;DR: Soit a fixe dans un corps quadratrique K.R.H., on montre que S a une densite. Nous donnons aussi des conditions necessaires and suffisantes for que cette densite soit strictement positive.
Abstract: Soit a fixe dans un corps quadratrique K. On note S l'ensemble des nombres premiers p pour lesquels a admet un ordre maximal modulo p. Sous G.R.H., on montre que S a une densite. Nous donnons aussi des conditions necessaires et suffisantes pour que cette densite soit strictement positive.
TL;DR: A computational method for testing the probable primality of a GFN is described, which can be used to support Bateman and Horn's quantitative form of "Hypothesis H" of Schinzel and Sierpinski.
Abstract: Numbers of the form Fb,n = b2n ±1 are called Generalized Fermat Numbers (GFN). A computational method for testing the probable primality of a GFN is described which is as fast as testing a number of the form 2m - 1. The theoretical distributions of GFN primes, for fixed n, are derived and compared to the actual distributions. The predictions are surprisingly accurate and can be used to support Bateman and Horn's quantitative form of "Hypothesis H" of Schinzel and Sierpinski. A list of the current largest known GFN primes is included.
TL;DR: In this paper, it was shown that the series of reciprocals of all prime divisors of Fermat numbers is convergent, and that for elite primes it is also convergent.
Patent•
16 Apr 2002
TL;DR: In this article, an information security device receives an input of prime q and generates prime N that is larger than prime q, and a primality judging unit judges the primality of number N, using numbers N and R generated by the judgement target generating unit.
Abstract: An information security device receives an input of prime q, and generates prime N that is larger than prime q. In the information security device, a partial information setting unit generates number u such that 2×u×q+1≠0 mod Li (i=1, 2, ... , n). A random number generating unit generates random number R'. A judgement target generating unit generates R=u+L1×L2×...×Ln×R' and N=2×R×q+1, using number u and random number R'. A primality judging unit judges the primality of number N, using numbers N and R generated by the judgement target generating unit.
Posted Content•
TL;DR: In this paper, the authors show how to produce congruences between forms of weights 2 and p+1, in terms of group cohomology, and also show how their method works in the contexts of quadratic imaginary fields and Hilbert modular forms over totally real fields of even degree.
Abstract: Let p be a prime number. The Hasse invariant is a modular form modulo p that is often used to produce congruences between modular forms of different weights. We show how to produce such congruences between forms of weights 2 and p+1, in terms of group cohomology. We also show how our method works in the contexts of quadratic imaginary fields (where there is no Hasse invariant available) and Hilbert modular forms over totally real fields of even degree.
TL;DR: The Pseudo-Smarandache Function is part of number theory and there are several formulas which make it easier to find the Z(n) values as discussed by the authors. But these formulas are not applicable to all numbers.
Abstract: The Pseudo-Smarandache Function is part of number theory. The function comes from the Smarandache Function. The Pseudo-Smarandache Function is represented by Z(n) where n represents any natural number. The value for a given Z(n) is the smallest integer such that 1+2+3+... + Z(n) is divisible by n. Within the Pseudo-Smarandache Function, there are several formulas which make it easier to find the Z(n) values.Formulas have been developed for most numbers including: a) p, where p equals a prime number greater than two; b) b, where p equals a prime number, x equals a natural number, and b=-px; c) x, where x equals a natural number, if x/2 equals an odd number greater than two; d) x, where x equals a natural number, if x/3 equals a prime number greater than three. Therefore, formulas exist in the Pseudo-Smarandache Function for all values of b except for the following: a) x, where x = a natural number, if x/3 = a nonprime number whose factorization is not 3x; b) multiples of four that are not powers of two. All of these formulas are proven, and their use greatly reduces the effort needed to find Z(n) values.
3 Apr 2002
TL;DR: This paper compares the algorithm of Tonelli and Shanks with an algorithm based in quadratic field extensions due to Cipolla, and gives an explicit condition on a prime number to decide which algorithm is faster.
Abstract: The algorithm of Tonelli and Shanks for computing square roots modulo a prime number is the most used, and probably the fastest among the known algorithms when averaged over all prime numbers. However, for some particular prime numbers, there are other algorithms which are considerably faster.In this paper we compare the algorithm of Tonelli and Shanks with an algorithm based in quadratic field extensions due to Cipolla, and give an explicit condition on a prime number to decide which algorithm is faster. Finally, we show that there exists an infinite sequence of prime numbers for which the algorithm of Tonelli and Shanks is asymptotically worse.
TL;DR: In this paper, it was shown that for every k, there is a positive integer l 0(k) such that for all integers l⩾l0(k), there exists a sequence A with length l which has no relative prime number.
Patent•
29 Oct 2002
TL;DR: In this article, a mod remainder table is initialized for the candidate prime number using conventional mod operations and all mod remainder entries in the table are non-zero, the candidate number is tested for primality.
Abstract: A method, apparatus, and article of manufacture provide the ability to rapidly generate a large prime number to be utilized in a cryptographic key of a cryptographic system. A candidate prime number is determined and a mod remainder table is initialized for the candidate prime number using conventional mod operations. If all mod remainder entries in the table are non-zero, the candidate number is tested for primality. If the candidate prime number tests positive for primality, the candidate number is utilized in a cryptographic key of a cryptographic system. If any of the table entries is zero, the candidate number and each mod remainder entry are decremented/incremented. If any mod remainder entry is less than zero or greater than the corresponding prime number, the corresponding prime number is added/subtracted to/from the mod remainder. The process then repeats until a satisfactory number is obtained.
TL;DR: The densities D(i) of prime numbers p having the least primitive root g(p) = i, where i is equal to one of the initial positive integers less than 32, have been numerically calculated.
Abstract: In this paper the densities D(i) of prime numbers p having the least primitive root g(p) = i, where i is equal to one of the initial positive integers less than 32, have been numerically calculated. The computations were carried out under the assumption of the Generalised Riemann Hypothesis. The results of these computations were compared with the results of numerical frequency estimations.
Patent•
9 Sep 2002
TL;DR: In this paper, a random prime number is generated within a predetermined interval by precalculating and storing a single value that functions as a universal parameter for generating prime numbers of any desired size.
Abstract: A random prime number is generated within a predetermined interval by precalculating and storing a single value that functions as a universal parameter for generating prime numbers of any desired size. The value, π, is chosen as a product of k prime numbers. A number a is also chosen such that is co-prime with π. Once the values for π and a have been determined they can be stored and used for all subsequent iterations of the prime number generating algorithm. To generate a prime number, a random number x is chosen with uniform distribution, and a candidate prime number within the predetermined interval is calculated on the basis of the random number. This candidate is tested for primality, and returned as the result if it is prime. If the candidate is not prime, the random number x is multiplied by a, and used to generate a new candidate. This procedure is repeated, until the candidate is prime. Since a single value, namely π, needs to be precalculated, economies of storage are achieved. In addition, the interval of interest is approximated with a higher degree of resolution. Moreover, it is possible to utilize the same value of π for a number of different intervals.
Posted Content•
TL;DR: In this paper, the automorphism group of the modular curve for all prime numbers was determined for all the prime numbers, where p is the number of prime numbers in the modular graph.
Abstract: We determine the automorphism group of the modular curve $X_0^*(p)$ for all prime numbers $p$.
TL;DR: It is shown that a sufficient condition for the existence of the K1,k-factorization of Km,n whenever k is any positive integer, is that (1) m ≤ kn, (2) n ≤ km, (3) km - n ≡ kn - m ≡ 0 (mod(k2 - 1)) and (4) (km - n)(kn - m) ≡0 (modk(k - 1)(k2- 1)(m + n)).
TL;DR: The notion of primality was introduced by Euclid around 300 BCE in the Elements of the Elements as discussed by the authors, and it has been used extensively in factoring and primality testing.
Abstract: Factoring and primality testing have become increasingly important in today's information based society, since they both have produced techniques used in the secure transmission of data. However, often lost in the modern-day shuffle of information are the contributions of the pioneers whose ideas ushered in the computer age and, as we shall see, some of whose ideas are still used today as the underpinnings of powerful algorithms for factoring and primality testing. We offer this brief history to help readers know more about these contributions and appreciate their significance. Virtually everyone who has graduated from high school knows the definition of a prime number, namely a p E N = {1, 2, 3, 4, . . .} such that p > 1 and if p = Em where X, m E , then either f = 1 or m = 1. (If n E N and n > 1 is not prime, then n is called composite.) Although we cannot be certain, the concept of primality probably arose with the ancient Greeks over two and one-half millennia ago. The first recorded definition of prime numbers was given by Euclid around 300 BCE in his Elements. However, there is some indirect evidence that the concept of primality might have been known far earlier, for instance, to Pythagoras and his followers. The Greeks of antiquity used the term arithmetic to mean what today we would call number theory, namely the study of the properties of the natural numbers and the relationships between them. The Greeks reserved the word logistics for the study of ordinary computations using the standard operations of addition/subtraction and multiplication/division, which we now call arithmetic. The Pythagoreans introduced the term mathematics, which to them meant the study of arithmetic, astronomy, geometry, and music. This curriculum became known as the quadrivium in the Middle Ages. Although we have enjoyed the notion of a prime for millennia, only very recently have we developed eJficient tests for primality. This seemingly trivial task is in fact much more difficult than it appears. A primality test is an algorithm (a methodology following a set of rules to achieve a goal), the steps of which verify that given some integer n, we may conclude "n is a prime number." A primality proof is a successful application of a primality test. Such tests are typically called true primality tests to distinguish them from probabilistic primality tests (which can only conclude that "n is prime" up to a specified likelihood). We will not discuss such algorithms here (see [9] for these). A concept used frequently in primality testing is the notion of a sieve. A "sieve" is a process to find numbers with particular characteristics (for instance primes) by searching among all integers up to a prescribed bound, and eliminating invalid candidates until only the desired numbers remain. Eratosthenes (ca. 284-204 scE) proposed the first sieve for finding primes. The following example illustrates the Sieve of Eratosthenes.
TL;DR: In this article, an explicit construction of the kernel J 0 (N )[I ] of the Eisenstein ideal was given. But this construction was only for the special case where N −1 is not divisible by 16.
TL;DR: In this paper, a complete list of bispectral operators whose order is a prime number is given and the main theorem is exactly the result of Duistermaat-Grunbaum.
Abstract: The aim of this paper is to solve the bispectral problem for bispectral operators whose order is a prime number. More precisely we give a complete list of such bispectral operators. We use systematically the operator approach and in particular - Dixmier ideas on the first Weyl algebra. When the order is 2 the main theorem is exactly the result of Duistermaat-Gr\"unbaum . On the other hand our proofs seem to be simpler.