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Large Character Sums
 CHARACTERS AND THE POLYAVINOGRADOV THEOREM 29
"... A central problem in analytic number theory is to gain an understanding of character sums χ(n), n≤x ..."
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A central problem in analytic number theory is to gain an understanding of character sums χ(n), n≤x
On the distribution of quadratic residues and nonresidues modulo a prime number
 Mathematics of Computation
, 1992
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you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, noncommercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at.
The Security of all RSA and Discrete Log Bits
, 2003
"... We study the security of individual bits in an RSA encrypted message EN (x). We show that given EN (x), predicting any single bit in x with only a nonnegligible advantage over the trivial guessing strategy, is (through a polynomial time reduction) as hard as breaking RSA. Moreover, we prove that bl ..."
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We study the security of individual bits in an RSA encrypted message EN (x). We show that given EN (x), predicting any single bit in x with only a nonnegligible advantage over the trivial guessing strategy, is (through a polynomial time reduction) as hard as breaking RSA. Moreover, we prove that blocks of O(log log N) bitsofxare computationally indistinguishable from random bits. The results carry over to the Rabin encryption scheme. Considering the discrete exponentiation function gx modulo p, with probability 1 − o(1) over random choices of the prime p, the analog results are demonstrated. The results do not rely on group representation, and therefore applies to general cyclic groups as well. Finally, we prove that the bits of ax + b modulo p give hard core predicates for any oneway function f. All our results follow from a general result on the chosen multiplier hidden number problem: givenanintegerN, and access to an algorithm Px that on input a random a ∈ ZN, returns a guess of the ith bit of ax mod N, recover x. We show that for any i, ifPx has at least a nonnegligible advantage in predicting the ith bit, we either recover x, or, obtain a nontrivial factor of N in polynomial time. The result also extends to prove the results about simultaneous security of blocks of O(log log N) bits.
Integers, without large prime factors, in arithmetic progressions, II
"... : We show that, for any fixed " ? 0, there are asymptotically the same number of integers up to x, that are composed only of primes y, in each arithmetic progression (mod q), provided that y q 1+" and log x=log q ! 1 as y ! 1: this improves on previous estimates. y An Alfred P. Sloan Research Fe ..."
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: We show that, for any fixed " ? 0, there are asymptotically the same number of integers up to x, that are composed only of primes y, in each arithmetic progression (mod q), provided that y q 1+" and log x=log q ! 1 as y ! 1: this improves on previous estimates. y An Alfred P. Sloan Research Fellow. Supported, in part, by the National Science Foundation Integers, without large prime factors, in arithmetic progressions, II Andrew Granville 1. Introduction. The study of the distribution of integers with only small prime factors arises naturally in many areas of number theory; for example, in the study of large gaps between prime numbers, of values of character sums, of Fermat's Last Theorem, of the multiplicative group of integers modulo m, of Sunit equations, of Waring's problem, and of primality testing and factoring algorithms. For over sixty years this subject has received quite a lot of attention from analytic number theorists and we have recently begun to attain a very pre...
Some Primality Testing Algorithms
 Notices of the AMS
, 1993
"... We describe the primality testing algorithms in use in some popular computer algebra systems, and give some examples where they break down in practice. 1 Introduction In recent years, fast primality testing algorithms have been a popular subject of research and some of the modern methods are now i ..."
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We describe the primality testing algorithms in use in some popular computer algebra systems, and give some examples where they break down in practice. 1 Introduction In recent years, fast primality testing algorithms have been a popular subject of research and some of the modern methods are now incorporated in computer algebra systems (CAS) as standard. In this review I give some details of the implementations of these algorithms and a number of examples where the algorithms prove inadequate. The algebra systems reviewed are Mathematica, Maple V, Axiom and Pari/GP. The versions we were able to use were Mathematica 2.1 for Sparc, copyright dates 19881992; Maple V Release 2, copyright dates 19811993; Axiom Release 1.2 (version of February 18, 1993); Pari/GP 1.37.3 (Sparc version, dated November 23, 1992). The tests were performed on Sparc workstations. Primality testing is a large and growing area of research. For further reading and comprehensive bibliographies, the interested re...
Several generalizations of Weil sums
 J. Number Theory
, 1994
"... We consider several generalizations and variations of the character sum inequalities of Weil and Burgess. A number of incomplete character sum inequalities are proved while further conjectures are formulated. These inequalities are motivated by extremal graph theory with applications to problems in ..."
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We consider several generalizations and variations of the character sum inequalities of Weil and Burgess. A number of incomplete character sum inequalities are proved while further conjectures are formulated. These inequalities are motivated by extremal graph theory with applications to problems in computer science. 1 1.
Elliptic Curve Normalization
 Crypto Group Technical Report Series CG2001/2 , Univ. Catholique de Louvain
, 2001
"... Let y 2 = x 3 + ax + b be an elliptic curve over F p , p a prime number greater than 3, and consider a, b # [1, p]. In this paper, we study elliptic curve isomorphisms, with a view towards reduction in the size of elliptic curves coefficients. We first consider reducing the ratio a/b. We then apply ..."
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Let y 2 = x 3 + ax + b be an elliptic curve over F p , p a prime number greater than 3, and consider a, b # [1, p]. In this paper, we study elliptic curve isomorphisms, with a view towards reduction in the size of elliptic curves coefficients. We first consider reducing the ratio a/b. We then apply these considerations to determine the number of elliptic curve isomorphism classes. Later we work on both coefficients. We introduce the number M(p) as the lower bound of all M # N such that each isomorphism class has a representative with max(a, b) < M . Using results from the theory of uniform distributions, we prove an upper and lower bound on M(p).
TORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION
"... Abstract. We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results may be viewed as refinements on torsion bounds of Silverberg and PrasadYogananda: our bounds take into account the class number of the CM orde ..."
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Abstract. We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results may be viewed as refinements on torsion bounds of Silverberg and PrasadYogananda: our bounds take into account the class number of the CM order and the splitting of the prime in the CM field. In many cases we can show that our refined bounds are optimal or asymptotically optimal. We also derive asymptotic upper and lower bounds on the least degree of a CMpoint on X1(N). These are compared to bounds for the least degree for which there exist infinitely many rational points on X1(N): we deduce that for (effectively computably) sufficiently large N, X1(N) will have a rational CM point of degree smaller than the degrees of at least all but finitely many nonCM points. 1.
Vinogradov's Method and Some Applications
, 1996
"... In this talk we consider in an elementary way some simple problems which relate to incomplete sums and can be studied by appealing to a classical method of Vinogradov and its modifications. Vinogradov's idea was to use finite Fourier transforms in order to estimate an incomplete sum by means of comp ..."
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In this talk we consider in an elementary way some simple problems which relate to incomplete sums and can be studied by appealing to a classical method of Vinogradov and its modifications. Vinogradov's idea was to use finite Fourier transforms in order to estimate an incomplete sum by means of complete but in some respects more complicated sums. A natural application was to estimate an incomplete sum of multiplicative characters by means of Gaussian sums. This application can be generalized in many natural ways; e.g., we may find bounds for the least nonnegative residue or nonresidue of a polynomial modulo a prime. Mordell and some others modified Vinogradov's method in order to find small solutions of congruences or small boxes containing solutions of a system of equations over finite fields. We survey these results briefly and in a very elementary way. Thereafter we consider a new interesting application. Code division multiple access systems require large families of sequences with...