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19
Synthesizers and Their Application to the Parallel Construction of PseudoRandom Functions
, 1995
"... A pseudorandom function is a fundamental cryptographic primitive that is essential for encryption, identification and authentication. We present a new cryptographic primitive called pseudorandom synthesizer and show how to use it in order to get a parallel construction of a pseudorandom function. ..."
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Cited by 42 (10 self)
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A pseudorandom function is a fundamental cryptographic primitive that is essential for encryption, identification and authentication. We present a new cryptographic primitive called pseudorandom synthesizer and show how to use it in order to get a parallel construction of a pseudorandom function. We show several NC¹ implementations of synthesizers based on concrete intractability assumptions as factoring and the DiffieHellman assumption. This yields the first parallel pseudorandom functions (based on standard intractability assumptions) and the only alternative to the original construction of Goldreich, Goldwasser and Micali. In addition, we show parallel constructions of synthesizers based on other primitives such as weak pseudorandom functions or trapdoor oneway permutations. The security of all our constructions is similar to the security of the underlying assumptions. The connection with problems in Computational Learning Theory is discussed.
Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem
, 2001
"... 1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are ..."
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Cited by 14 (3 self)
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1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1
Limits to List Decodability of Linear Codes
 In Proc. 34th ACM Symp. on Theory of Computing
, 2002
"... We consider the problem of the best possible relation between the list decodability of a binary linear code and its minimum distance. We prove, under a widelybelieved numbertheoretic conjecture, that the classical "Johnson bound" gives, in general, the best possible relation between the list decod ..."
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Cited by 10 (5 self)
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We consider the problem of the best possible relation between the list decodability of a binary linear code and its minimum distance. We prove, under a widelybelieved numbertheoretic conjecture, that the classical "Johnson bound" gives, in general, the best possible relation between the list decoding radius of a code and its minimum distance. The analogous result is known to hold by a folklore random coding argument for the case of nonlinear codes, but the linear case is more subtle and has remained open.
On Hooley's Theorem With Weights
, 1995
"... We adapt Hooley's proof that the Generalized Riemann Hypothesis implies the Artin Conjecture for primitive roots to various other problems. We consider the sum p#x f (i p ) where i p is the index of 2 modulo p and f is a given function. In various cases we establish asymptotic formulas for such a ..."
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Cited by 7 (2 self)
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We adapt Hooley's proof that the Generalized Riemann Hypothesis implies the Artin Conjecture for primitive roots to various other problems. We consider the sum p#x f (i p ) where i p is the index of 2 modulo p and f is a given function. In various cases we establish asymptotic formulas for such a sum and analyse the constants. While we claim no originality, we outline the method to approach this problem in a fairly general case.
A twovariable Artin conjecture
 J. Number Theory
"... Abstract. Let a, b ∈ Q ∗ be rational numbers that are multiplicatively independent. We study the natural density δ(a, b) of the set of primes p for which the subgroup of F ∗ p generated by (a mod p) contains (b mod p). It is shown that, under assumption of the generalized Riemann hypothesis, the den ..."
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Cited by 7 (3 self)
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Abstract. Let a, b ∈ Q ∗ be rational numbers that are multiplicatively independent. We study the natural density δ(a, b) of the set of primes p for which the subgroup of F ∗ p generated by (a mod p) contains (b mod p). It is shown that, under assumption of the generalized Riemann hypothesis, the density δ(a, b) exists and equals a positive rational multiple of the universal constant S = � p prime (1 − p/(p3 − 1)). An explicit value of δ(a, b) is given under mild conditions on a and b. This extends and corrects earlier work of Stephens [13]. We also discuss the relevance of the result in the context of second order linear recurrent sequences and some numerical aspects of the determination of δ(a, b). 1.
Fast Elliptic Curve Point Counting Using Gaussian Normal Basis
 Proc. of ANTS V, Lecture Notes in Comput. Sci. 2369
, 2002
"... In this paper we present an improved algorithm for counting points on elliptic curves over finite fields. It is mainly based on SatohSkjernaa Taguchi algorithm [SST01], and uses a Gaussian Normal Basis (GNB) of small type t 4. In practice, about 42% (36% for prime N) of fields in cryptograph ..."
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Cited by 6 (0 self)
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In this paper we present an improved algorithm for counting points on elliptic curves over finite fields. It is mainly based on SatohSkjernaa Taguchi algorithm [SST01], and uses a Gaussian Normal Basis (GNB) of small type t 4. In practice, about 42% (36% for prime N) of fields in cryptographic context (i.e., for p = 2 and 160 < N < 600) have such bases. They can be lifted from F p N to Z p N in a natural way. From the specific properties of GNBs, e#cient multiplication and the Frobenius substitution are available. Thus a fast norm computation algorithm is derived, which runs in O(N log N) with O(N ) space, where the time complexity of multiplying two nbit objects is O(n ). As a result, for all small characteristic p, we reduced the time complexity of the SSTalgorithm ) and the space complexity still fits in O(N ). Our approach is expected to be applicable to the AGM since the exhibited improvement is not restricted to only [SST01].
THE GENERALIZED ARTIN CONJECTURE AND ARITHMETIC ORBIFOLDS
"... Let K be a number field with positive unit rank, and let OK denote the ring of integers of K. A generalization of Artin’s primitive root conjecture is that that O × K is a primitive root set for infinitely many prime ideals. We prove this with additional conjugacy conditions in the case when K is G ..."
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Cited by 3 (3 self)
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Let K be a number field with positive unit rank, and let OK denote the ring of integers of K. A generalization of Artin’s primitive root conjecture is that that O × K is a primitive root set for infinitely many prime ideals. We prove this with additional conjugacy conditions in the case when K is Galois with unit rank greater than three. This was previously known under the assumption of the Generalized Riemann Hypothesis. From our result, we deduce a topological corollary about the structure of quotients of PSL2(OK).
Compatible systems of mod p Galois representations II
"... Compatible systems of ndimensional, mod p representations of absolute Galois groups of number fields were considered by Serre in his study of openness of images of adelic Galois representations arising from elliptic curves in [S2]. ..."
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Cited by 2 (1 self)
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Compatible systems of ndimensional, mod p representations of absolute Galois groups of number fields were considered by Serre in his study of openness of images of adelic Galois representations arising from elliptic curves in [S2].
CONJECTURES INVOLVING ARITHMETICAL SEQUENCES
"... Abstract. We pose thirty conjectures on arithmetical sequences, most of which are about monotonicity of sequences of the form ( n √ an)n�1 or the form ( n+1 √ an+1 / n √ an)n�1, where (an)n�1 is a numbertheoretic or combinatorial sequence of positive integers. This material might stimulate further ..."
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Cited by 1 (1 self)
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Abstract. We pose thirty conjectures on arithmetical sequences, most of which are about monotonicity of sequences of the form ( n √ an)n�1 or the form ( n+1 √ an+1 / n √ an)n�1, where (an)n�1 is a numbertheoretic or combinatorial sequence of positive integers. This material might stimulate further research. 1.
Distribution of Residues Modulo p
, 2006
"... The distribution of quadratic residues and nonresidues modulo p has been of intrigue to the number theorists of the last several decades. Although Gauss ’ celebrated Quadratic Reciprocity Law gives a beautiful criterion to decide whether a given number is a quadratic residue modulo p or not, it ..."
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The distribution of quadratic residues and nonresidues modulo p has been of intrigue to the number theorists of the last several decades. Although Gauss ’ celebrated Quadratic Reciprocity Law gives a beautiful criterion to decide whether a given number is a quadratic residue modulo p or not, it