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We present a stream cipher based on mathematical considerations which is much faster then many other mathematical ciphers. Its security is based on the uniformity of the distribution of the quadratic symbol in function fields.
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Cryptographic primitives by P. Garrett Cryptography in the real world today by D. Lieman Public-key cryptography and proofs of security by N. Howgrave-Graham Elliptic curves and cryptography by J. H. Silverman Towards faster cryptosystems... more
Cryptographic primitives by P. Garrett Cryptography in the real world today by D. Lieman Public-key cryptography and proofs of security by N. Howgrave-Graham Elliptic curves and cryptography by J. H. Silverman Towards faster cryptosystems I by W. Whyte Towards faster cryptosystems, II by W. D. Banks Playing "hide-and-seek" with numbers: The hidden number problem, lattices and exponential sums by I. E. Shparlinski Index.
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ABSTRACT
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has a rational solution. An extensive compilation of the older history of the problem is given in Dickson [6]. Within the past century, researchers have tried to exploit (either explicitly or unknowingly) the fact that the curve (0.1) is... more
has a rational solution. An extensive compilation of the older history of the problem is given in Dickson [6]. Within the past century, researchers have tried to exploit (either explicitly or unknowingly) the fact that the curve (0.1) is in fact an elliptic curve. During the nineteenth century, Lucas, and later Sylvester, used a descent argument to prove that (0.1) had no solution for infinitely many D in certain congruence classes mod 9 and 18 (see [6], Ch. XXI). Zagier and Kramarz [19] have produced a great deal of numerical evidence about the L-series of the curves; based on these computations, they have argued heuristically that for
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All the notation in this appendix will be as in the preceding paper. Let f be a Maass form which is a newform for Fo(N), with eigenvalue A and central character X, normalized so that (f, f) = 1. We have seen that the size of p(l), the... more
All the notation in this appendix will be as in the preceding paper. Let f be a Maass form which is a newform for Fo(N), with eigenvalue A and central character X, normalized so that (f, f) = 1. We have seen that the size of p(l), the first Fourier coefficient of f, is intimately related to ...
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Progress in Computer Science and Applied Logic, Vol. 20 © 2001 BirkhauserVerlag Basel/Switzerland Polynomial Rings and Efficient Public Key Authentication II Jeffrey Hoffstein and Joseph H. Silverman Abstract. In a recent paper [3] a... more
Progress in Computer Science and Applied Logic, Vol. 20 © 2001 BirkhauserVerlag Basel/Switzerland Polynomial Rings and Efficient Public Key Authentication II Jeffrey Hoffstein and Joseph H. Silverman Abstract. In a recent paper [3] a highly efficient public key authentication scheme called ...
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Cryptographic primitives by P. Garrett Cryptography in the real world today by D. Lieman Public-key cryptography and proofs of security by N. Howgrave-Graham Elliptic curves and cryptography by J. H. Silverman Towards faster cryptosystems... more
Cryptographic primitives by P. Garrett Cryptography in the real world today by D. Lieman Public-key cryptography and proofs of security by N. Howgrave-Graham Elliptic curves and cryptography by J. H. Silverman Towards faster cryptosystems I by W. Whyte Towards faster cryptosystems, II by W. D. Banks Playing "hide-and-seek" with numbers: The hidden number problem, lattices and exponential sums by I. E. Shparlinski Index.
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We obtain an exponential lower bound on the non-linear complexity of the new pseudo-random function, introduced recently by M. Naor and O. Reingold. This bound is an extension of the lower bound on the linear complexity of this function... more
We obtain an exponential lower bound on the non-linear complexity of the new pseudo-random function, introduced recently by M. Naor and O. Reingold. This bound is an extension of the lower bound on the linear complexity of this function that has been obtained by F. Gri-n and I. E. Shparlinski.
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This paper surveys the connection between the elliptic curve E_D: x^3 + y^3 = D and a certain metaplectic form on the cubic cover of GL(3) which has the property that its m,n^th Whittaker--Fourier coefficient is essentially the L--series... more
This paper surveys the connection between the elliptic curve E_D: x^3 + y^3 = D and a certain metaplectic form on the cubic cover of GL(3) which has the property that its m,n^th Whittaker--Fourier coefficient is essentially the L--series of the curve E_m^2n. One may obtain information about the collective behavior the curves E_D by exploiting this connection; for example, one can prove: Theorem: Fix any prime p 3, and any congruence class c mod p. Then there are infinitely many D congruent to c mod p such that the curve E_D has no rational solutions. This paper is fairly self-contained; no prior knowledge of algebraic number theory, analytic number theory or metaplectic forms is assumed. On the other hand, this paper is a survey, no proofs are included.
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This paper surveys the connection between the elliptic curve E_D: x^3 + y^3 = D and a certain metaplectic form on the cubic cover of GL(3) which has the property that its m,n^{th} Whittaker--Fourier coefficient is essentially the... more
This paper surveys the connection between the elliptic curve E_D: x^3 + y^3 = D and a certain metaplectic form on the cubic cover of GL(3) which has the property that its m,n^{th} Whittaker--Fourier coefficient is essentially the L--series of the curve E_{m^2n}. One may obtain information about the collective behavior the curves E_D by exploiting this connection; for example, one can prove: Theorem: Fix any prime p \ne 3, and any congruence class c mod p. Then there are infinitely many D congruent to c mod p such that the curve E_D has no rational solutions. This paper is fairly self-contained; no prior knowledge of algebraic number theory, analytic number theory or metaplectic forms is assumed. On the other hand, this paper is a survey, no proofs are included.
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Let n ≥ 3 be a fixed integer and let F be a global field containing the n-th roots of unity. In this paper we study the collective behavior of the n-th order twists of a fixed Hecke L-series for F . To do so, we introduce a double... more
Let n ≥ 3 be a fixed integer and let F be a global field containing the n-th roots of unity. In this paper we study the collective behavior of the n-th order twists of a fixed Hecke L-series for F . To do so, we introduce a double Dirichlet series in two complex variables (s, w) which is a weighted sum of the twists, and obtain its meromorphic continuation. We also study related sums of n-th order Gauss sums. These objects together satisfy a nonabelian group of functional equations in (s, w) of order 32. Mathematics Subject Classification (1991): 11R42, 11F66, 11F70, 11M41, 11R47.
This paper gives a new example of exploiting the idea of using polynomials with restricted coefficients over finite fields and rings to construct reliable cryptosystems and identification schemes.
has a rational solution. An extensive compilation of the older history of the problem is given in Dickson [6]. Within the past century, researchers have tried to exploit (either explicitly or unknowingly) the fact that the curve (0.1) is... more
has a rational solution. An extensive compilation of the older history of the problem is given in Dickson [6]. Within the past century, researchers have tried to exploit (either explicitly or unknowingly) the fact that the curve (0.1) is in fact an elliptic curve. During the nineteenth century, Lucas, and later Sylvester, used a descent argument to prove that (0.1) had no solution for infinitely many D in certain congruence classes mod 9 and 18 (see [6], Ch. XXI). Zagier and Kramarz [19] have produced a great deal of numerical evidence about the L-series of the curves; based on these computations, they have argued heuristically that for
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The main result in this paper is the explicit computation of the functional equation satisfied by the GL(3) Mellin transform of a twisted non-cuspidal metaplectic form of non-trivial level. For concreteness, we work with one particular... more
The main result in this paper is the explicit computation of the functional equation satisfied by the GL(3) Mellin transform of a twisted non-cuspidal metaplectic form of non-trivial level. For concreteness, we work with one particular metaplectic form, automorphic under Λ(3), although our methods extend without change to any form automorphic with respect to Λ(p),p an odd prime. We clearly
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... Following the established tradition, we call the communicating parties Alice and Bob. ... Bob then computes the reduction $ of the polynomial figi+...+ fkdk modulo the ideal generated by and sends the polynomial $= TO-I-#. Step 5 To... more
... Following the established tradition, we call the communicating parties Alice and Bob. ... Bob then computes the reduction $ of the polynomial figi+...+ fkdk modulo the ideal generated by and sends the polynomial $= TO-I-#. Step 5 To decrypt the message,. ...
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This paper surveys the connection between the elliptic curve E_D: x^3 + y^3 = D and a certain metaplectic form on the cubic cover of GL(3) which has the property that its m,n^{th} Whittaker--Fourier coefficient is essentially the... more
This paper surveys the connection between the elliptic curve E_D: x^3 + y^3 = D and a certain metaplectic form on the cubic cover of GL(3) which has the property that its m,n^{th} Whittaker--Fourier coefficient is essentially the L--series of the curve E_{m^2n}. One may obtain information about the collective behavior the curves E_D by exploiting this connection; for example, one can prove: Theorem: Fix any prime p \ne 3, and any congruence class c mod p. Then there are infinitely many D congruent to c mod p such that the curve E_D has no rational solutions. This paper is fairly self-contained; no prior knowledge of algebraic number theory, analytic number theory or metaplectic forms is assumed. On the other hand, this paper is a survey, no proofs are included.