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Computing zeta functions over finite fields
 Contemporary Math
, 1999
"... Abstract. In this report, we discuss the problem of computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo p m of the zeta function of a hypersurface, where p is the characteristic of the finite field. 1991 Mathematics Subj ..."
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Cited by 14 (3 self)
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Abstract. In this report, we discuss the problem of computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo p m of the zeta function of a hypersurface, where p is the characteristic of the finite field. 1991 Mathematics Subject Classification: 11Y16, 11T99, 14Q15. 1.
POINT COUNTING IN FAMILIES OF HYPERELLIPTIC CURVES IN CHARACTERISTIC 2
"... Let ĒΓ be a family of hyperelliptic curves over F2 alg cl with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm for computing the zeta function of Ē¯γ, with ¯γ in a degree n extension field of F, which has as time complexity Õ(n3) bit operations ..."
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Cited by 12 (5 self)
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Let ĒΓ be a family of hyperelliptic curves over F2 alg cl with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm for computing the zeta function of Ē¯γ, with ¯γ in a degree n extension field of F, which has as time complexity Õ(n3) bit operations and memory requirements O(n2) bits. With a slightly different algorithm we can get time O(n2.667) and memory O(n2.5), and the computation for n curves of the family can be done in time Õ(n3.376). All of these algorithms are polynomialtime in the genus.
Semirings and Semigroup Actions in PublicKey Cryptography
, 2002
"... by Christopher J. Monico In this dissertation, several generalizations of cryptographic protocols based on the Discrete Logarithm Problem (DLP) are examined. ..."
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Cited by 9 (3 self)
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by Christopher J. Monico In this dissertation, several generalizations of cryptographic protocols based on the Discrete Logarithm Problem (DLP) are examined.
Classical and quantum polynomial reconstruction via Legendre symbol evaluation
 Journal of Complexity
, 2004
"... We consider the problem of recovering a hidden monic polynomial f(X) of degree d ≥ 1 over a finite field Fp of p elements given a black box which, for any x ∈ Fp, evaluates the quadratic character of f(x). We design a classical algorithm of complexity O(d 2 p d+ε) and also show that the quantum quer ..."
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Cited by 3 (2 self)
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We consider the problem of recovering a hidden monic polynomial f(X) of degree d ≥ 1 over a finite field Fp of p elements given a black box which, for any x ∈ Fp, evaluates the quadratic character of f(x). We design a classical algorithm of complexity O(d 2 p d+ε) and also show that the quantum query complexity of this problem is O(d). Some of our results extend those of Wim van Dam, Sean Hallgren and Lawrence Ip obtained in the case of a linear polynomial f(X) = X +s (with unknown s); some are new even in this case. 1 1
Computing Jacobi Symbols Modulo Sparse Integers And Polynomials And Some Applications
 J. Algorithms
"... We describe a polynomial time algorithm to compute Jacobi symbols of exponentially large integers of special form, including socalled sparse integers which are exponentially large integers with only polynomially many nonzero binary digits. In a number of papers sequences of Jacobi symbols have bee ..."
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Cited by 1 (0 self)
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We describe a polynomial time algorithm to compute Jacobi symbols of exponentially large integers of special form, including socalled sparse integers which are exponentially large integers with only polynomially many nonzero binary digits. In a number of papers sequences of Jacobi symbols have been proposed as generators of cryptographically secure pseudorandom bits. Our algorithm allows us to use much larger moduli in such constructions. We also use our algorithm to design a probabilistic polynomial time test which decides if a given integer of the aforementioned type is a perfect square (assuming the Extended Riemann Hypothesis). We also obtain analogues of these results for polynomials over finite fields. Moreover, in this case the perfect square testing algorithm is unconditional. These results can be compared with many known NPhardness results for some natural problems on sparse integers and polynomials.
An Extremely Small And Efficient Identification Scheme
"... . We present a new identification scheme which is based on Legendre symbols modulo a certain hidden prime and which is naturally suited for low power, low memory applications. 1 Overview One of the most desirable cryptographic functions is a secure, small, zeroknowledge publickey identification sc ..."
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. We present a new identification scheme which is based on Legendre symbols modulo a certain hidden prime and which is naturally suited for low power, low memory applications. 1 Overview One of the most desirable cryptographic functions is a secure, small, zeroknowledge publickey identification scheme. The applications are many and obvious  even the single application of credit/smartcard security is enough to stimulate research. In this paper, we present a scheme that requires extremely little computing power to perform a verification and to which we refer as FLIP (Fast Legendre Identification Protocol). Our scheme is unbalanced by design: the party proving his/her identity needs almost no computing power, while the party to whom the identity is being proved needs only a very small amount. Our scheme is based on the assumption that integer factorization is a "hard problem." In fact, we believe that the only feasible attack on our scheme is via the factorization of a certain modulus...
unknown title
, 2006
"... If an eavesdropper Eve is equipped with quantum computers, she can easily break the public key exchange protocols used today. In this paper we will discuss the postquantum DiffieHellman key exchange and private key exchange protocols. ..."
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If an eavesdropper Eve is equipped with quantum computers, she can easily break the public key exchange protocols used today. In this paper we will discuss the postquantum DiffieHellman key exchange and private key exchange protocols.
unknown title
, 2006
"... If an eavesdropper Eve is equipped with quantum computers, she can easily break the public key exchange protocols used today. In this paper we will discuss the postquantum DiffieHellman key exchange and private key exchange protocols. ..."
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If an eavesdropper Eve is equipped with quantum computers, she can easily break the public key exchange protocols used today. In this paper we will discuss the postquantum DiffieHellman key exchange and private key exchange protocols.