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Pairingbased Cryptography at High Security Levels
 Proceedings of Cryptography and Coding 2005, volume 3796 of LNCS
, 2005
"... Abstract. In recent years cryptographic protocols based on the Weil and Tate pairings on elliptic curves have attracted much attention. A notable success in this area was the elegant solution by Boneh and Franklin [7] of the problem of efficient identitybased encryption. At the same time, the secur ..."
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Abstract. In recent years cryptographic protocols based on the Weil and Tate pairings on elliptic curves have attracted much attention. A notable success in this area was the elegant solution by Boneh and Franklin [7] of the problem of efficient identitybased encryption. At the same time, the security standards for public key cryptosystems are expected to increase, so that in the future they will be capable of providing security equivalent to 128, 192, or 256bit AES keys. In this paper we examine the implications of heightened security needs for pairingbased cryptosystems. We first describe three different reasons why highsecurity users might have concerns about the longterm viability of these systems. However, in our view none of the risks inherent in pairingbased systems are sufficiently serious to warrant pulling them from the shelves. We next discuss two families of elliptic curves E for use in pairingbased cryptosystems. The first has the property that the pairing takes values in the prime field Fp over which the curve is defined; the second family consists of supersingular curves with embedding degree k = 2. Finally, we examine the efficiency of the Weil pairing as opposed to the Tate pairing and compare a range of choices of embedding degree k, including k = 1 and k = 24. Let E be the elliptic curve 1.
Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms
, 2001
"... The fundamental operation in elliptic curve cryptographic schemes is that of point multiplication of an elliptic curve point by an integer. This paper describes a new method for accelerating this operation on classes of elliptic curves that have efficientlycomputable endomorphisms. One advantage of ..."
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Cited by 75 (0 self)
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The fundamental operation in elliptic curve cryptographic schemes is that of point multiplication of an elliptic curve point by an integer. This paper describes a new method for accelerating this operation on classes of elliptic curves that have efficientlycomputable endomorphisms. One advantage of the new method is that it is applicable to a larger class of curves than previous such methods.
The two faces of lattices in cryptology
 In Cryptography and lattices
, 2001
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Analysis of PSLQ, An Integer Relation Finding Algorithm
 Mathematics of Computation
, 1999
"... Let K be either the real, complex, or quaternion number system and let O(K) be the corresponding integers. Let × = (Xl, • • • , ×n) be a vector in K n. The vector × has an integer relation if there exists a vector m = (ml,..., mn) E O(K) n, m = _ O, such that mlx I + m2x 2 +... + mnXn = O. In th ..."
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Cited by 69 (27 self)
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Let K be either the real, complex, or quaternion number system and let O(K) be the corresponding integers. Let × = (Xl, • • • , ×n) be a vector in K n. The vector × has an integer relation if there exists a vector m = (ml,..., mn) E O(K) n, m = _ O, such that mlx I + m2x 2 +... + mnXn = O. In this paper we define the parameterized integer relation construction algorithm PSLQ(r), where the parameter rcan be freely chosen in a certain interval. Beginning with an arbitrary vector X = (Xl,..., Xn) _ K n, iterations of PSLQ(r) will produce lower bounds on the norm of any possible relation for X. Thus PS/Q(r) can be used to prove that there are no relations for × of norm less than a given size. Let M x be the smallest norm of any relation for ×. For the real and complex case and each fixed parameter rin a certain interval, we prove that PSLQ(r) constructs a relation in less than O(fl 3 + n 2 log Mx) iterations.
A Construction of a SpaceTime Code Based on Number Theory
 IEEE Trans. Inform. Theory
, 2002
"... We construct a full data rate spacetime block code over M =2 transmit antennas and T =2 symbol periods, and we prove that it achieves a transmit diversity of 2 over all constellations carved from Z[i] . Further, we optimize the coding gain of the proposed code and then compare it to the Alamouti co ..."
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Cited by 69 (2 self)
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We construct a full data rate spacetime block code over M =2 transmit antennas and T =2 symbol periods, and we prove that it achieves a transmit diversity of 2 over all constellations carved from Z[i] . Further, we optimize the coding gain of the proposed code and then compare it to the Alamouti code. It is shown that the new code outperforms the Alamouti code at low and high SNR when the number of receive antennas N>1. The performance improvement is further enhanced when N or the size of the constellation increases. We relate the problem of spacetime diversity gain to algebraic number theory, and the coding gain optimization to the theory of simultaneous Diophantine approximation in the geometry of numbers. We find that the coding gain optimization is equivalent to find irrational numbers "the furthest" from any simultaneous rational approximations.
Efficient Algorithms for Elliptic Curve Cryptosystems
, 1997
"... Elliptic curves are the basis for a relative new class of publickey schemes. It is predicted that elliptic curves will replace many existing schemes in the near future. It is thus of great interest to develop algorithms which allow efficient implementations of elliptic curve crypto systems. This th ..."
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Cited by 69 (9 self)
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Elliptic curves are the basis for a relative new class of publickey schemes. It is predicted that elliptic curves will replace many existing schemes in the near future. It is thus of great interest to develop algorithms which allow efficient implementations of elliptic curve crypto systems. This thesis deals with such algorithms. Efficient algorithms for elliptic curves can be classified into lowlevel algorithms, which deal with arithmetic in the underlying finite field and highlevel algorithms, which operate with the group operation. This thesis describes three new algorithms for efficient implementations of elliptic curve cryptosystems. The first algorithm describes the application of the KaratsubaOfman Algorithm to multiplication in composite fields GF ((2 n ) m ). The second algorithm deals with efficient inversion in composite Galois fields of the form GF ((2 n ) m ). The third algorithm is an entirely new approach which accelerates the multiplication of points which i...
Fully homomorphic encryption with relatively small key and ciphertext sizes
 In Public Key Cryptography — PKC ’10, Springer LNCS 6056
, 2010
"... Abstract. We present a fully homomorphic encryption scheme which has both relatively small key and ciphertext size. Our construction follows that of Gentry by producing a fully homomorphic scheme from a “somewhat ” homomorphic scheme. For the somewhat homomorphic scheme the public and private keys c ..."
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Cited by 67 (6 self)
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Abstract. We present a fully homomorphic encryption scheme which has both relatively small key and ciphertext size. Our construction follows that of Gentry by producing a fully homomorphic scheme from a “somewhat ” homomorphic scheme. For the somewhat homomorphic scheme the public and private keys consist of two large integers (one of which is shared by both the public and private key) and the ciphertext consists of one large integer. As such, our scheme has smaller message expansion and key size than Gentry’s original scheme. In addition, our proposal allows efficient fully homomorphic encryption over any field of characteristic two. 1
Mahler's Measure and Special Values of Lfunctions
, 1998
"... this paper is to describe an attempt to understand and generalize a recent formula of Deninger [1997] by means of systematic numerical experiment. This conjectural formula, ..."
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Cited by 65 (1 self)
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this paper is to describe an attempt to understand and generalize a recent formula of Deninger [1997] by means of systematic numerical experiment. This conjectural formula,
The Tate pairing and the discrete logarithm applied to elliptic curve cryptosystems
 IEEE Transactions on Information Theory
, 1999
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Speeding Up Computations via Molecular Biology
, 1994
"... : We show how to extend the recent result of Adleman [1] to use biological experiments to directly solve any NP problem. We, then, show how to use this method to speedup a large class of important problems. 1. Introduction In a recent breakthrough Adleman [1] showed how to use biological experiment ..."
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: We show how to extend the recent result of Adleman [1] to use biological experiments to directly solve any NP problem. We, then, show how to use this method to speedup a large class of important problems. 1. Introduction In a recent breakthrough Adleman [1] showed how to use biological experiments to solve instances of the famous Hamiltonian Path Problem (HPP). Since this problem is known to be NPcomplete it follows that biology can be used to solve any problem from NP. Recall that all problems in NP can be reduced to any NPcomplete one. However, this does not mean that all instances of NP problems can be solved in a feasible sense. Adleman solves the HPP in a totally brute force way: he designs a biological system that "tries" all possible tours of the given cities. The speed of any computer, biological or not, is determined by two factors: (i) how many parallel processes it has; (ii) how many steps each can perform per unit time. The exciting point about biology is that the firs...