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242
K.: Signatures for network coding
 In: International Journal on Information and Coding
, 2006
"... Abstract. This paper presents a practical digital signature scheme to be used in conjunction with network coding. This signature scheme seems to be the first example of a homomorphic signature scheme. Furthermore, our idea simultaneously provides authentication and detects malicious nodes that inten ..."
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Cited by 81 (3 self)
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Abstract. This paper presents a practical digital signature scheme to be used in conjunction with network coding. This signature scheme seems to be the first example of a homomorphic signature scheme. Furthermore, our idea simultaneously provides authentication and detects malicious nodes that intentionally corrupt content on the network. 1.
Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms
 ACTA ARITHMETICA
, 1994
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Ternary Diophantine equations via Galois representations and modular forms
 CANAD J. MATH
, 2004
"... In this paper, we develop techniques for solving ternary Diophantine equations of the shape Ax n + By n = Cz 2, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters A, B and C ..."
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Cited by 48 (8 self)
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In this paper, we develop techniques for solving ternary Diophantine equations of the shape Ax n + By n = Cz 2, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters A, B and C. We conclude with an application of our results to certain classical polynomialexponential equations, such as those of Ramanujan–Nagell type.
Ranks of twists of elliptic curves and Hilbert’s tenth problem, arxiv:0904.3709v2 [math.NT
"... Abstract. In this paper we investigate the 2Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2Selmer rank, and we give lower bounds for the number of twists (with bound ..."
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Cited by 35 (4 self)
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Abstract. In this paper we investigate the 2Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial MordellWeil group, and (assuming the ShafarevichTate conjecture) many others with infinite cyclic MordellWeil group. Using work of Poonen and Shlapentokh, it follows from our results that if the ShafarevichTate conjecture holds, then Hilbert’s Tenth Problem has a negative answer over the ring of integers of every number field. 1. Introduction and
Chinese Remaindering Based Cryptosystems in the Presence of Faults
 TO APPEAR IN JOURNAL OF CRYPTOLOGY.
"... We present some observations on publickey cryptosystems that use the Chinese remaindering algorithm. Our results imply that careless implementations of such systems could be vulnerable. Only one faulty signature, in some explained context, is enough to recover the secret key. ..."
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Cited by 33 (5 self)
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We present some observations on publickey cryptosystems that use the Chinese remaindering algorithm. Our results imply that careless implementations of such systems could be vulnerable. Only one faulty signature, in some explained context, is enough to recover the secret key.
Elliptic curves with complex multiplication and the conjecture of Birch and
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Primes Generated by Elliptic Curves
, 2003
"... For a rational elliptic curve in Weierstrass form, Chudnovsky and Chudnovsky considered the likelihood that the denominators of the xcoordinates of the multiples of a rational point are squares of primes. Assuming the point is the image of a rational point under an isogeny, we use Siegel’s Theorem t ..."
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Cited by 31 (9 self)
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For a rational elliptic curve in Weierstrass form, Chudnovsky and Chudnovsky considered the likelihood that the denominators of the xcoordinates of the multiples of a rational point are squares of primes. Assuming the point is the image of a rational point under an isogeny, we use Siegel’s Theorem to prove that only finitely many primes will arise. The same question is considered for elliptic curves in homogeneous form, prompting a visit to Ramanujan’s famous taxicab equation. Finiteness is provable for these curves with no extra assumptions. Finally, consideration is given to the possibilities for prime generation in higher rank.
Elliptic Curve Discrete Logarithms and the Index Calculus
"... . The discrete logarithm problem forms the basis of numerous cryptographic systems. The most effective attack on the discrete logarithm problem in the multiplicative group of a finite field is via the index calculus, but no such method is known for elliptic curve discrete logarithms. Indeed, Miller ..."
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Cited by 29 (4 self)
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. The discrete logarithm problem forms the basis of numerous cryptographic systems. The most effective attack on the discrete logarithm problem in the multiplicative group of a finite field is via the index calculus, but no such method is known for elliptic curve discrete logarithms. Indeed, Miller [23] has given a brief heuristic argument as to why no such method can exist. IN this note we give a detailed analysis of the index calculus for elliptic curve discrete logarithms, amplifying and extending miller's remarks. Our conclusions fully support his contention that the natural generalization of the index calculus to the elliptic curve discrete logarithm problem yields an algorithm with is less efficient than a bruteforce search algorithm. 0. Introduction The discrete logarithm problem for the multiplicative group F q of a finite field can be solved in subexponential time using the Index Calculus method, which appears to have been first discovered by Kraitchik [14, 15] in the 192...
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 28 (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
TWOCOVER DESCENT ON HYPERELLIPTIC CURVES
, 2009
"... We describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve, with the property that any rational point will lift to one of the covers. In particular, if the algorithm returns an empty set, then the hyperelliptic curve has no rational points. This provides a re ..."
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Cited by 28 (8 self)
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We describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve, with the property that any rational point will lift to one of the covers. In particular, if the algorithm returns an empty set, then the hyperelliptic curve has no rational points. This provides a relatively efficiently tested criterion for solvability ofhyperelliptic curves. We also discuss applications of this algorithm to curves ofgenus 1 and to curves with rational points.