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121
Integer Factorization
, 2005
"... Many public key cryptosystems depend on the difficulty of factoring large integers. This thesis serves as a source for the history and development of integer factorization algorithms through time from trial division to the number field sieve. It is the first description of the number field sieve fro ..."
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Cited by 113 (8 self)
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Many public key cryptosystems depend on the difficulty of factoring large integers. This thesis serves as a source for the history and development of integer factorization algorithms through time from trial division to the number field sieve. It is the first description of the number field sieve from an algorithmic point of view making it available to computer scientists for implementation. I have implemented the general number field sieve from this description and it is made publicly available from the Internet. This means that a reference implementation is made available for future developers which also can be used as a framework where some of the sub
Integral points on elliptic curves and 3torsion in class groups
"... We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasiorthogonality in the MordellWeil lattice ([Sil6], [GS], [He]). We apply our results to break previous ..."
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Cited by 25 (6 self)
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We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasiorthogonality in the MordellWeil lattice ([Sil6], [GS], [He]). We apply our results to break previous
The Xedni Calculus And The Elliptic Curve Discrete Logarithm Problem
 Designs, Codes and Cryptography
, 1999
"... . Let E=Fp be an elliptic curve defined over a finite field, and let S ..."
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Cited by 22 (1 self)
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. Let E=Fp be an elliptic curve defined over a finite field, and let S
Elliptic curve cryptosystems on reconfigurable hardware
 MASTER’S THESIS, WORCESTER POLYTECHNIC INST
, 1998
"... Security issues will play an important role in the majority of communication and computer networks of the future. As the Internet becomes more and more accessible to the public, security measures will have to be strengthened. Elliptic curve cryptosystems allow for shorter operand lengths than other ..."
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Cited by 21 (0 self)
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Security issues will play an important role in the majority of communication and computer networks of the future. As the Internet becomes more and more accessible to the public, security measures will have to be strengthened. Elliptic curve cryptosystems allow for shorter operand lengths than other publickey schemes based on the discrete logarithm in finite fields and the integer factorization problem and are thus attractive for many applications. This thesis describes an implementation of a crypto engine based on elliptic curves. The underlying algebraic structures are composite Galois fields GF((2 n) m) in a standard base representation. As a major new feature, the system is developed for a reconfigurable platform based on Field Programmable Gate Arrays (FPGAs). FPGAs combine the flexibility of software solutions with the security of traditional hardware implementations. In particular, it is possible to easily change all algorithm parameters such as curve coefficients, field order, or field representation. The thesis deals with the design and implementation of elliptic curve point multiplicationarchitectures. The architectures are described in VHDL and mapped to Xilinx FPGA devices. Architectures over Galois fields of different order and representation were implemented and compared. Area and timing measurements are provided for all architectures. It is shown that a full point multiplication on elliptic curves of realworld size can be implemented on commercially available FPGAs.
Euler’s concordant forms
 Acta Arith
, 1996
"... In [6] Euler asks for a classification of those pairs of distinct nonzero integers M and N for which there are integer solutions (x, y, t, z) with xy ̸ = 0 to (1) x 2 + My 2 = t 2 and x 2 + Ny 2 = z 2. This is known as Euler’s concordant forms problem, and when M = −N Euler’s problem ..."
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Cited by 21 (1 self)
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In [6] Euler asks for a classification of those pairs of distinct nonzero integers M and N for which there are integer solutions (x, y, t, z) with xy ̸ = 0 to (1) x 2 + My 2 = t 2 and x 2 + Ny 2 = z 2. This is known as Euler’s concordant forms problem, and when M = −N Euler’s problem
The rank of elliptic curves over real quadratic number fields of class number 1
 Math. Comp
, 1995
"... Abstract. In this paper we describe an algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one. This algorithm extends the one originally described by Birch and SwinnertonDyer for curves over Q. Several examples are included. 1. ..."
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Cited by 16 (4 self)
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Abstract. In this paper we describe an algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one. This algorithm extends the one originally described by Birch and SwinnertonDyer for curves over Q. Several examples are included. 1.
Algebras associated to elliptic curves
 Trans. Amer. Math. Soc
, 1997
"... Abstract. This paper completes the classication of ArtinSchelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree ..."
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Cited by 14 (0 self)
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Abstract. This paper completes the classication of ArtinSchelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree one. We show that there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism, just as in the case where the algebras are assumed to be generated in degree one. In particular, we study the elliptic algebras A(+), A(−), and A(a), where a 2 P2, which were rst dened in an earlier paper. We omit a set S P2 consisting of 11 specied points where the algebras A(a) become too degenerate to be regular. Theorem. Let A represent A(+), A(−) or A(a), where a 2 P2 n S. Then A is an ArtinSchelter regular algebra of global dimension three. Moreover, A is a Noetherian domain with the same Hilbert series as the (appropriately graded) commutative polynomial ring in three variables. This, combined with our earlier results, completes the classication. 1.
Average Frobenius distributions for elliptic curves with nontrivial rational torsion
 TO APPEAR IN ACTA ARITHMETICA
, 2005
"... In this paper we consider the LangTrotter conjecture (Conjecture 1 below) for various families of elliptic curves with prescribed torsion structure. We prove that the LangTrotter conjecture holds in an average sense for these families of curves (see Theorem 3). ..."
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Cited by 14 (4 self)
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In this paper we consider the LangTrotter conjecture (Conjecture 1 below) for various families of elliptic curves with prescribed torsion structure. We prove that the LangTrotter conjecture holds in an average sense for these families of curves (see Theorem 3).
Multiples of integral points on elliptic curves
 J. Number Theory
"... Abstract. We show that if Lang’s conjectured lower bound on heights of points on elliptic curves exists, then there is an absolute constant C such that for any integral point P on a minimal elliptic curve with integral coefficients, nP is integral for at most one value of n> C. As a corollary, we ..."
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Cited by 10 (1 self)
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Abstract. We show that if Lang’s conjectured lower bound on heights of points on elliptic curves exists, then there is an absolute constant C such that for any integral point P on a minimal elliptic curve with integral coefficients, nP is integral for at most one value of n> C. As a corollary, we show that if E/Q is a fixed elliptic curve, then for all twists E ′ of E of sufficient height, and all torsionfree, rankone subgroups Γ ⊆ E ′ (Q), Γ contains at most 6 integral points. Explicit computations for congruent number curves are included.
ON THE SQUAREFREE SIEVE
, 2004
"... A squarefree sieve is a result that gives an upper bound for how often a squarefree polynomial may adopt values that are not squarefree. More generally, we may wish to control the behavior of a function depending on the largest square factor of P(x1,..., xn), where P is a squarefree polynomial. ..."
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Cited by 9 (5 self)
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A squarefree sieve is a result that gives an upper bound for how often a squarefree polynomial may adopt values that are not squarefree. More generally, we may wish to control the behavior of a function depending on the largest square factor of P(x1,..., xn), where P is a squarefree polynomial.