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Improved CRT algorithm for class polynomials in genus 2.” In: Algorithmic Number Theory — ANTSX. Edited by Everett Howe and Kiran Kedlaya
 Mathematical Science Publishers
"... Abstract. We present a generalization to genus 2 of the probabilistic algorithm in Sutherland [28] for computing Hilbert class polynomials. The improvement over the algorithm presented in [5] for the genus 2 case, is that we do not need to find a curve in the isogeny class with endomorphism ring whi ..."
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Abstract. We present a generalization to genus 2 of the probabilistic algorithm in Sutherland [28] for computing Hilbert class polynomials. The improvement over the algorithm presented in [5] for the genus 2 case, is that we do not need to find a curve in the isogeny class with endomorphism ring which is the maximal order: rather we present a probabilistic algorithm for “going up ” to a maximal curve (a curve with maximal endomorphism ring), once we find any curve in the right isogeny class. Then we use the structure of the Shimura class group and the computation of (ℓ, ℓ)isogenies to compute all isogenous maximal curves from an initial one. This article is an extended version of the version published at ANTS X. 1.
On Quadratic Polynomials for the Number Field Sieve
 Australian Computer Science Communications
, 1997
"... . The newest, and asymptotically the fastest known integer factorisation algorithm is the number field sieve. The area in which the number field sieve has the greatest capacity for improvement is polynomial selection. The best known polynomial selection method finds quadratic polynomials. In this pa ..."
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. The newest, and asymptotically the fastest known integer factorisation algorithm is the number field sieve. The area in which the number field sieve has the greatest capacity for improvement is polynomial selection. The best known polynomial selection method finds quadratic polynomials. In this paper we examine the smoothness properties of integer values taken by these polynomials. Given a quadratic NFS polynomial f , let \Delta be its discriminant. We show that a prime p can divide values taken by f only if (\Delta=p) = 1. We measure the effect of this residuosity property on the smoothness of fvalues by adapting a parameter ff, developed for analysis of MPQS, to quadratic NFS polynomials. We estimate the yield of smooth values for these polynomials as a function of ff, and conclude that practical changes in ff might bring significant changes in the yield of smooth and almost smooth polynomial values. Keywords: integer factorisation, number field sieve 1
Approximating the number of integers without large prime factors
 Mathematics of Computation
, 2004
"... Abstract. Ψ(x, y) denotes the number of positive integers ≤ x and free of prime factors>y. Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ(x, y) in the range (log x) 1+ɛ
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Abstract. Ψ(x, y) denotes the number of positive integers ≤ x and free of prime factors>y. Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ(x, y) in the range (log x) 1+ɛ <y ≤ x,whereɛ is a fixed positive number ≤ 1/2. In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate Ψ(x, y). The computational complexity of this algorithm is O ( � (log x)(log y)). We give numerical results which show that this algorithm provides accurate estimates for Ψ(x, y) andisfaster than conventional methods such as algorithms exploiting Dickman’s function. 1.
The twoparameter PoissonDirichlet point process
, 2007
"... The twoparameter PoissonDirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (i.e., the random point process obtai ..."
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The twoparameter PoissonDirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (i.e., the random point process obtained by regarding the masses as points in the positive real line) is given in terms of the correlation functions. Relying on this, we apply the theory of point processes to reveal mathematical structure of the twoparameter PoissonDirichlet distribution. Also, developing the Laplace transform approach due to Pitman and Yor, we will be able to extend several results previously known for the oneparameter case, and the MarkovKrein identity for the generalized Dirichlet process is discussed from a point of view of functional analysis based on the twoparameter PoissonDirichlet distribution. 1
ECC: Do We Need to Count?
, 1999
"... A prohibitive barrier faced by elliptic curve users is the difficulty of computing the curves' cardinalities. Despite recent theoretical breakthroughs, point counting still remains very cumbersome and intensively time consuming. ..."
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A prohibitive barrier faced by elliptic curve users is the difficulty of computing the curves' cardinalities. Despite recent theoretical breakthroughs, point counting still remains very cumbersome and intensively time consuming.
A GENERIC APPROACH TO SEARCHING FOR JACOBIANS
 MATHEMATICS OF COMPUTATION
, 2009
"... We consider the problem of finding cryptographically suitable Jacobians. By applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family, we can search for Jacobians containing a large subgroup of prime order. For a suitable distribution ..."
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We consider the problem of finding cryptographically suitable Jacobians. By applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family, we can search for Jacobians containing a large subgroup of prime order. For a suitable distribution of curves, the complexity is subexponential in genus 2, and O(N 1/12) in genus 3. We give examples of genus 2 and genus 3 hyperelliptic curves over prime fields with group orders over 180 bits in size, improving previous results. Our approach is particularly effective over lowdegree extension fields, where in genus 2 we find Jacobians over F p 2 and trace zero varieties over F p 3 with nearprime orders up to 372 bits in size. For p =2 61 − 1, the average time to find a group with 244bit nearprime order is under an hour on a PC.
On a combinatorial method for counting smooth numbers in sets of integers
 J. Number Theory
"... In this paper we prove a result for determining the number of integers without large prime factors lying in a given set S. We will apply it to give an easy proof that certain sufficiently dense sets A and B always produce the expected number of “smooth ” sums a + b, a ∈ A, b ∈ B. The proof of this r ..."
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In this paper we prove a result for determining the number of integers without large prime factors lying in a given set S. We will apply it to give an easy proof that certain sufficiently dense sets A and B always produce the expected number of “smooth ” sums a + b, a ∈ A, b ∈ B. The proof of this result is completely combinatorial and elementary. 1
Cryptanalysis of ISO/IEC 97961
"... Abstract. We describe two different attacks against the iso/iec 97961 signature standard for RSA and Rabin. Both attacks consist in an existential forgery under a chosenmessage attack: the attacker asks for the signature of some messages of his choice, and is then able to produce the signature of ..."
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Abstract. We describe two different attacks against the iso/iec 97961 signature standard for RSA and Rabin. Both attacks consist in an existential forgery under a chosenmessage attack: the attacker asks for the signature of some messages of his choice, and is then able to produce the signature of a message that was never signed by the legitimate signer. The first attack is a variant of Desmedt and Odlyzko’s attack and requires a few hundreds of signatures. The second attack is more powerful and requires only three signatures.
On Positive Integers ≤x with Prime Factors ≤t log x
"... . It is not difficult to estimate the function /(x; y), which counts integers x, free of prime factors ? y, by "smooth" functions whenever y log 1=2 x or y is a fixed power of x. This can be extended to y ! log 3=4 x, and y ? log 2+" x under the assumption of the Riemann Hypothesis. The re ..."
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. It is not difficult to estimate the function /(x; y), which counts integers x, free of prime factors ? y, by "smooth" functions whenever y log 1=2 x or y is a fixed power of x. This can be extended to y ! log 3=4 x, and y ? log 2+" x under the assumption of the Riemann Hypothesis. The real difficulty lies when y is a fixed multiple of log x and, in this paper, we investigate the set of integers x, free of prime factors ? t log x, by estimating various functions related to /(x; t log x). 1. INTRODUCTION. Define S(x; y) to be the set of positive integers x, composed only of prime factors y. The cardinality of this set, /(x; y), is called the DickmanDe Bruijn function and has been extensively investigated by many authors (see [14] for a review). In this section we will give some wellknown results about /(x; y) and sketch proofs of smooth asymptotic estimates when y ! log 1=2 x and when y is a fixed power of x. We also indicate how, in the literature, these have been ...
THEOREM
"... If n 2 2 is an integer, let p(n) denote the largest prime factor of IZ. For every x>Oandevery t,O~ttl,letA(x,t)denotethenumberof nix withP(n)rx’. A wellknown result due to Dickmaa [4] and others is ..."
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If n 2 2 is an integer, let p(n) denote the largest prime factor of IZ. For every x>Oandevery t,O~ttl,letA(x,t)denotethenumberof nix withP(n)rx’. A wellknown result due to Dickmaa [4] and others is