7 research outputs found
Counting Points on Genus 2 Curves with Real Multiplication
We present an accelerated Schoof-type point-counting algorithm for curves of
genus 2 equipped with an efficiently computable real multiplication
endomorphism. Our new algorithm reduces the complexity of genus 2 point
counting over a finite field (\F_{q}) of large characteristic from
(\widetilde{O}(\log^8 q)) to (\widetilde{O}(\log^5 q)). Using our algorithm we
compute a 256-bit prime-order Jacobian, suitable for cryptographic
applications, and also the order of a 1024-bit Jacobian
Deterministic Encoding and Hashing to Odd Hyperelliptic Curves
The original publication is available at www.springerlink.comInternational audienceIn this paper we propose a very simple and efficient encoding function from Fq to points of a hyperelliptic curve over Fq of the form H : y2 = f(x) where f is an odd polynomial. Hyperelliptic curves of this type have been frequently considered in the literature to obtain Jacobians of good order and pairing-friendly curves. Our new encoding is nearly a bijection to the set of Fq -rational points on H . This makes it easy to construct well-behaved hash functions to the Jacobian J of H , as well as injective maps to J (Fq ) which can be used to encode scalars for such applications as ElGamal encryption. The new encoding is already interesting in the genus 1 case, where it provides a well-behaved encoding to Joux?s supersingular elliptic curves
Secret-sharing with a class of ternary codes
Theoretical Computer Science2461-2285-298TCSC
MPQS with three large primes
We report the factorization of a 135-digit integer by the triple-large-prime variation of the multiple polynomial quadratic sieve. Previous workers [6][10] had suggested that using more than two large primes would be counterproductive, because of the greatly increased number of false reports from the sievers. We provide evidence that, for this number and our implementation, using three large primes is approximately 1.7 times as fast as using only two. The gain in efficiency comes from a sudden growth in the number of cycles arising from relations which contain three large primes. This effect, which more than compensates for the false reports, was not anticipated by the authors of [6] [10] but has become quite familiar from factorizations obtained using the number field sieve. We characterize the various types of cycles present, and give a semi-quantitative description of their rather mysterious behaviour
Classical and quantum algorithms for exponential congruences
We discuss classical and quantum algorithms for solvability testing and finding integer solutions x,y of equations of the form af x + bg y = c over finite fields Fq. A quantum algorithm with time complexity q 3/8 (logq) O(1) is presented. While still superpolynomial in logq, this quantum algorithm is significantly faster than the best known classical algorithm, which has time complexity q 9/8 (logq) O(1). Thus it gives an example of a natural problem where quantum algorithms provide about a cubic speed-up over classical ones.10 page(s
Constructions of approximately mutually unbiased bases
We construct systems of bases of ℂn which are mutually almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on bounds of classical exponential sums and exponential sums over elliptic curves.7 page(s