Guillaume Hanrot

MPFR: A multiple-precision binary floating-point library with correct rounding

This article presents a multiple-precision binary floating-point library, written in the ISO C language, and based on the GNU MP library. Its particularity is to extend to arbitrary-precision, ideas from the IEEE 754 standard, by providing correct rounding and exceptions. We demonstrate how these strong semantics are achieved---with no significant slowdown with respect to other arbitrary-precision tools---and discuss a few applications where such a library can be useful.

Improved analysis of Kannan's shortest lattice vector algorithm

The security of lattice-based cryptosystems such as NTRU, GGH and Ajtai-Dwork essentially relies upon the intractability of computing a shortest non-zero lattice vector and a closest lattice vector to a given target vector in high dimensions. The best algorithms for these tasks are due to Kannan, and, though remarkably simple, their complexity estimates have not been improved since over twenty years. Kannans algorithm for solving the shortest vector problem (SVP) is in particular crucial in Schnorrs celebrated block reduction algorithm, on which rely the best known generic attacks against the lattice-based encryption schemes mentioned above. In this paper we improve the complexity upper-bounds of Kannans algorithms. The analysis provides new insight on the practical cost of solving SVP, and helps progressing towards providing meaningful key-sizes.

The Middle Product Algorithm I

Abstract.We present new algorithms for the inverse, division, and square root of power series. The key trick is a new algorithm – MiddleProduct or, for short, MP – computing the n middle coefficients of a (2n−1)×n full product in the same number of multiplications as a full n×n product. This improves previous work of Brent, Mulders, Karp and Markstein, Burnikel and Ziegler. These results apply both to series and polynomials.

Primality proving with elliptic curves

Elliptic curves are fascinating mathematical objects. In this paper, we present the way they have been represented inside the Coq system, and how we have proved that the classical composition law on the points is internal and gives them a group structure. We then describe how having elliptic curves inside a prover makes it possible to derive a checker for proving the primality of natural numbers.

Solvability by radicals from an algorithmic point of view

Any textbook on Galois theory contains a proof that a polynomial equation with solvable Galois group can be solved by radicals. From a practical point of view, we need to find suitable representations of the group and the roots of the polynomial. We first reduce the problem to that of cyclic extensions of prime degree and then work out the radicals, using the work of Girstmair. We give numerical examples of Abelian and non-Abelian solvable equations and apply the general framework to the construction of Hilbert Class fields of imaginary quadratic fields.

Proposal for a Standardization of Mathematical Function Implementation in Floating-Point Arithmetic

Some aspects of what a standard for the implementation of the mathematical functions could be are presented. Firstly, the need for such a standard is motivated. Then the proposed standard is given. The question of roundings constitutes an important part of this paper: three levels are proposed, ranging from a level relatively easy to attain (with fixed maximal relative error) up to the best quality one, with correct rounding on the whole range of every function. We do not claim that we always suggest the right choices, or that we have thought about all relevant issues. The mere goal of this paper is to raise questions and to launch the discussion towards a standard.

Worst Cases of a Periodic Function for Large Arguments

One considers the problem of finding hard to round cases of a periodic function for large floating-point inputs, more precisely when the function cannot be efficiently approximated by a polynomial. This is one of the last few issues that prevents from guaranteeing an efficient computation of correctly rounded transcendentals for the whole IEEE-754 double precision format. The first non-naive algorithm for that problem is presented, with a heuristic complexity of O(20.676p) for a precision of p bits. The efficiency of the algorithm is shown on the largest IEEE-754 double precision binade for the sine function, and some corresponding bad cases are given. We can hope that all the worst cases of the trigonometric functions in their whole domain will be found within a few years, a task that was considered out of reach until now.

Time-and space-efficient evaluation of some hypergeometric constants

The currently best known algorithms for the numerical evaluation of hypergeometric constants such as Ç(3) to <i>d</i> decimal digits have time complexity <i>O</i>(<i>M</i>(<i>d</i>) log²<i>d</i>) and space complexity of <i>O</i>(<i>d</i> log <i>d</i>) or <i>O</i>(<i>d</i>). Following work from Cheng, Gergel, Kim and Zima, we present a new algorithm with the same asymptotic complexity, but more efficient in practice. Our implementation of this algorithm improves over existing programs for the computation of Π, and we announce a new record of 2 billion digits for Ç(3).

Algorithmic number theory : 9th International Symposium, ANTS-IX, Nancy, France, July 19-23, 2010 : proceedings

Invited papers.- Putting the Hodge and Tate Conjectures to the Test.- Curves of Genus 3 with a Group of Automorphisms Isomorphic to S3.- Learning with Errors over Rings.- Lattices and Spherical Designs.- Fixed Points for Discrete Logarithms.- Contributed papers.- Explicit Coleman Integration for Hyperelliptic Curves.- Smallest Reduction Matrix of Binary Quadratic Forms.- Practical Improvements to Class Group and Regulator Computation of Real Quadratic Fields.- On the Use of the Negation Map in the Pollard Rho Method.- An O(M(n) logn) Algorithm for the Jacobi Symbol.- New Families of ECM Curves for Cunningham Numbers.- Visualizing Elements of Sha[3] in Genus 2 Jacobians.- On Weil Polynomials of K3 Surfaces.- Class Invariants by the CRT Method.- Short Bases of Lattices over Number Fields.- On the Complexity of the Montes Ideal Factorization Algorithm.- Congruent Number Theta Coefficients to 1012.- Pairing the Volcano.- A Subexponential Algorithm for Evaluating Large Degree Isogenies.- Huffs Model for Elliptic Curves.- Efficient Pairing Computation with Theta Functions.- Small-Span Characteristic Polynomials of Integer Symmetric Matrices.- Decomposition Attack for the Jacobian of a Hyperelliptic Curve over an Extension Field.- Factoring Polynomials over Local Fields II.- On a Problem of Hajdu and Tengely.- Sieving for Pseudosquares and Pseudocubes in Parallel Using Doubly-Focused Enumeration and Wheel Datastructures.- On the Extremality of an 80-Dimensional Lattice.- Computing Automorphic Forms on Shimura Curves over Fields with Arbitrary Class Number.- Improved Primality Proving with Eisenstein Pseudocubes.- Hyperbolic Tessellations Associated to Bianchi Groups.Mazur proved that any element xi of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that xi lies in the kernel of the natural homomorphism between the cohomology groups H^1(k,E) -> H^1(k,A). However, the abelian surface in Mazurs construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get explicit models of the genus 2 curves involved.

Density results on floating-point invertible numbers

Let Fk denote the k-bit mantissa floating-point (FP) numbers. We prove a conjecture of Muller according to which the proportion of numbers in Fk with no FP-reciprocal (for rounding to the nearest element) approaches ½ - 3/2log4/3 ≈ 0.06847689 as k → ∞. We investigate a similar question for the inverse square root.