Homomorphic encryption is one of the main solutions for building secure and privacy-preserving solutions for Machine Learning as a Service, a major challenge in a society where AI becomes more and more pervasive. This motivates the development of homomorphic algorithms for the main building blocks of AI, typically for the components of the various types of neural networks architectures.

The 2008 revision of the IEEE-754 standard, which governs floating-point arithmetic, recommends that a certain set of elementary functions should be correctly rounded. Successful attempts for solving the Table Maker's Dilemma in binary64 made it possible to design CRlibm, a library which offers correctly rounded evaluation in binary64 of some functions of the usual libm. It evaluates functions using a two step strategy, which relies on a folklore heuristic that is well spread in the community of mathematical functions designers. Under this heuristic, one can compute the distribution of the lengths of runs of zeros/ones after the rounding bit of the value of the function at a given floating-point number. The goal of this paper is to change, whenever possible, this heuristic into a rigorous statement. The underlying mathematical problem amounts to counting integer points in the neighborhood of a curve, which we tackle using so-called exponential sums techniques, a tool from analytic number theory.

To be considered for an IEEE Jack Keil Wolf ISIT Student Paper Award. Sampling from a lattice Gaussian distribution is emerging as an important problem in various areas such as coding and cryptography. The default sampling algorithm - Klein's algorithm yields a distribution close to the lattice Gaussian only if the standard deviation is sufficiently large. In this paper, we propose the Markov chain Monte Carlo (MCMC) method for lattice Gaussian sampling when this condition is not satisfied. In particular, we present a sampling algorithm based on Gibbs sampling, which converges to the target lattice Gaussian distribution for any value of the standard deviation. To improve the convergence rate, a more efficient algorithm referred to as Gibbs-Klein sampling is proposed, which samples block by block using Klein's algorithm. We show that Gibbs-Klein sampling yields a distribution close to the target lattice Gaussian, under a less stringent condition than that of the original Klein algorithm.

In the present paper, we investigate the approximation of a function by a polynomial with floating-point coefficients; we are looking for the best approximation in the L2sense. Finding a best polynomial L2-approximation with real coefficients is an easy exercise about orthogonal projections. However, truncating the coefficients to floating-point numbers, which is needed for further computations, makes the approximation way worse. Hence, we study the problem of computing best approximations under the constraint that coefficients are floating-point numbers. We show that the corresponding problem is NP-hard, by reduction to the CVP problem. We investigate the practical behaviour of exact and approximate algorithms for this problem. The conclusion is that it is possible in a short amount of time to obtain a relative or absolute best L2-approximation. The main applications are for large dimension, as a preliminary step of finding Linfin-approximations and for functions with large variations, for which relative best approximation is by far more interesting than absolute.

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.

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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.

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.