18/06/2015

Intervenant(s) : Olga Kupriianova, Anastasia Lozanova Volkova, Julien Eynard (Pequan)

Trois exposés courts cette semaine :

- Code generators for mathematical functions*, par Olga Kupriianova,
- Reliable evaluation of the Worst-Case Peak Gain matrix in multiple precision*, par Anastasia Lozanova Volkova,
- RNS Arithmetic Approach in Lattice-based Cryptography - Accelerating the "Rounding-off" Core Procedure*, par Julien Eynard.

A typical floating-point environment includes support for a small set of about 30 mathematical functions. These functions are provided by mathematical software libraries (libm), typically in IEEE754 single, double and quad precision.

We suggest to replace this libm paradigm by a more general approach: the on-demand generation of numerical function code, on arbitrary domains and with arbitrary accuracies.

First, such code generation opens up the libm function space available to programmers. It may capture a much wider set of functions, and may capture even standard functions on non-standard domains and accuracy/performance points.

Second, writing libm code requires fine-tuned instruction selection and scheduling for performance, and sophisticated floating-point techniques for accuracy. Automating this task through code generation improves confidence in the code while enabling better design space exploration, and therefore better time to market, even for the libm functions.

This article discusses the new challenges of this paradigm shift, and presents the current state of open-source function code generators available on http://www.metalibm.org/.

The worst-case peak gain (WCPG) of an LTI filter is an important measure for the implementation of signal processing algorithms. It is used in the error propagation analysis for filters, thus a reliable evaluation with controlled precision is required. The WCPG is computed as an infinite sum and has matrix powers in each summand. We propose a direct formula for the lower bound on truncation order of the infinite sum in dependency of desired truncation error. Several multiprecision methods for complex matrix operations are developed and their error analysis performed. A multiprecision matrix powering method is presented. All methods yield a rigorous solution with an absolute error bounded by an a priori given value. The results are illustrated with numerical examples.

Residue Number Systems (RNS) are naturally considered as an interesting candidate to provide efficient arithmetic for implementations of cryptosystems such as RSA, ECC (Elliptic Curve Cryptography), pairings, etc. More recently, RNS have been used to accelerate fully homomorphic encryption as lattice-based cryptogaphy. In this paper, we present an RNS algorithm resolving the Closest Vector Problem (CVP). This algorithm is particularly efficient for a certain class of lattice basis. It provides a full RNS Babai round-off procedure without any costly conversion into alternative positional number system such as Mixed Radix System (MRS). An optimized Cox-Rower architecture adapted to the proposed algorithm is also presented. The main modifications reside in the Rower unit whose feature is to use only one multiplier. This allows to free two out of three multipliers from the Rower unit by reusing the same one with an overhead of 3 more cycles per inner reduction. An analysis of feasibility of implementation within FPGA is also given.

marc (at) nullmezzarobba.net