Integer and Floating-Point Constant Multipliers for FPGAs

International audienceReconfigurable circuits now have a capacity that allows them to be used as floating-point accelerators. They offer massive parallelism, but also the opportunity to design optimised floating-point hardware operators not available in microprocessors. Multiplication by a constant is an important example of such an operator. This article presents an architecture generator for the correctly rounded multiplication of a floating-point number by a constant. This constant can be a floating-point value, but also an arbitrary irrational number. The multiplication of the significands is an instance of the well-studied problem of constant integer multiplication, for which improvement to existing algorithms are also proposed and evaluated

Hardware Implementation of the GPS authentication

In this paper, we explore new area/throughput trade- offs for the Girault, Poupard and Stern authentication protocol (GPS). This authentication protocol was selected in the NESSIE competition and is even part of the standard ISO/IEC 9798. The originality of our work comes from the fact that we exploit a fixed key to increase the throughput. It leads us to implement GPS using the Chapman constant multiplier. This parallel implementation is 40 times faster but 10 times bigger than the reference serial one. We propose to serialize this multiplier to reduce its area at the cost of lower throughput. Our hybrid Chapman's multiplier is 8 times faster but only twice bigger than the reference. Results presented here allow designers to adapt the performance of GPS authentication to their hardware resources. The complete GPS prover side is also integrated in the network stack of the PowWow sensor which contains an Actel IGLOO AGL250 FPGA as a proof of concept.Comment: ReConFig - International Conference on ReConFigurable Computing and FPGAs (2012

Multiplication by rational constants: LIP research report 2011-3

International audienceMultiplications by simple rational constants often appear in fixed-point or floating-point application code, for instance in the form of division by an integer constant. The hardware implementation of such operations is of practical interest to FPGA-accelerated computing. It is well known that the binary representation of rational constants is eventually periodic. This article shows how this feature can be exploited to implement multiplication by a rational constant in a number of additions that is logarithmic in the precision. An open-source implementation of these techniques is provided, and is shown to be practically relevant for constants with small numerators and denominators, where it provides improvements of 20 to 40\% in area with respect to the state of the art. It is also shown that for such constants, the additional cost for a correctly rounded result is very small, and that correct rounding very often comes for free in practice

Block Floating Point Implementations for DSP Computations in Reconfigurable Computing

This allows the use of just the right number of bits and the right number of operations on these bit

Hardware division by small integer constants

International audienceThis article studies the design of custom circuits for division by a small positive constant. Such circuits can be useful to specific FPGA and ASIC applications. The first problem studied is the Euclidean division of an unsigned integer by a constant, computing a quotient and a remainder. Several new solutions are proposed and compared against the state of the art. As the proposed solutions use small look-up tables, they match well the hardware resources of an FPGA. The article then studies whether the division by the product of two constants is better implemented as two successive dividers or as one atomic divider. It also considers the case when only a quotient or only a remainder are needed. Finally, it addresses the correct rounding of the division of a floating-point number by a small integer constant. All these solutions, and the previous state of the art, are compared in terms of timing, area, and area-timing product. In general, the relevance domains of the various techniques are very different on FPGA and on ASIC

Hardware division by small integer constants

International audienceThis article studies the design of custom circuits for division by a small positive constant. Such circuits can be useful to specific FPGA and ASIC applications. The first problem studied is the Euclidean division of an unsigned integer by a constant, computing a quotient and a remainder. Several new solutions are proposed and compared against the state of the art. As the proposed solutions use small look-up tables, they match well the hardware resources of an FPGA. The article then studies whether the division by the product of two constants is better implemented as two successive dividers or as one atomic divider. It also considers the case when only a quotient or only a remainder are needed. Finally, it addresses the correct rounding of the division of a floating-point number by a small integer constant. All these solutions, and the previous state of the art, are compared in terms of timing, area, and area-timing product. In general, the relevance domains of the various techniques are very different on FPGA and on ASIC

Hardware Implementation of Barrett Reduction Exploiting Constant Multiplication

The efficient realization of an Elliptic Curve Cryptosystem is contingent on the efficiency of scalar multiplication. These systems can be improved by optimizing the underlying finite field arithmetic operations which are the most costly such as modular reduction. There are elliptic curves over prime fields for which very efficient reduction formulas are possible due to the special structure of the moduli. For prime moduli of arbitrary form, however, use of general reduction formulas, such as Barrett's reduction algorithm, are necessary. Barrett's algorithm performs modular reduction efficiently by using multiplication as opposed to division, an operation which is generally expensive to realize in hardware. We note, however, that when an Elliptic Curve Cryptosystem is defined over a fixed prime field, all multiplication steps in Barrett's scheme can be realized through constant multiplications; this allows for further optimization. In this thesis, we study the influence using constant multipliers has on four different Barrett reduction variants targeting the Virtex-7 (xc7vx485tffg1157-1). We use the FloPoCo core generator to construct constant multiplier implementations for the different multiplication steps required in each scheme. Then, we create a hybrid constant multiplier circuit based on Karatsuba multiplication which uses smaller FloPoCo-generated base multipliers. It is shown that for certain multiplication steps, the hybrid design provides an improvement in the resource utilization of the constant multiplier circuit at the cost of an increase in the critical path delay. A performance comparison of different Barrett reduction circuits using different combinations of constant multiplier architectures is presented. Additionally, a fully pipelined implementation of each Barrett reduction variant is also designed capable of achieving operational frequencies in the range of 496-504MHz depending on the Barrett scheme considered. With the addition of a 256-bit pipelined Karatsuba multiplier circuit, we also present a compact and fully pipelined modular multiplier based on these Barrett architectures capable of achieving very high throughput compared to others in the literature without the use of embedded multipliers

Optimisations arithmétiques et synthèse de haut niveau

High-level synthesis (HLS) tools offer increased productivity regarding FPGA programming.However, due to their relatively young nature, they still lack many arithmetic optimizations.This thesis proposes safe arithmetic optimizations that should always be applied.These optimizations are simple operator specializations, following the C semantic.Other require to a lift the semantic embedded in high-level input program languages, which are inherited from software programming, for an improved accuracy/cost/performance ratio.To demonstrate this claim, the sum-of-product of floating-point numbers is used as a case study. The sum is performed on a fixed-point format, which is tailored to the application, according to the context in which the operator is instantiated.In some cases, there is not enough information about the input data to tailor the fixed-point accumulator.The fall-back strategy used in this thesis is to generate an accumulator covering the entire floating-point range.This thesis explores different strategies for implementing such a large accumulator, including new ones.The use of a 2's complement representation instead of a sign+magnitude is demonstrated to save resources and to reduce the accumulation loop delay.Based on a tapered precision scheme and an exact accumulator, the posit number systems claims to be a candidate to replace the IEEE floating-point format.A throughout analysis of posit operators is performed, using the same level of hardware optimization as state-of-the-art floating-point operators.Their cost remains much higher that their floating-point counterparts in terms of resource usage and performance. Finally, this thesis presents a compatibility layer for HLS tools that allows one code to be deployed on multiple tools.This library implements a strongly typed custom size integer type along side a set of optimized custom operators.À cause de la nature relativement jeune des outils de synthèse de haut-niveau (HLS), de nombreuses optimisations arithmétiques n'y sont pas encore implémentées. Cette thèse propose des optimisations arithmétiques se servant du contexte spécifique dans lequel les opérateurs sont instanciés.Certaines optimisations sont de simples spécialisations d'opérateurs, respectant la sémantique du C.D'autres nécéssitent de s'éloigner de cette sémantique pour améliorer le compromis précision/coût/performance.Cette proposition est démontré sur des sommes de produits de nombres flottants.La somme est réalisée dans un format en virgule-fixe défini par son contexte.Quand trop peu d’informations sont disponibles pour définir ce format en virgule-fixe, une stratégie est de générer un accumulateur couvrant l'intégralité du format flottant.Cette thèse explore plusieurs implémentations d'un tel accumulateur.L'utilisation d'une représentation en complément à deux permet de réduire le chemin critique de la boucle d'accumulation, ainsi que la quantité de ressources utilisées. Un format alternatif aux nombres flottants, appelé posit, propose d'utiliser un encodage à précision variable.De plus, ce format est augmenté par un accumulateur exact.Pour évaluer précisément le coût matériel de ce format, cette thèse présente des architectures d'opérateurs posits, implémentés avec le même degré d'optimisation que celui de l'état de l'art des opérateurs flottants.Une analyse détaillée montre que le coût des opérateurs posits est malgré tout bien plus élevé que celui de leurs équivalents flottants.Enfin, cette thèse présente une couche de compatibilité entre outils de HLS, permettant de viser plusieurs outils avec un seul code. Cette bibliothèque implémente un type d'entiers de taille variable, avec de plus une sémantique strictement typée, ainsi qu'un ensemble d'opérateurs ad-hoc optimisés