This article describes an extensive study of the use of DSP48E2 Slices in Ultrascale FPGAs to design hardware versions of the Montgomery Multiplication algorithm for the hardware acceleration of modular multiplications. Our fully scalable systolic architectures result in parallelized, DSP48E2-optimized scheduling of operations analogous to the FIOS block variant of the Montgomery Multiplication. We explore the impacts of different pipelining strategies within DSP blocks, scheduling of operations, processing element configurations, global design structures and their tradeoffs in terms of performance and resource costs. We discuss the application of our methodology to multiple types of DSP primitives. We provide ready-to-use fast, efficient, and fully parametrizable designs, which can adapt to a wide range of requirements and applications. Implementations are scalable to any operand width. Our most efficient designs can perform 128, 256, 512, 1024, 2048, and 4096 bits Montgomery modular multiplications in 0.0992 μs, 0.2032 μs, 0.3952 μs, 0.7792μs, 1.550 μs, and 3.099 μs using 4, 6, 11, 21, 41, and 82 DSP blocks, respectively.

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The Polynomial Modular Number System (PMNS) is an integer number system designed to speed up arithmetic operations modulo a prime p. Such a system is defined by a tuple B = (p, n, γ, ρ, E) where E ε Z[X] and E(γ) = 0 mod p. In a PMNS, an element a of Z/pZ is represented by a polynomial A such that: A(γ) = a mod p, deg A <; n ||A||∞ <; p. In [6], the authors mentioned that PMNS can be highly redundant but they didn't really take advantage of this possibility. In this paper we use, for the first time, the redundancy of PMNS to protect algorithms against Side Channel Attacks (SCA). More precisely, we focus on elliptic curve cryptography. We show how to randomize the modular multiplication in order to be safe against existing SCA and we demonstrate the resistance of our construction. We describe the generation of a PMNS while guaranteeing, for all elements of Z/pZ, the minimum number of distinct representations we want. We also show how to reach all these representations.

In this paper, we improved the method proposed in [1] which was designed to detect errors in the elliptic curve scalar multiplication algorithm (ECSM). The algorithm we propose is SPA-secure and safe against recent data and instruction cache timing attacks.

We study Goppa codes, /spl Gamma/(L,g), defined by the polynomial g(z)=a(z)TrF/sub p/ms:F/sub p/s(b(z)). It is shown that the dimension of these codes never reaches the general, well-known, bound for Goppa codes. New bounds are proposed depending on the value of m and p. Furthermore, we prove that when p=2 these codes have only even weights.