Abstract. We improve our proposal of a new variant of the McEliece cryptosystem based on QC-LDPC codes. The original McEliece cryptosystem, based on Goppa codes, is still unbroken up to now, but has two major drawbacks: long key and low transmission rate. Our variant is based on QC-LDPC codes and is able to overcome such drawbacks, while avoiding the known attacks. Recently, however, a new attack has been discovered that can recover the private key with limited complexity. We show that such attack can be avoided by changing the form of some constituent matrices, without altering the remaining system parameters. We also propose another variant that exhibits an overall increased security level. We analyze the complexity of the encryption and decryption stages by adopting efficient algorithms for processing large circulant matrices. The Toom-Cook algorithm and the short Winograd convolution are considered, that give a significant speed-up in the cryptosystem operations.

Abstract. Toom-Cook strategy is a well-known method for building algorithms to efficiently multiply dense univariate polynomials. Efficiency of the algorithm depends on the choice of interpolation points and on the exact sequence of operations for evaluation and interpolation. If carefully tuned, it gives the fastest algorithm for a wide range of inputs. This work smoothly extends the Toom strategy to polynomial rings, with a focus on GF2 [x]. Moreover a method is proposed to find the faster Toom multiplication algorithm for any given splitting order. New results found with it, for polynomials in characteristic 2, are presented. A new extension for multivariate polynomials is also introduced; through a new definition of density leading Toom strategy to be efficient.

Abstract. Strassen method is not the asymptotically fastest known matrix multiplication algorithm, but it is the most widely used for large matrices on finite fields. Since his manuscript was published, a number of variants have been proposed with various addition complexities. Here we describe a new one. The new variant is as good as those already known for a simple matrix multiplication, but can save operations either when more than two matrices are to be multiplied or for squaring. Moreover it can be proved optimal for this tasks. The biggest gain is shown for n th -power computation, in this scenario the additive complexity can be halved, with respect to original Strassen's.

In this paper the problem of univariate polynomial evaluation is considered. When both polynomial coefficients and the evaluation "point" are integers, unbalanced multiplications (one factor having many more digits than the other one) in classical Ruffini-Horner rule do not let computations completely benefit of subquadratic methods, like Karatsuba, Toom-Cook and Schönhage-Strassen's.We face this problem by applying an approach originally proposed by Estrin to augment parallelism exploitation in computation. We show that it is also effective in the sequential case, whenever data dimensions grow, e.g. in the long integer case. We add some adjustments to Estrin's proposal obtaining a smoother behavior around corner cases, and to avoid performance degradation when most of the coefficients are zero.This way, a new general algorithm is obtained, improving both theoretical complexity and actual performance. The algorithm itself is very simple, and its use can be usefully extended to evaluation of polynomials on rationals or on polynomials (polynomial composition).Some tests, results and comparisons obtained with PARI/GP are also presented, for both dense and "sparse" polynomials.

We present a simple and efficient methodology for the development, tuning, and installation of matrix algorithms such as the hybrid Strassen's and Winograd's fast matrix multiply or their combination with the 3M algorithm for complex matrices (i.e., hybrid: a recursive algorithm as Strassen's until a highly tuned BLAS matrix multiplication allows performance advantages). We investigate how modern Symmetric Multiprocessor (SMP) architectures present old and new challenges that can be addressed by the combination of an algorithm design with careful and natural parallelism exploitation at the function level (optimizations) such as function-call parallelism, function percolation, and function software pipelining. We have three contributions: first, we present a performance overview for double- and double-complex-precision matrices for state-of-the-art SMP systems; second, we introduce new algorithm implementations: a variant of the 3M algorithm and two new different schedules of Winograd's matrix multiplication (achieving up to 20% speedup with respect to regular matrix multiplication). About the latter Winograd's algorithms: one is designed to minimize the number of matrix additions and the other to minimize the computation latency of matrix additions; third, we apply software pipelining and threads allocation to all the algorithms and we show how this yields up to 10% further performance improvements.