Alberto Zanoni

Integer and polynomial multiplication: towards optimal toom-cook matrices

Karatsuba and Toom-Cook are well-known methods used to multiply efficiently long integers. There have been different proposal about the interpolating values used to determine the matrix to be inverted and the sequence of operations to invert it. A deffinitive word about which is the optimal matrix (values) and the (number of) basic operations to invert it seems still not to have been said. In this paper we present some particular examples of useful matrices and a method to generate automatically, by means of optimised exhaustive searches on a graph, the best sequence of basic operations to invert them.

Hilbert stratification and parametric gröbner bases

In this paper we generalize a method to analyze inhomogeneous polynomial systems containing parameters. In particular, the Hilbert function is used as a tool to check that the specialization of a “generic” Grobner basis of the parametric polynomial system (computed in a polynomial ring having both parameters and unknowns as variables) is a Grobner basis of the specialized system. Extending the analysis, we can also build the so-called Hilbert stratification of the associated variety. We classify the possible specializations according to the value of the Hilbert function of the specialized system. Some computation examples with the PoSSoLib are reported.

An algebraic interpretation of AES -128

We analyze an algebraic representation of

Intervals, syzygies, numerical gröbner bases: a mixed study

\mathcal{AES}

Iterative Toom-Cook methods for very unbalanced long integer multiplication

128 as an embedding in

Looking Inside Aεs and Bεs

\mathcal{BES}

Gröbner bases specialization through Hilbert functions: the homogeneous case

, due to Murphy and Robshaw. We present two systems of equations S⋆ and K⋆ concerning encryption and key generation processes. After some simple but rather cumbersome substitutions, we should obtain two new systems

Long Integers and Polynomial Evaluation with Estrin's Scheme

{\mathcal{C}}_{1}

Intervals, syzygies, numerical Gröbner bases : A mixed study

and

Hilbert stratification and parametric Gröbner bases

{\mathcal{C}}_{2}