Let L be a number field and let E/L be an elliptic curve with potential supersingular reduction at a prime ideal ℘ of L above a rational prime p. In this article we describe a formula for the slopes of the Newton polygon associated to the multiplication-by-p map in the formal group of E, that only depends on the congruence class of p mod 12, the ℘-adic valuation of the discriminant of a model f...

L et I be an ideal, homogeneous with respect to the usual grading, in a polynomial ring R = k[x0, . . . , xn] in n+ 1 variables (over an algebraically closed field k). Denote the graded component of I of degree d by Id, and likewise the k-vector space of homogeneous forms of R of degree d by Rd. Since I is a graded R-module, we have k-linear maps μd,i : Id ⊗ Ri → Id+i given for each i and d by ...

We propose a numerical linear algebra based method to find the multiplication operators of the quotient ring C[x]/I associated to a zero-dimensional ideal I generated by n C-polynomials in n variables. We assume that the polynomials are generic in the sense that the number of solutions in Cn equals the Bézout number. The main contribution of this paper is an automated choice of basis for C[x]/I...

Advances in their technologies have positioned FPGAs and embedded processors to compete with digital signal processors (DSPs). In this paper, we evaluate the performance in terms of both latency and energy-efficiency of FPGAs, embedded processors, and DSPs in multiplying two matrices. As specific examples, we have chosen a representative of each type of device. Our results show that the FPGAs c...

A novel algorithm is proposed for the interpolation step of the Guruswami-Sudan list decoding algorithm. The proposed method is based on the binary exponentiation algorithm, and can be considered as an extension of the Lee-O’Sullivan method. The algorithm is shown to achieve both asymptotical and practical performance gain compared to the case of iterative interpolation algorithm. Further compl...

We consider the exact computation of matrix eigenproblems in residue class rings for solving systems of algebraic equations. We construct multiplication tables using a Gröbner basis of a zero-dimensional ideal. Then, we analyze the tables by exactly computing their Frobenius normal forms. The derogatoriness and the diagonalizability are determined by the normal forms, and the problem is divided...

We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain number-theoretic predicates such as ‘divisible by’, ‘congruent’ and ‘coprime’; one notable example in this class is the Chinese Remaind...

Let S be a set of n ideals of a commutative ring A and let Geven (respectively Godd) denote the product of all the sums of even (respectively odd) number of ideals of S. If n ≤ 6 the product of Geven and the intersection of all ideals of S is included in Godd. In the case A is an Noetherian integral domain, this inclusion is replaced by equality if and only if A is a Dedekind domain.

The concept of a 0-distributive poset is introduced. It is shown that a section semicomplemented poset is distributive if and only if it is 0-distributive. It is also proved that every pseudocomplemented poset is 0-distributive. Further, 0-distributive posets are characterized in terms of their ideal lattices.

This paper deals with Fountain codes, and especially with their encoding matrices, which are required here to be invertible. A result is stated that an encoding matrix induces a permutation. Also, a result is that encoding matrices form a group with multiplication operation. An encoding is a transformation, which reduces the entropy of an initially high-entropy input vector. A special encoding ...