In this paper we study evaluation codes arising from plane quotients of the Hermitian curve, defined by affine equations of the form $$y^q+y=x^m,\,q$$ being a prime power and $$m$$ a positive integer which divides $$q+1$$ . The dual minimum distance and minimum weight of such codes are studied from a geometric point of view. In many cases we completely describe the minimum-weight codewords of their dual codes through a geometric characterization of the supports, and provide their number. Finally, we apply our results to describe Goppa codes of classical interest on such curves.

The correctness in decrypting a ciphertext after some operations in the DGVH scheme depends heavily on the dimension of the secret key. In this paper we compute two bounds on the size of the secret key for the DGHV scheme to decrypt correctly a ciphertext after a fixed number of additions and a fixed number of multiplication. Moreover we improve the original bound on the dimension of the secret key for a general circuit.

Various optimizations in the Canetti–Krawczyk model for secure protocol design are proven to preserve security. In particular it is shown that multiple authenticators may be safely used together; that certain message components generated by authenticators may be reordered (to be sent at a different time) or replaced with other values with certain precautions; and that protocols may be defined in the ideal world with session identifiers constructed during protocol runs. Consequently protocol designers now have a set of clear rules to optimize and customize their designs without fear of breaking the security proof. In order to obtain the required proofs, we find it necessary to slightly revise the authenticated links part of the Canetti–Krawczyk model.

We prove a theorem, which provides a formula for the computation of the Poincaré series of a monomial ideal ink[X₁,⋯, X_n], via the computation of the Poincaré series of some monomial ideals ink[X₁,⋯, X_i,⋯, X_n]. The complexity of our algorithm is optimal for Borel-normed ideals and an implementation in CoCoA strongly confirms its efficiency. An easy extension computes the Poincaré series of graded modules over standard algebras.

We study genus g hyperelliptic curves with reduced automorphism group A₅ and give equations y² = f(x) for such curves in both cases where f(x) is a decomposable polynomial in x² or x⁵. For any fixed genus the locus of such curves is a rational variety. We show that for every point in this locus the field of moduli is a field of definition. Moreover, there exists a rational model y² = F(x) or y² = xF(x) of the curve over its field of moduli where F(x) can be chosen to be decomposable in x² or x⁵. While similar equations have been given in (Bujalance et al. in Mm. Soc. Math. Fr. No. 86, 2001) over $${\mathbb R}$$ , this is the first time that these equations are given over the field of moduli of the curve.

Herein, it is shown that by exploiting integral definitions of well known special functions, through generalizations and differentiations, broad classes of definite integrals can be solved in closed form or in terms of special functions. This is especially useful when there is no closed form solution to the indefinite form of the integral. In this paper, three such classes of definite integrals are presented. Two of these classes incorporate and supercede all of Kölbig's integration formulae [11], including his formulation for the computation of Cauchy principal values. Also presented are the mathematical derivations that support the implementation of a third class which exploits the incomplete Gamma function. The resulting programs, based on pattern matching, differentiation, and occasionally limits, are very efficient.

In this paper, we present a complete algorithm to decompose nonlinear differential polynomials in one variable and with coefficients in a computable differential field $${\mathcal K}$$ of characteristic zero. The algorithm provides an efficient reduction of the problem to the factorization of LODOs over the same coefficient field. Besides arithmetic operations, the algorithm needs decomposition of algebraic polynomials, factorization of multi-variable polynomials, and solution of algebraic linear equation systems. The algorithm is implemented in Maple for the constant field case. The program can be used to decompose differential polynomials with thousands of terms effectively.