The aim of this work is to compare in terms of performance, area and power dissipation, the implementations of complex FIR filters based on the following number representations: traditional two’s complement (TCS), Quadratic Residue Number System (QRNS) and radix-2j Redundant Complex Number Systems (RCNS)). The resulting implementations, designed to work at the same clock rate, show that filters...

It is shown that an even partition A∪B of the set R = {1, 2, . . . , p− 1} of positive residues modulo an odd prime p is the partition into quadratic residues and quadratic non-residues if and only if the elements of A and B satisfy certain additive properties, thus providing a purely additive characterization of the set of quadratic residues. 1 Additive properties of quadratic residues An inte...

Let E1 and E2 be Q-nonisogenous, semistable elliptic curves over Q, having respective conductors NE1 and NE2 and both without complex multiplication. For each prime p, denote by aEi(p) := p+1−#Ei(Fp) the trace of Frobenius. Under the assumption of the Generalized Riemann Hypothesis (GRH) for the convolved symmetric power L-functions L(s, SymE1 ⊗SymE2) where i, j ∈ {0, 1, 2}, we prove an explici...

Type II 4-codes are introduced as self-dual codes over the integers modulo 4 containing the all-one vector and with Euclidean weights multiple of 8. Their weight enumerators are characterized by means of invariant theory. A notion of extremality for the Euclidean weight is introduced. Their binary images under the Gray map are formally self-dual with even weights. Extended quadratic residue 4-c...

Based on the quadratic residuosity assumption we present a non-interactive crypto-computing protocol for the greater-than function, i.e., a non-interactive procedure between two parties such that only the relation of the parties’ inputs is revealed. In comparison to previous solutions our protocol reduces the number of modular multiplications significantly. We also discuss applications to condi...

We show how the theory of real quadratic congruence function elds can be used to produce a secure key distribution protocol. The technique is similar to that advocated by Diie and Hellman in 1976, but instead of making use of a group for its underlying structure, makes use of a structure which is \almost" a group. The method is an extension of the recent ideas of Scheidler, Buchmann and William...

Lower bound~ on the capacity and on the product of capacity and computation time are obtained for machines which recognize the set of squares. The bound on capacity is approached to within a factor of four by a specific machine which carries out a test based on the fact that every non-square is a quadratic non-residue of some rational prime. A machine which carries out a te st based on the stan...

It is based on the intractability of Quadratic Residuosity Assumption modulo a composite N . Very roughly, we select uniformly at random Quadratic Residues from ZN to encrypt 0 bit and to encrypt 1 bit we select quadratic non-residue from ZN . However, since the distribution of quadratic residues and quadratic non-residues are not same in ZN we confine ourselves to a subset of ZN where the numb...