Fluorescence depolarization of the exposed tryptophanyl residue in the peptide melittin and the buried tryptophanyl residue in the protein azurin (from the bacteria Pseudomonos aeruginoso) was measured with subpicosecond resolution at various excitation and emission wavelengths. A short time component was found in both the fluorescence decay and fluorescence anisotropy decay for the tryptophany...

For an odd prime p and each non-empty subset S ⊂ GF (p), consider the hyperelliptic curve XS defined by y = fS(x), where fS(x) = Q a∈S(x− a). Using a connection between binary quadratic residue codes and hyperelliptic curves over GF (p), this paper investigates how coding theory bounds give rise to bounds such as the following example: for all sufficiently large primes p there exists a subset S...

Legendre was the first to state the law of quadratic reciprocity in the form that we know it and he was able to prove it in some but not all cases, with the first complete proof being given by Gauss. In this paper we trace the evolution of Legendre’s work on quadratic reciprocity in his four great works on number theory. As is well known, Adrien-Marie Legendre (1752-1833) was the first to state...

In this paper we present a cyclotomic approach to the construction of all binary duadic codes of prime lengths. We calculate the number of all binary duadic codes for a given prime length and that of all duadic codes that are not quadratic residue codes. We give necessary and sufficient conditions for p such that all binary duadic codes of length p are quadratic residue (Q.R.) codes. We also sh...

The distribution of quadratic residues and non-residues modulo p has been of intrigue to the number theorists of the last several decades. Although Gauss’ celebrated Quadratic Reciprocity Law gives a beautiful criterion to decide whether a given number is a quadratic residue modulo p or not, it is still an open problem to find a small upper bound on the least quadratic non-residue mod p as a fu...

In this paper, we prove that for any polynomial function f of fixed degree without multiple roots, the probability that all the (f(x+ 1), f(x+ 2), ..., f(x+ κ)) are quadratic non-residue is ≈ 1 2 . In particular for f(x) = x + ax+ b corresponding to the elliptic curve y = x + ax+ b, it implies that the quadratic residues (f(x + 1), f(x + 2), . . . in a finite field are sufficiently randomly dis...

Abstrac t . We consider the acyclic subgraph polytope and define the notion of strength of a relaxation as the maximum improvement obtained by using this relaxation instead of the most trivial relaxation of the problem. We show that the strength of a relaxation is the maximum of the strengths of the relaxations obtained by simply adding to the trivial relaxation each valid inequality separately...

This paper addresses the problem of improving the unknown syndrome representations to develop algebraic decoding of the (17,9,5) and (23,12,7) binary quadratic residue codes up to true minimum distance, respectively. The proposed unknown syndrome representations are expressed as binary polynomials in terms of the single known syndrome, which is different from the known syndrome in [Chang-Lee, A...

We revise Boneh-Gentry-Hamburg’s identity-based encryption schemes and we show that we can renounce to the use of pseudo-random functions. We then prove IND-ID-CPA and ANON-IND-ID-CPA security of these schemes by showing that the advantage of any efficient adversary against these schemes is less than or equal to the quadratic residuosity advantage of some efficient adversary against the RSA gen...

We study the quadratic residue problem known as an NP complete problem by way of the prime number and show that a nondeterministic polynomial process does not belong to the class P because of a random distribution of solutions for the quadratic residue problem.