This paper studies self-dual and maximal self-orthogonal codes over GF(3). First, a number of Gleason-type theorems are given, describing the weight enumerators of such codes. Second, a table of all such codes of length _-<12 is constructed. Finally, the complete weight enumerators of various quadratic residue and symmetry codes of length =<60.are obtained.

We propose a remote password authentication scheme based on quadratic residues. In our scheme, any legal user can freely choose his own password in the card initialization phase. Using his password and smart card which contains identity and other information, he can then log into the system successfully. According to our analysis, intruders cannot obtain any secret information from the public i...

In previous work, the author has extended the concept of regular and irregular primes to the setting of arbitrary totally real number fields k0, using the values of the zeta function ζk0 at negative integers as our “higher Bernoulli numbers”. In the case where k0 is a real quadratic field, Siegel presented two formulas for calculating these zeta-values: one using entirely elementary methods and...

Vorlesungen iiber Zahlentheorie. By Edmund Landau. Volume I: Aus der elementaren unà additiven Zahlentheorie. xii+360 pp. Volume II: Aus der analytischen und geometrischen Zahlentheorie. vii+308 pp. Volume III : Aus der algebraischen Zahlentheorie und iiber die Fermatsche Vermutung. vii+341 pp. Leipzig, S. Hirzel, 1927. These three excellently printed and arranged volumes form an addition of th...

We analyze a large class of subset membership problems related to integer factorization. We show that there is no algorithm solving these problems efficiently without exploiting properties of the given representation of ring elements, unless factoring integers is easy. Our results imply that problems with high relevance for a large number of cryptographic applications, such as the quadratic res...

We study residues on a complete toric variety X , which are defined in terms of the homogeneous coordinate ring of X . We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on ...

In this note, we consider arithmetic properties of the function K(n) = (2n)!(2n + 2)! (n− 1)!(n + 1)!2(n + 2)! which counts the number of two–legged knot diagrams with one self– intersection and n − 1 tangencies. This function recently arose in a paper by Jacobsen and Zinn–Justin on the enumeration of knots via a transfer matrix approach. Using elementary number theoretic techniques, we prove v...

In this correspondence, we consider quadratic double and bordered double circulant construction methods over the ring R := F2 + uF2 + uF2, where u = 1. Among other examples, extremal binary self-dual codes of length 66 are obtained by these constructions. These are extended by using extension theorems for self-dual codes and as a result 8 new extremal binary self-dual codes of length 68 are obt...

Let n = pq, p, q distinct odd primes. We can divide the numbers in Zn into four classes depending on their membership in QRp and QRq. Let Q11 n be those numbers that are quadratic residues mod both p and q; let Q10 n be those numbers that are quadratic residues mod p but not mod q; let Q 01 n be those numbers that are quadratic residues mod q but not mod p; and let Q00 n be those numbers that a...

This paper presents a selection of lexicographic codes over a prime field Fp , codes constructed so as to be linear and to have a prescribed minimum distance and divisor. The development of the code when the minimum distance is 3p and the divisor is p leads to questions involving the distribution of quadratic residues and non-residues modulo p. In the course of events, the Hall planeH(p) emerge...