In this paper, we consider self-dual codes over the finite ring Zps of integer modulo p for any prime p and for an integer s ≥ 4. We start with any self-dual code in lower modulo and give an necessary and sufficient condition for the self-duality of induced codes. Then we can give an inductive algorithm for construction of all self-dual codes and the mass formula in case of odd prime p.

We calculate the negative integer moments of the (regularized) characteristic polynomials of N × N random matrices taken from the Gaussian Orthogonal Ensemble (GOE) in the limit as N → ∞. The results agree nontrivially with a recent conjecture of Berry & Keating motivated by techniques developed in the theory of singularity-dominated strong fluctuations. This is the first example where nontrivi...

This paper examines with elementary proofs some interesting properties of numbers in the binary quadratic form a + ab+ b, where a and b are non-negative integers. Key findings of this paper are (i) a prime number p can be represented as a + ab + b if and only if p is of the form 6k + 1, with the only exception of 3, (ii) any positive integer can be represented as a +ab+ b if and only if its all...

Let [Formula: see text] be an integer and odd square-free integer. We compute the rank of 2-class group some fields form when all prime divisors are congruent to or text].

Let q denote a large prime number, a be any integer. Let N(q, a) denote the smallest positive integer n for which φ(n) ≡ a (mod q). The number N(q, a) exists. Indeed, if a+1 ≡ 0 (mod q) then one can take n = q. Otherwise, one can take n to be a prime from the arithmetical progression a + 1 (mod q). The problem of upper bound estimates for N(q, a) has been a subject of study of the work of Fried...

let $g={rm sl}_2(p^f)$ be a special linear group and $p$ be a sylow‎ ‎$2$-subgroup of $g$‎, ‎where $p$ is a prime and $f$ is a positive‎ ‎integer such that $p^f>3$‎. ‎by $n_g(p)$ we denote the normalizer of‎ ‎$p$ in $g$‎. ‎in this paper‎, ‎we show that $n_g(p)$ is nilpotent (or‎ ‎$2$-nilpotent‎, ‎or supersolvable) if and only if $p^{2f}equiv‎ ‎1,({rm mod},16)$‎.

We present novel realizations of the transitive signature primitive introduced by Micali andRivest [11]. Our first scheme, FBTS-1, is proven transitively unforgeable under adaptive chosen-message attack assuming factoring is hard. We then present a hash-based modification, FBTS-2achieving shorter signatures by eliminating the need for “node certificates”, and provable underthe s...

let r be a prime ring with extended centroid c, h a generalized derivation of r and n ⩾ 1 a xed integer. in this paper we study the situations: (1) if (h(xy))n = (h(x))n(h(y))n for all x; y 2 r; (2) obtain some related result in case r is a noncommutative banach algebra and h is continuous or spectrally bounded.

1 A simple paradigm Generating a large prime is an important step in the RSA algorithm. We can generate a large prime by repeatedly selecting a large random integer and testing it for primality. We stop when a prime number is found. Therefore, assume that a large integer n is given, and choose a rational number α < 1 to perform the following algorithm: repeat generate a random integer p ∈ [n α ...

let g be a transitive permutation group on a set ? and let m be a positive integer. if no element of g moves any subset of ? by more than m points, then |? | [2mp i (p-1)] wherep is the least odd primedividing |g |. when the bound is attained, we show that | ? | = 2 p q ….. q where ? is a non-negative integer with 2 < p, r 1 and q is a prime satisfying p < q < 2p, ? = 0 or 1, i i n. furthermore...