Primitive root is a fundamental concept in modern cryptography as well as in modern number theory. Fermat prime numbers have practical uses in several branches of number theory. As of today, there is no simple general way to compute the primitive roots of a given prime, though there exists methods to find a primitive root that are faster than simply trying every possible number. We prove the eq...

A characterization of predicate encryption (PE) with support for homomorphic operations is presented and we describe the homomorphic properties of some existing PE constructions. Even for the special case of IBE, there are few known group-homomorphic cryptosystems. Our main construction is an XOR-homomorphic IBE scheme based on the quadratic residuosity problem (variant of the Cocks’ scheme), w...

Let p be a prime number, Fp be a finite field, and let k ∈ Fp. In this paper, we consider the number of rational points on conics Cp,k : x − ky = 1 over Fp. We proved that the order of Cp,k over Fp is p− 1 if k is a quadratic residue mod p and is p+1 if k is not a quadratic residue mod p. Later we derive some results concerning the sums ∑ C [x] p,k(Fp) and ∑ C [y] p,k(Fp), the sum of x− and y−c...

Digital watermarking for multimedia information has been widely adopted as a measure to detect for copyrights infringements. In many watermarking schemes, often only one watermarking signature is used in the embedding of hidden information. This paper introduces new multiple sequence families with good autoand cross-correlation properties, which are ideal in the application of digital watermark...

Using the Newton’s identities, we give the inductive formula for the generator polynomials of the p-adic quadratic residue codes.

In this paper we investigate the index theory of transversally elliptic pseudo-differential operators in the framework of noncommutative geometry. We give examples of spectral triples in the sense of Alain Connes and Henri Moscovici in [12] that are transversally elliptic but nonelliptic. We prove that these spectral triples satisfy the conditions which ensure the Connes-Moscovici local index f...

The degenerate Stirling numbers and degenerate Eulerian polynomials are intimately connected to the arithmetic of generalized factorials. In this article we show that these numbers and similar sequences may in fact be expressed as p-adic integrals of generalized factorials. As an application of this identiication we deduce systems of congruences which are analogues and generalizations of the Ku...