a power digraph, denoted by $g(n,k)$, is a directed graph with $z_{n}={0,1,..., n-1}$ as the set of vertices and $l={(x,y):x^{k}equiv y~(bmod , n)}$ as the edge set, where $n$ and $k$ are any positive integers. in this paper, the structure of $g(2q+1,k)$, where $q$ is a sophie germain prime is investigated. the primality tests for the integers of the form $n=2q+1$ are established in terms of th...

A power digraph, denoted by $G(n,k)$, is a directed graph with $Z_{n}={0,1,..., n-1}$ as the set of vertices and $L={(x,y):x^{k}equiv y~(bmod , n)}$ as the edge set, where $n$ and $k$ are any positive integers. In this paper, the structure of $G(2q+1,k)$, where $q$ is a Sophie Germain prime is investigated. The primality tests for the integers of the form $n=2q+1$ are established in terms of th...

Safe primes are prime numbers of the form p = 2q +1 where q is prime. This note introduces a simple method for doubling the speed of safe prime generation. The method is particularly suited to settings where a large number of RSA moduli must be generated. keywords : safe primes, key-generation, prime-generation, RSA.

A number p is a safe prime if both p and (p − 1)/2 are prime. This note describes a method of generating safe primes that is considerably faster than repeatedly generating random primes q until p = 2q + 1 is also prime.

In this paper, we provide a generalization of Proth’s theorem for integers the form Kp^n+1. particular, primality test that requires modular exponentiation (with proper base a) similar to Fermat’s without computation any GCD’s. We also two tests increase chances proving Kp^n+1 primes. As corollaries, three families N whose can be certified only by a^{N-1} \equiv 1 \pmod (Fermat’s test). One the...

In this paper we present some interesting connections between primitive roots and quadratic non-residues modulo a prime. Using these correlations, we propose some polynomial deterministic algorithms for generating primitive roots for primes with special forms (for example, for safe primes).

We present eecient zero-knowledge proof systems for quasi-safe prime products and other related languages. Quasi-safe primes are a relaxation of safe primes, a class of prime numbers useful in many cryptographic applications. Our proof systems achieve higher security and better eeciency than all previously known ones. In particular, all our proof systems are perfect or statistical zero-knowledg...

We present a new protocol for efficient distributed computation modulo a shared secret. We further present a protocol to distributively generate a random shared prime or safe prime that is much more efficient than previously known methods. This allows to distributively compute shared RSA keys, where the modulus is the product of two safe primes, much more efficiently than was previously known.

The generation of prime numbers underlies the use of most public-key schemes, essentially as a major primitive needed for the creation of key pairs or as a computation stage appearing during various cryptographic setups. Surprisingly, despite decades of intense mathematical studies on primality testing and an observed progressive intensification of cryptographic usages, prime number generation ...