This paper presents the first efficient statistical zero-knowledge protocols to prove statements such as: A committed number is a pseudo-prime. A committed (or revealed) number is the product of two safe primes, i.e., primes p and q such that (p1)=2 and (q 1)=2 are primes as well. A given value is of large order modulo a composite number that consists of two safe prime factors. So far, nomethod...

The purpose of this paper is to justify the claim that a method for generating primes presented at EUROCRYPT'89 generates primes with virtually uniform distribution. Using convincing heuristic arguments, the conditional probability distributions of the size of the largest prime factor p 1 (n) of a number n on the order of N is derived, given that n satisses one of the conditions 2n+1 is prime, ...

We present a new algorithm that nds all primes up to n using at most O(n= log log n) arithmetic operations and O(n=(log n log log n)) space. This algorithm is an improvement of a linear prime number sieve due to Pritchard. Our new algorithm matches the running time of the best previous prime number sieve, but uses less space by a factor of (log n). In addition, we present the results of our imp...

We present the rst eecient statistical zero-knowledge protocols to prove statements such as: { A committed number is a prime. { A committed (or revealed) number is the product of two safe primes, i.e., primes p and q such that (p ? 1)=2 and (q ? 1)=2 are prime. { A given integer has large multiplicative order modulo a composite number that consists of two safe prime factors. The main building b...

in this paper we classify fuzzy subgroups of the dihedral group $d_{pqrs}$  for distinct primes  $p$, $q$, $r$ and $s$. this follows similar work we have done on distinct fuzzy subgroups of some dihedral groups.we present formulae for the number of (i) distinct maximal chains of subgroups, (ii) distinct fuzzy subgroups and (iii) non-isomorphic classes of fuzzy subgroups under our chosen equival...