Proving in Zero-Knowledge that a Number Is the Product of Two Safe Primes

Abstract. We present the first efficient 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 blocks of our protocols are statistical zero-knowledge proofs of knowledge that are of independent interest. We show how to prove the correct computation of a modular addition, a modular multiplication, and a modular exponentiation, where all values including the modulus are committed to but not publicly known. Apart from the validity of the equations, no other information about the modulus (e.g., a generator whose order equals the modulus) or any other operand is exposed. Our techniques can be generalized to prove that any multivariate modular polynomial equation is satis...

• Type: Conference
• Year: 1999
• Where: EUROCRYPT
• Authors: Jan Camenisch, Markus Michels
• Keywords: Building Blocks, Cryptology, EUROCRYPT 1999, Safe Prime, Statistical Zero-knowledge Protocols