A Public-Key Cryptosystem Using Purely Cubic Fields

Abstract

This paper presents an RSA-like public-key cryptosystem that can only be broken by factoring its modulus. Messages are encoded as units in a purely cubic field, and the encryption exponent is a multiple of 3. Similar systems with encryption powers of the form 2e as well as 3e were designed by Rabin, Williams, and Loxton et al. Our scheme is more general than previously developed methods in that it allows a broader class of primes for its modulus, namely any pair of distinct primes $ p, q \equiv 1 ( {\rm mod}3 )$ rather than $ p \equiv 4 ( {\rm mod} 9 )$ and $ q \equiv 7 ( {\rm mod} 9 )$ . The system employs several number theoretic techniques in the cyclotomic field $ \mbox{\bf Q}(\sqrt{-3})$ , including Euclidean division, rapid evaluation of cubic residuacity characters, and the computation of prime divisors of rational primes.