Some Public-Key Crypto-Functions as Intractable as Factorization

In the RSA public-key crypto system a message M (<R) is encrypted by calculating K≡me (mod R), where 0<K<R and R, e are integers which are made public. The recipient of K can decipher it by raising it to a power d and reducing modulo R. Only the recipient knows the values of two large primes p, q such that R=pq; consequently, only he possesses d, as e is preselected such that gcd (e, φ(R))=1 and ed≡1 (mod φ(R)). In this paper we discuss an RSA-like public-key cryptosystem in which we raise a certain quadratic irrational to the power e modulo R. We show that this cryptosystem is as difficult to break as it is to find the factors of R. Further, this scheme, like the RSA scheme, can also be used to produce signatures.