Digital Signature Schemes over the Ring Z[e2πi/5

Abstract

This paper proposes an extended RSA digital signature scheme and extended ElGamal digital signature schemes with appendix and with message recovery over the algebraic integer ring Z[e2πi/5] of the cyclotomic field Q[e2πi/5]. In these digital signature schemes, the extended Euler-phi function is φ(n) = (p4 - 1) (q4 - 1) compared to the classical Euler-phi function φ(n)=(p-1)(q-1) where n is a product of two large primes p and q. The encryption exponent is chosen to be coprime to the extended Euler-phi function. Thus, the encryption exponents have more options, which provides more security than that of the classical case. The securities of digital signature schemes depend on the difficulty of factoring large integers into a product of primes, and logarithmic computations in the ring Z[e2πi/5] Some numerical examples are given.