Cryptographic Schemes based on Elliptic Curves over Integer Rings

Public key cryptographic schemes are normally based on the difficulty of solving a trapdoor one-way function. Schemes based on integer factorization, discrete logarithm, quadratic residuosity and subset-sum problem are quite common in practice. Cryptographic schemes based on elliptic curves over finite fields have recently gained importance due to their attractive key size requirements and their ability to provide desired levels of security with lesser computations. Security of such schemes relies on the difficulty of solving the discrete log problem on an elliptic curve. Properties of elliptic curve over algebraic rings have also been studied in the past. However, elliptic curves over integer rings have been very recently explored for their applications in cryptography. Unlike previous ECC schemes, the points on such curves do not form a group and hence require different construction of algebraic operations and trapdoor functions. In this paper we report and analyze the newly constructed trapdoors and use these for design and implementation of encryption schemes and digital signature. We also analyze the security properties of these cryptographic schemes based on elliptic over integer rings.