Symmetric Key Image Encryption Scheme with Key Sequences Derived from Random Sequence of Cyclic Elliptic Curve Points over GF(p)

Abstract

Until recently, Cryptography has been of interest primarily to the military and diplomatic communities. But the dawning of the information age has revealed an urgent need for cryptography in the private sector too. Cryptography is the study of techniques for ensuring the secrecy and authentication of the information. In this paper, cyclic elliptic curve of the form y2 = x3 + ax + b, a, b ∈ GF(p) with order M is considered and key Sequences are derived from random sequence of cyclic elliptic Curve points. Elliptic Curve is a cubic equation in two variables, x and y, with coefficients from a field satisfying certain conditions. For cryptographic applications the coefficients are chosen from finite fields. A point on the Elliptic curve is a pair of (x,y) which satisfies the Elliptic curve equation. The total number of points (x,y) which satisfy the elliptic curve equation along with x=∞,y=∞ is called the Order of the curve ‘M’. The least integer N for which NP is equal to point at infinity O is called order of the point P. Elliptic curves for which there exists a point P having the same order N, as that of the curve M are called cyclic elliptic curves. A pseudorandom sequence generator based on chaotic function and Elliptic Curve arithmetic over GF(p) is proposed here. The logistic Map is used as a chaotic function which generates a random sequence of real numbers. This random real sequence is converted to binary which drives an Elliptic Curve arithmetic module generating a random sequence of Elliptic Curve points. The sequence of points {P, 2P, …, NP} is calculated from a base point P, and stored in a file. Every element in this sequence is a point on the cyclic elliptic curve. The Chaotic binary sequence selects x or y-coordinates of elliptic curve points, pre-computed and stored. This forms a random integer sequence. The randomness properties of this sequence have been tested using various techniques like, autocorrelation distribution, crosscorrelation distribution and first return map. It is observed that the sequence generated satisfies the required randomness properties. These sequences find applications in Stream Cipher Systems. An additive Stream Cipher system is designed using this sequence as the key sequence to encrypt images. Results of image encryption and decryption for a medical image is discussed and analyzed in this paper. The results are also compared with the scheme proposed by Lap-Piu Lee and Kwok-Wo Wong[1]. The security analysis of the proposed system is also discussed. It is interesting to observe that, proposed algorithm is superior compared to Lap-Piu Lee scheme[1].