Symmetric Key Image Encryption Scheme with Key Sequences Derived from Random Sequence of Cyclic Elliptic Curve Points.

In this paper, cyclic elliptic curves of the form y^2+xy=x^3+ax^2+b,a,b  GF(2(superscript m)) with order M is considered in the design of a Symmetric Key Image Encryption Scheme with Key Sequence derived from random sequence of cyclic elliptic Curve points. P with co-ordinates (xP, yP) which satisfy the elliptic curve equation is called a point on elliptic curve. The order M is the total number of points (x, y) along with x=∞, y=∞ which satisfy the elliptic curve equation. Least integer N for which NP is equal to point at infinity O is called order of point P. Then P, 2P,…. (N-1) P are distinct points on elliptic curve. In case of cyclic elliptic Curve there exists a point P having the same order N as elliptic curve order M. A finite field GF (p) (p≥N) is considered. Random sequence {k(subscript i)} of integers is generated using Linear Feedback Shift Register (LFSR) over GF (p) for maximum period. Such sequences are called maximal length sequences and their properties are well established. Every element in sequence {k(subscript i)} is mapped to k(subscript i) P which is a point on cyclic elliptic Curve with co-ordinates say (x(subscript i), y(subscript i)). The sequence {k(subscript i)} is a random sequence of elliptic curve points. From the sequence (x(subscript i), y(subscript i)) several binary and non-binary sequences are derived. These sequences find applications in Stream Cipher Systems. Two encryption algorithms-Additive Cipher and Affine Cipher are considered. Results of Image Encryption for a medical image is discussed in this paper. Here, cyclic elliptic Curve over GF(2^8) is chosen for analysis.