Random Binary and Non-Binary Sequences Derived from Random Sequence of Points on Cyclic Elliptic Curve Over Finite Field GF(2 m ) and Their Properties

Abstract

In this paper, Cyclic Elliptic Curves of the form y 2 + xy = x 3 + ax 2 + b,a,b ∈ GF(2 m ) with order M is considered. A finite field GF(p) (p ≥ N, where N is the order of point P) is considered. Random sequence {k i } of integers is generated using Linear Feedback Shift Register (LFSR) over GF(p) for maximum period. Every element in sequence {k i } is mapped to k i P which is a point on Cyclic Elliptic Curve with co-ordinates say (x i , y i ). The sequence {k i P} is a random sequence of elliptic curve points. From the sequence (x i , y i ) several binary and non-binary sequences are derived and their randomness properties are investigated. The results are discussed. It is found that these sequences pass FIPS-140, NIST tests and exhibit good Hamming Correlation properties. These sequences find applications in Stream Cipher Systems. Here, Cyclic Elliptic Curve over GF(28) is chosen for analysis.