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

摘要

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_iP} 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(2~8) is chosen for analysis.