Linear equation on polynomial single cycle T-functions

Abstract

Polynomial functions are widely used in the design of cryptographic transformations such as block ciphers, hash functions and stream ciphers, which belong to the category of T-functions. When a polynomial function is used as state transition function in a pseudorandom generator, it is usually required that the polynomial function generates a single cycle. In this paper, we first present another proof of the sufficient and necessary condition on a polynomial function being a single cycle T-function. Then we give a general linear equation on the sequences {x i } generated by these T-functions, that is, where A i,2 is a sequence of period 4, a and b are constants determined by the coefficients c i . This equation shows that the sequences generated by polynomial single cycle T-functions have potential secure problems. © 2008 Springer-Verlag Berlin Heidelberg.