Linear models for a time-variant permutation generator

A keystream generator, known as RC4, consisting of a permutation table that slowly varies in time under the control of itself, is analyzed by the linear model approach. The objective is to find linear relations among the keystream bits that hold with probability different from one half by using the linear sequential circuit approximation method. To estimate the corresponding correlation coefficients, some interesting correlation properties of random Boolean functions are derived. It is thus shown that the second binary derivative of the least significant bit output sequence is correlated to 1 with the correlation coefficient close to 15 2 3n where n is the variable word size of RC4. The output sequence length required for the linear statistical weakness detection is then around 64n=225. The result can be used to distinguish RC4 from other keystream generators and to determine the unknown parameter n, as well as for the plaintext uncertainty reduction if n is small.