Generalization of Roos bias in RC4 and some results on key-keystream relations

RC4 has attracted many cryptologists due to its simple structure. In [9], Paterson, Poettering and Schuldt reported the results of a large scale computation of RC4 biases. Among the biases reported by them, we try to theoretically analyze a few which show very interesting visual patterns. We first study the bias which relates the key stream byte, where k is the first byte of the secret key. We then present a generalization of the Roos bias. In 1995, Roos observed the bias of initial bytes S of the permutation after KSA towards f. Here we study the probability of S. Our generalization provides a complete correlation between z i. We also analyze the key-keystream relation z i = f i - 1 which was studied by Maitra and Paul [6] in FSE 2008. We provide more accurate formulas for the probability of both z i = i - f i {z-{i}=i-f-{i}} and z i = f i - 1 {z-{i}=f-{i-1}} for different i's than the existing works.