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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 bytez  i{z_{i}}    withi  -   k  ⁢   [  0  ]{i-k[0]}   , wherek  ⁢   [  0  ]{k[0]}    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 bytesS  ⁢   [  i  ]{S[i]}    of the permutation after KSA towardsf  i   =∑   r  =  1   i   r   +    ∑   r  =  0   i    K  ⁢   [  r  ]{f_{i}=\sum_{r=1}^{i}r+\sum_{r=0}^{i}K[r]}   . Here we study the probability ofS  ⁢   [  i  ]{S[i]}    equalingf  y   =∑   r  =  1   y   r   +    ∑   r  =  0   y    K  ⁢   [  r  ]{f_{y}=\sum_{r=1}^{y}r+\sum_{r=0}^{y}K[r]}    fori  ≠  y{i\neq y}   . Our generalization provides a complete correlation betweenz  i{z_{i}}    andi  -   f  y{i-f_{y}}   . We also analyze the key-keystream relationz  i   =   f   i  -  1{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 bothz  i   =   i  -   f  i{z_{i}=i-f_{i}}    andz  i   =   f   i  -  1{z_{i}=f_{i-1}}    for different  i ’s than the existing works.

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