An Analysis of the RC4 Family of Stream Ciphers against Algebraic Attacks

To date, most applications of algebraic analysis and attacks on stream ciphers are on those based on linear feedback shift registers (LFSRs). In this paper, we extend algebraic analysis to non-LFSR based stream ciphers. Specifically, we perform an algebraic analysis on the RC4 family of stream ciphers, an example of stream ciphers based on dynamic tables, and investigate its implications to potential algebraic attacks on the cipher. This is, to our knowledge, the first paper that evaluates the security of RC4 against algebraic attacks through providing a full set of equations that describe the complex word manipulations in the system. For an arbitrary word size, we derive algebraic representations for the three main operations used in RC4, namely state extraction, word addition and state permutation. Equations relating the internal states and keystream of RC4 are then obtained from each component of the cipher based on these algebraic representations, and analysed in terms of their contributions to the security of RC4 against algebraic attacks. Interestingly, it is shown that each of the three main operations contained in the components has its own unique algebraic properties, and when their respective equations are combined, the resulting system becomes infeasible to solve. This results in a high level of security being achieved by RC4 against algebraic attacks. On the other hand, the removal of an operation from the cipher could compromise this security. Experiments on reduced versions of RC4 have been performed, which confirms the validity of our algebraic analysis and the conclusion that the full RC4 stream cipher seems to be immune to algebraic attacks at present.