Paper 2013/184

The Vernam cipher is robust to small deviations from randomness

Boris Ryabko

Abstract

The Vernam cipher (or one-time pad) has played an important rule in cryptography because it is a perfect secrecy system. For example, if an English text (presented in binary system) $X_1{X}_2...$ is enciphered according to the formula $Z_i=(X_i+Y_i)\mathrm{mod}2$, where $Y_1{Y}_2...$ is a key sequence generated by the Bernoulli source with equal probabilities of 0 and 1, anyone who knows $Z_1{Z}_2...$ has no information about $X_1{X}_2...$ without the knowledge of the key $Y_1{Y}_2...$. (The best strategy is to guess $X_1{X}_2...$ not paying attention to $Z_1{Z}_2...$.) But what should one say about secrecy of an analogous method where the key sequence $Y_1{Y}_2...$ is generated by the Bernoulli source with a small bias, say, $$ $$? To the best of our knowledge, there are no theoretical estimates for the secrecy of such a system, as well as for the general case where $$ (the plaintext) and key sequence are described by stationary ergodic processes. We consider the running-key ciphers where the plaintext and the key are generated by stationary ergodic sources and show how to estimate the secrecy of such systems. In particular, it is shown that, in a certain sense, the Vernam cipher is robust to small deviations from randomness.