Information-Theoretic Indistinguishability via the Chi-Squared Method

Proving tight bounds on information-theoretic indistinguishability is a central problem in symmetric cryptography. This paper introduces a new method for information-theoretic indistinguishability proofs, called “the chi-squared method”. At its core, the method requires upperbounds on the so-called χ2 divergence (due to Neyman and Pearson) between the output distributions of two systems being queries. The method morally resembles, yet also considerably simplifies, a previous approach proposed by Bellare and Impagliazzo (ePrint, 1999), while at the same time increasing its expressiveness and delivering tighter bounds. We showcase the chi-squared method on some examples. In particular: (1) We prove an optimal bound of q/2 for the XOR of two permutations, and our proof considerably simplifies previous approaches using the H-coefficient method, (2) we provide improved bounds for the recently proposed encrypted Davies-Meyer PRF construction by Cogliati and Seurin (CRYPTO ’16), and (3) we give a tighter bound for the Swap-or-not cipher by Hoang, Morris, and Rogaway (CRYPTO ’12).