International Association
for Cryptologic Research

A substantial effort has been devoted to proving optimal bounds for the security of key-alternating ciphers with independent sub-keys in the random permutation model (e.g., Chen and Steinberger, EUROCRYPT '14; Hoang and Tessaro, CRYPTO '16). While common in the study of multi-round constructions, the assumption that sub-keys are truly independent is not realistic, as these are generally highly correlated and generated from shorter keys. In this paper, we show the existence of non-trivial distributions of limited independence for which a t-round key-alternating cipher achieves optimal security. Our work is a natural continuation of the work of Chen et al. (CRYPTO '14) which considered the case of t = 2 when all-subkeys are identical. Here, we show that key-alternating ciphers remain secure for a large class of (t-1)-wise and (t-2)-wise independent distribution of sub-keys. Our proofs proceed by generalizations of the so-called Sum-Capture Theorem, which we prove using Fourier-analytic techniques.

We build symmetric encryption schemes from a pseudorandom function/permutation with domain size $N$ which have very high security -- in terms of the amount of messages $q$ they can securely encrypt -- assuming the adversary has $S < N$ bits of memory. We aim to minimize the number of calls $k$ we make to the underlying primitive to achieve a certain $q$, or equivalently, to maximize the achievable $q$ for a given $k$. We target in particular $q \gg N$, in contrast to recent works (Jaeger and Tessaro, EUROCRYPT '19; Dinur, EUROCRYPT '20) which aim to beat the birthday barrier with one call when $S < \sqrt{N}$. Our first result gives new and explicit bounds for the Sample-then-Extract paradigm by Tessaro and Thiruvengadam (TCC '18). We show instantiations for which $q =\Omega((N/S)^{k})$. If $S < N^{1- \alpha}$, Thiruvengadam and Tessaro's weaker bounds only guarantee $q > N$ when $k = \Omega(\log N)$. In contrast, here, we show this is true already for $k = O(1/\alpha)$. We also consider a scheme by Bellare, Goldreich and Krawczyk (CRYPTO '99) which evaluates the primitive at $k$ independent random strings, and masks the message with the XOR of the outputs. Here, we show $q= \Omega((N/S)^{k/2})$, using new combinatorial bounds on the list-decodability of XOR codes which are of independent interest. We also study best-possible attacks against this construction.