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Publication# Efficient Non-malleable Codes and Key-Derivation for Poly-size Tampering Circuits

Abstract

Non-malleable codes, defined by Dziembowski, Pietrzak and Wichs (ICS ’10), provide roughly the following guarantee: if a codeword c encoding some message x is tampered to c' = f(c) such that c c, then the tampered message x contained in c reveals no information about x. Non-malleable codes have applications to immunizing cryptosystems against tampering attacks and related-key attacks. One cannot have an efficient non-malleable code that protects against all efficient tampering functions f. However, in this work we show “the next best thing”: for any polynomial bound s given a-priori, there is an efficient non-malleable code that protects against all tampering functions f computable by a circuit of size s. More generally, for any family of tampering functions F of size |F| ≤ 2s, there is an efficient nonmalleable code that protects against all f ∈ F. The rate of our codes, defined as the ratio of message to codeword size, approaches 1. Our results are information-theoretic and our main proof technique relies on a careful probabilistic method argument using limited independence. As a result, we get an efficiently samplable family of efficient codes, such that a random member of the family is non-malleable with overwhelming probability. Alternatively, we can view the result as providing an efficient non-malleable code in the “common reference string” (CRS) model. We also introduce a new notion of non-malleable key derivation, which uses randomness x to derive a secret key y = h(x) in such a way that, even if x is tampered to a different value x = f(x), the derived key y = h(x) does not reveal any information about y. Our results for non-malleable key derivation are analogous to those for non-malleable codes. As a useful tool in our analysis, we rely on the notion of “leakage-resilient storage” of Davı, Dziembowski and Venturi (SCN ’10) and, as a result of independent interest, we also significantly improve on the parameters of such schemes.

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Related-key attack

In cryptography, a related-key attack is any form of cryptanalysis where the attacker can observe the operation of a cipher under several different keys whose values are initially unknown, but where some mathematical relationship connecting the keys is known to the attacker. For example, the attacker might know that the last 80 bits of the keys are always the same, even though they don't know, at first, what the bits are.

Information theory

Information theory is the mathematical study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field, in applied mathematics, is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering. A key measure in information theory is entropy.

Randomness

In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if the probability distribution is known, the frequency of different outcomes over repeated events (or "trials") is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4.