Non-Malleable Codes for Bounded Depth, Bounded Fan-in Circuits

We show how to construct efficient, unconditionally secure non-malleable codes for bounded output locality. In particular, our scheme is resilient against functions such that any output bit is dependent on at most $n^{\delta}$ bits, where $n$ is the total number of bits in a codeword and $0 \leq \delta < 1$ a constant. Notably, this tampering class includes $\mathsf{NC}^0$

Non-Malleable Codes for Small-Depth Circuits

We construct efficient, unconditional non-malleable codes that are secure against tampering functions computed by small-depth circuits. For constant-depth circuits of polynomial size (i.e. $\mathsf{AC^0}$ tampering functions), our codes have codeword length $n = k^{1+o(1)}$ for a $k$-bit message. This is an exponential improvement of the previous best construction due to Chattopadhyay and Li (STOC 2017), which had codeword length $2^{O(\sqrt{k})}$. Our construction remains efficient for circuit depths as large as $\Theta(\log(n)/\log\log(n))$ (indeed, our codeword length remains $n\leq k^{1+\epsilon})$, and extending our result beyond this would require separating $\mathsf{P}$ from $\mathsf{NC^1}$. We obtain our codes via a new efficient non-malleable reduction from small-depth tampering to split-state tampering. A novel aspect of our work is the incorporation of techniques from unconditional derandomization into the framework of non-malleable reductions. In particular, a key ingredient in our analysis is a recent pseudorandom switching lemma of Trevisan and Xue (CCC 2013), a derandomization of the influential switching lemma from circuit complexity; the randomness-efficiency of this switching lemma translates into the rate-efficiency of our codes via our non-malleable reduction.Comment: 26 pages, 4 figure

Limits to Non-Malleability

There have been many successes in constructing explicit non-malleable codes for various classes of tampering functions in recent years, and strong existential results are also known. In this work we ask the following question: When can we rule out the existence of a non-malleable code for a tampering class ?? First, we start with some classes where positive results are well-known, and show that when these classes are extended in a natural way, non-malleable codes are no longer possible. Specifically, we show that no non-malleable codes exist for any of the following tampering classes: - Functions that change d/2 symbols, where d is the distance of the code; - Functions where each input symbol affects only a single output symbol; - Functions where each of the n output bits is a function of n-log n input bits. Furthermore, we rule out constructions of non-malleable codes for certain classes ? via reductions to the assumption that a distributional problem is hard for ?, that make black-box use of the tampering functions in the proof. In particular, this yields concrete obstacles for the construction of efficient codes for NC, even assuming average-case variants of P ? NC

Algebra in Computational Complexity

At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples. The algebraic theme continues in some of the most exciting recent progress in computational complexity. There have been significant recent advances in algebraic circuit lower bounds, and the so-called "chasm at depth 4" suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model, and these are tied to central questions regarding the power of randomness in computation. Representation theory has emerged as an important tool in three separate lines of work: the "Geometric Complexity Theory" approach to P vs. NP and circuit lower bounds, the effort to resolve the complexity of matrix multiplication, and a framework for constructing locally testable codes. Coding theory has seen several algebraic innovations in recent years, including multiplicity codes, and new lower bounds. This seminar brought together researchers who are using a diverse array of algebraic methods in a variety of settings. It plays an important role in educating a diverse community about the latest new techniques, spurring further progress

Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas

We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: an $\varepsilon$-PRG for the class of size-$M$ depth-$d$ $\mathsf{AC}^0$ circuits with seed length $\log(M)^{d+O(1)}\cdot \log(1/\varepsilon)$, and an $\varepsilon$-PRG for the class of $S$-sparse $\mathbb{F}_2$ polynomials with seed length $2^{O(\sqrt{\log S})}\cdot \log(1/\varepsilon)$. These results bring the state of the art for unconditional derandomization of these classes into sharp alignment with the state of the art for computational hardness for all parameter settings: improving on the seed lengths of either PRG would require breakthrough progress on longstanding and notorious circuit lower bounds. The key enabling ingredient in our approach is a new \emph{pseudorandom multi-switching lemma}. We derandomize recently-developed \emph{multi}-switching lemmas, which are powerful generalizations of H{\aa}stad's switching lemma that deal with \emph{families} of depth-two circuits. Our pseudorandom multi-switching lemma---a randomness-efficient algorithm for sampling restrictions that simultaneously simplify all circuits in a family---achieves the parameters obtained by the (full randomness) multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and H{\aa}stad [H{\aa}s14]. This optimality of our derandomization translates into the optimality (given current circuit lower bounds) of our PRGs for $\mathsf{AC}^0$ and sparse $\mathbb{F}_2$ polynomials

Non-malleable codes for space-bounded tampering

Non-malleable codes—introduced by Dziembowski, Pietrzak and Wichs at ICS 2010—are key-less coding schemes in which mauling attempts to an encoding of a given message, w.r.t. some class of tampering adversaries, result in a decoded value that is either identical or unrelated to the original message. Such codes are very useful for protecting arbitrary cryptographic primitives against tampering attacks against the memory. Clearly, non-malleability is hopeless if the class of tampering adversaries includes the decoding and encoding algorithm. To circumvent this obstacle, the majority of past research focused on designing non-malleable codes for various tampering classes, albeit assuming that the adversary is unable to decode. Nonetheless, in many concrete settings, this assumption is not realistic

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness and randomness extraction. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes

Continuously non-malleable codes with split-state refresh

Non-malleable codes for the split-state model allow to encode a message into two parts, such that arbitrary independent tampering on each part, and subsequent decoding of the corresponding modified codeword, yields either the same as the original message, or a completely unrelated value. Continuously non-malleable codes further allow to tolerate an unbounded (polynomial) number of tampering attempts, until a decoding error happens. The drawback is that, after an error happens, the system must self-destruct and stop working, otherwise generic attacks become possible. In this paper we propose a solution to this limitation, by leveraging a split-state refreshing procedure. Namely, whenever a decoding error happens, the two parts of an encoding can be locally refreshed (i.e., without any interaction), which allows to avoid the self-destruct mechanism. An additional feature of our security model is that it captures directly security against continual leakage attacks. We give an abstract framework for building such codes in the common reference string model, and provide a concrete instantiation based on the external Diffie-Hellman assumption. Finally, we explore applications in which our notion turns out to be essential. The first application is a signature scheme tolerating an arbitrary polynomial number of split-state tampering attempts, without requiring a self-destruct capability, and in a model where refreshing of the memory happens only after an invalid output is produced. This circumvents an impossibility result from a recent work by Fuijisaki and Xagawa (Asiacrypt 2016). The second application is a compiler for tamper-resilient RAM programs. In comparison to other tamper-resilient compilers, ours has several advantages, among which the fact that, for the first time, it does not rely on the self-destruct feature

Efficient public-key cryptography with bounded leakage and tamper resilience

We revisit the question of constructing public-key encryption and signature schemes with security in the presence of bounded leakage and tampering memory attacks. For signatures we obtain the first construction in the standard model; for public-key encryption we obtain the first construction free of pairing (avoiding non-interactive zero-knowledge proofs). Our constructions are based on generic building blocks, and, as we show, also admit efficient instantiations under fairly standard number-theoretic assumptions. The model of bounded tamper resistance was recently put forward by Damgård et al. (Asiacrypt 2013) as an attractive path to achieve security against arbitrary memory tampering attacks without making hardware assumptions (such as the existence of a protected self-destruct or key-update mechanism), the only restriction being on the number of allowed tampering attempts (which is a parameter of the scheme). This allows to circumvent known impossibility results for unrestricted tampering (Gennaro et al., TCC 2010), while still being able to capture realistic tampering attack