TY - JOUR
AB - We introduce the notion of “non-malleable codes” which relaxes the notion of error correction and error detection. Informally, a code is non-malleable if the message contained in a modified codeword is either the original message, or a completely unrelated value. In contrast to error correction and error detection, non-malleability can be achieved for very rich classes of modifications. We construct an efficient code that is non-malleable with respect to modifications that affect each bit of the codeword arbitrarily (i.e., leave it untouched, flip it, or set it to either 0 or 1), but independently of the value of the other bits of the codeword. Using the probabilistic method, we also show a very strong and general statement: there exists a non-malleable code for every “small enough” family F of functions via which codewords can be modified. Although this probabilistic method argument does not directly yield efficient constructions, it gives us efficient non-malleable codes in the random-oracle model for very general classes of tampering functions—e.g., functions where every bit in the tampered codeword can depend arbitrarily on any 99% of the bits in the original codeword. As an application of non-malleable codes, we show that they provide an elegant algorithmic solution to the task of protecting functionalities implemented in hardware (e.g., signature cards) against “tampering attacks.” In such attacks, the secret state of a physical system is tampered, in the hopes that future interaction with the modified system will reveal some secret information. This problem was previously studied in the work of Gennaro et al. in 2004 under the name “algorithmic tamper proof security” (ATP). We show that non-malleable codes can be used to achieve important improvements over the prior work. In particular, we show that any functionality can be made secure against a large class of tampering attacks, simply by encoding the secret state with a non-malleable code while it is stored in memory.
AU - Dziembowski, Stefan
AU - Pietrzak, Krzysztof Z
AU - Wichs, Daniel
ID - 107
IS - 4
JF - Journal of the ACM
TI - Non-malleable codes
VL - 65
ER -
TY - CONF
AB - Proofs of space (PoS) were suggested as more ecological and economical alternative to proofs of work, which are currently used in blockchain designs like Bitcoin. The existing PoS are based on rather sophisticated graph pebbling lower bounds. Much simpler and in several aspects more efficient schemes based on inverting random functions have been suggested, but they don’t give meaningful security guarantees due to existing time-memory trade-offs. In particular, Hellman showed that any permutation over a domain of size N can be inverted in time T by an algorithm that is given S bits of auxiliary information whenever (Formula presented). For functions Hellman gives a weaker attack with S2· T≈ N2 (e.g., S= T≈ N2/3). To prove lower bounds, one considers an adversary who has access to an oracle f: [ N] → [N] and can make T oracle queries. The best known lower bound is S· T∈ Ω(N) and holds for random functions and permutations. We construct functions that provably require more time and/or space to invert. Specifically, for any constant k we construct a function [N] → [N] that cannot be inverted unless Sk· T∈ Ω(Nk) (in particular, S= T≈ (Formula presented). Our construction does not contradict Hellman’s time-memory trade-off, because it cannot be efficiently evaluated in forward direction. However, its entire function table can be computed in time quasilinear in N, which is sufficient for the PoS application. Our simplest construction is built from a random function oracle g: [N] × [N] → [ N] and a random permutation oracle f: [N] → N] and is defined as h(x) = g(x, x′) where f(x) = π(f(x′)) with π being any involution without a fixed point, e.g. flipping all the bits. For this function we prove that any adversary who gets S bits of auxiliary information, makes at most T oracle queries, and inverts h on an ϵ fraction of outputs must satisfy S2· T∈ Ω(ϵ2N2).
AU - Abusalah, Hamza M
AU - Alwen, Joel F
AU - Cohen, Bram
AU - Khilko, Danylo
AU - Pietrzak, Krzysztof Z
AU - Reyzin, Leonid
ID - 559
SN - 978-331970696-2
TI - Beyond Hellman’s time-memory trade-offs with applications to proofs of space
VL - 10625
ER -
TY - CONF
AB - Position based cryptography (PBC), proposed in the seminal work of Chandran, Goyal, Moriarty, and Ostrovsky (SIAM J. Computing, 2014), aims at constructing cryptographic schemes in which the identity of the user is his geographic position. Chandran et al. construct PBC schemes for secure positioning and position-based key agreement in the bounded-storage model (Maurer, J. Cryptology, 1992). Apart from bounded memory, their security proofs need a strong additional restriction on the power of the adversary: he cannot compute joint functions of his inputs. Removing this assumption is left as an open problem. We show that an answer to this question would resolve a long standing open problem in multiparty communication complexity: finding a function that is hard to compute with low communication complexity in the simultaneous message model, but easy to compute in the fully adaptive model. On a more positive side: we also show some implications in the other direction, i.e.: we prove that lower bounds on the communication complexity of certain multiparty problems imply existence of PBC primitives. Using this result we then show two attractive ways to “bypass” our hardness result: the first uses the random oracle model, the second weakens the locality requirement in the bounded-storage model to online computability. The random oracle construction is arguably one of the simplest proposed so far in this area. Our results indicate that constructing improved provably secure protocols for PBC requires a better understanding of multiparty communication complexity. This is yet another example where negative results in one area (in our case: lower bounds in multiparty communication complexity) can be used to construct secure cryptographic schemes.
AU - Brody, Joshua
AU - Dziembowski, Stefan
AU - Faust, Sebastian
AU - Pietrzak, Krzysztof Z
ED - Kalai, Yael
ED - Reyzin, Leonid
ID - 605
SN - 978-331970499-9
TI - Position based cryptography and multiparty communication complexity
VL - 10677
ER -
TY - CONF
AB - Several cryptographic schemes and applications are based on functions that are both reasonably efficient to compute and moderately hard to invert, including client puzzles for Denial-of-Service protection, password protection via salted hashes, or recent proof-of-work blockchain systems. Despite their wide use, a definition of this concept has not yet been distilled and formalized explicitly. Instead, either the applications are proven directly based on the assumptions underlying the function, or some property of the function is proven, but the security of the application is argued only informally. The goal of this work is to provide a (universal) definition that decouples the efforts of designing new moderately hard functions and of building protocols based on them, serving as an interface between the two. On a technical level, beyond the mentioned definitions, we instantiate the model for four different notions of hardness. We extend the work of Alwen and Serbinenko (STOC 2015) by providing a general tool for proving security for the first notion of memory-hard functions that allows for provably secure applications. The tool allows us to recover all of the graph-theoretic techniques developed for proving security under the older, non-composable, notion of security used by Alwen and Serbinenko. As an application of our definition of moderately hard functions, we prove the security of two different schemes for proofs of effort (PoE). We also formalize and instantiate the concept of a non-interactive proof of effort (niPoE), in which the proof is not bound to a particular communication context but rather any bit-string chosen by the prover.
AU - Alwen, Joel F
AU - Tackmann, Björn
ED - Kalai, Yael
ED - Reyzin, Leonid
ID - 609
SN - 978-331970499-9
TI - Moderately hard functions: Definition, instantiations, and applications
VL - 10677
ER -
TY - JOUR
AB - PMAC is a simple and parallel block-cipher mode of operation, which was introduced by Black and Rogaway at Eurocrypt 2002. If instantiated with a (pseudo)random permutation over n-bit strings, PMAC constitutes a provably secure variable input-length (pseudo)random function. For adversaries making q queries, each of length at most l (in n-bit blocks), and of total length σ ≤ ql, the original paper proves an upper bound on the distinguishing advantage of Ο(σ2/2n), while the currently best bound is Ο (qσ/2n).In this work we show that this bound is tight by giving an attack with advantage Ω (q2l/2n). In the PMAC construction one initially XORs a mask to every message block, where the mask for the ith block is computed as τi := γi·L, where L is a (secret) random value, and γi is the i-th codeword of the Gray code. Our attack applies more generally to any sequence of γi’s which contains a large coset of a subgroup of GF(2n). We then investigate if the security of PMAC can be further improved by using τi’s that are k-wise independent, for k > 1 (the original distribution is only 1-wise independent). We observe that the security of PMAC will not increase in general, even if the masks are chosen from a 2-wise independent distribution, and then prove that the security increases to O(q<2/2n), if the τi are 4-wise independent. Due to simple extension attacks, this is the best bound one can hope for, using any distribution on the masks. Whether 3-wise independence is already sufficient to get this level of security is left as an open problem.
AU - Gazi, Peter
AU - Pietrzak, Krzysztof Z
AU - Rybar, Michal
ID - 6196
IS - 2
JF - IACR Transactions on Symmetric Cryptology
TI - The exact security of PMAC
VL - 2016
ER -
TY - CONF
AB - Memory-hard functions (MHFs) are hash algorithms whose evaluation cost is dominated by memory cost. As memory, unlike computation, costs about the same across different platforms, MHFs cannot be evaluated at significantly lower cost on dedicated hardware like ASICs. MHFs have found widespread applications including password hashing, key derivation, and proofs-of-work. This paper focuses on scrypt, a simple candidate MHF designed by Percival, and described in RFC 7914. It has been used within a number of cryptocurrencies (e.g., Litecoin and Dogecoin) and has been an inspiration for Argon2d, one of the winners of the recent password-hashing competition. Despite its popularity, no rigorous lower bounds on its memory complexity are known. We prove that scrypt is optimally memory-hard, i.e., its cumulative memory complexity (cmc) in the parallel random oracle model is Ω(n2w), where w and n are the output length and number of invocations of the underlying hash function, respectively. High cmc is a strong security target for MHFs introduced by Alwen and Serbinenko (STOC’15) which implies high memory cost even for adversaries who can amortize the cost over many evaluations and evaluate the underlying hash functions many times in parallel. Our proof is the first showing optimal memory-hardness for any MHF. Our result improves both quantitatively and qualitatively upon the recent work by Alwen et al. (EUROCRYPT’16) who proved a weaker lower bound of Ω(n2w/ log2 n) for a restricted class of adversaries.
AU - Alwen, Joel F
AU - Chen, Binchi
AU - Pietrzak, Krzysztof Z
AU - Reyzin, Leonid
AU - Tessaro, Stefano
ED - Coron, Jean-Sébastien
ED - Buus Nielsen, Jesper
ID - 635
SN - 978-331956616-0
TI - Scrypt is maximally memory hard
VL - 10212
ER -
TY - CONF
AB - For many cryptographic primitives, it is relatively easy to achieve selective security (where the adversary commits a-priori to some of the choices to be made later in the attack) but appears difficult to achieve the more natural notion of adaptive security (where the adversary can make all choices on the go as the attack progresses). A series of several recent works shows how to cleverly achieve adaptive security in several such scenarios including generalized selective decryption (Panjwani, TCC ’07 and Fuchsbauer et al., CRYPTO ’15), constrained PRFs (Fuchsbauer et al., ASIACRYPT ’14), and Yao garbled circuits (Jafargholi and Wichs, TCC ’16b). Although the above works expressed vague intuition that they share a common technique, the connection was never made precise. In this work we present a new framework that connects all of these works and allows us to present them in a unified and simplified fashion. Moreover, we use the framework to derive a new result for adaptively secure secret sharing over access structures defined via monotone circuits. We envision that further applications will follow in the future. Underlying our framework is the following simple idea. It is well known that selective security, where the adversary commits to n-bits of information about his future choices, automatically implies adaptive security at the cost of amplifying the adversary’s advantage by a factor of up to 2n. However, in some cases the proof of selective security proceeds via a sequence of hybrids, where each pair of adjacent hybrids locally only requires some smaller partial information consisting of m ≪ n bits. The partial information needed might be completely different between different pairs of hybrids, and if we look across all the hybrids we might rely on the entire n-bit commitment. Nevertheless, the above is sufficient to prove adaptive security, at the cost of amplifying the adversary’s advantage by a factor of only 2m ≪ 2n. In all of our examples using the above framework, the different hybrids are captured by some sort of a graph pebbling game and the amount of information that the adversary needs to commit to in each pair of hybrids is bounded by the maximum number of pebbles in play at any point in time. Therefore, coming up with better strategies for proving adaptive security translates to various pebbling strategies for different types of graphs.
AU - Jafargholi, Zahra
AU - Kamath Hosdurg, Chethan
AU - Klein, Karen
AU - Komargodski, Ilan
AU - Pietrzak, Krzysztof Z
AU - Wichs, Daniel
ED - Katz, Jonathan
ED - Shacham, Hovav
ID - 637
SN - 978-331963687-0
TI - Be adaptive avoid overcommitting
VL - 10401
ER -
TY - CONF
AB - Data-independent Memory Hard Functions (iMHFS) are finding a growing number of applications in security; especially in the domain of password hashing. An important property of a concrete iMHF is specified by fixing a directed acyclic graph (DAG) Gn on n nodes. The quality of that iMHF is then captured by the following two pebbling complexities of Gn: – The parallel cumulative pebbling complexity Π∥cc(Gn) must be as high as possible (to ensure that the amortized cost of computing the function on dedicated hardware is dominated by the cost of memory). – The sequential space-time pebbling complexity Πst(Gn) should be as close as possible to Π∥cc(Gn) (to ensure that using many cores in parallel and amortizing over many instances does not give much of an advantage). In this paper we construct a family of DAGs with best possible parameters in an asymptotic sense, i.e., where Π∥cc(Gn) = Ω(n2/ log(n)) (which matches a known upper bound) and Πst(Gn) is within a constant factor of Π∥cc(Gn). Our analysis relies on a new connection between the pebbling complexity of a DAG and its depth-robustness (DR) – a well studied combinatorial property. We show that high DR is sufficient for high Π∥cc. Alwen and Blocki (CRYPTO’16) showed that high DR is necessary and so, together, these results fully characterize DAGs with high Π∥cc in terms of DR. Complementing these results, we provide new upper and lower bounds on the Π∥cc of several important candidate iMHFs from the literature. We give the first lower bounds on the memory hardness of the Catena and Balloon Hashing functions in a parallel model of computation and we give the first lower bounds of any kind for (a version) of Argon2i. Finally we describe a new class of pebbling attacks improving on those of Alwen and Blocki (CRYPTO’16). By instantiating these attacks we upperbound the Π∥cc of the Password Hashing Competition winner Argon2i and one of the Balloon Hashing functions by O (n1.71). We also show an upper bound of O(n1.625) for the Catena functions and the two remaining Balloon Hashing functions.
AU - Alwen, Joel F
AU - Blocki, Jeremiah
AU - Pietrzak, Krzysztof Z
ED - Coron, Jean-Sébastien
ED - Buus Nielsen, Jesper
ID - 640
SN - 978-331956616-0
TI - Depth-robust graphs and their cumulative memory complexity
VL - 10212
ER -
TY - CONF
AB - Pseudoentropy has found a lot of important applications to cryptography and complexity theory. In this paper we focus on the foundational problem that has not been investigated so far, namely by how much pseudoentropy (the amount seen by computationally bounded attackers) diﬀers from its information-theoretic counterpart (seen by unbounded observers), given certain limits on attacker’s computational power? We provide the following answer for HILL pseudoentropy, which exhibits a threshold behavior around the size exponential in the entropy amount:– If the attacker size (s) and advantage () satisfy s (formula presented) where k is the claimed amount of pseudoentropy, then the pseudoentropy boils down to the information-theoretic smooth entropy. – If s (formula presented) then pseudoentropy could be arbitrarily bigger than the information-theoretic smooth entropy. Besides answering the posted question, we show an elegant application of our result to the complexity theory, namely that it implies the clas-sical result on the existence of functions hard to approximate (due to Pippenger). In our approach we utilize non-constructive techniques: the duality of linear programming and the probabilistic method.
AU - Skórski, Maciej
ED - Jäger, Gerhard
ED - Steila, Silvia
ID - 648
SN - 978-331955910-0
TI - On the complexity of breaking pseudoentropy
VL - 10185
ER -
TY - CONF
AB - In this work we present a short and unified proof for the Strong and Weak Regularity Lemma, based on the cryptographic tech-nique called low-complexity approximations. In short, both problems reduce to a task of finding constructively an approximation for a certain target function under a class of distinguishers (test functions), where dis-tinguishers are combinations of simple rectangle-indicators. In our case these approximations can be learned by a simple iterative procedure, which yields a unified and simple proof, achieving for any graph with density d and any approximation parameter the partition size. The novelty in our proof is: (a) a simple approach which yields both strong and weaker variant, and (b) improvements when d = o(1). At an abstract level, our proof can be seen a refinement and simplification of the “analytic” proof given by Lovasz and Szegedy.
AU - Skórski, Maciej
ED - Jäger, Gerhard
ED - Steila, Silvia
ID - 650
SN - 03029743
TI - A cryptographic view of regularity lemmas: Simpler unified proofs and refined bounds
VL - 10185
ER -