We study the memory-tightness of security reductions in public-key cryptography, focusing in particular on Hashed ElGamal. We prove that any {\em straightline} (i.e., without rewinding) black-box reduction needs memory which grows linearly with the number of queries of the adversary it has access to, as long as this reduction treats the underlying group generically. This makes progress towards proving a conjecture by Auerbach {\em et al.} (CRYPTO 2017), and is also the first lower bound on memory-tightness for a concrete cryptographic scheme (as opposed to generalized reductions across security notions). Our proof relies on compression arguments in the generic group model.

Concrete security proofs give upper bounds on the attacker’s advantage as a function of its time/query complexity. Cryptanalysis suggests however that other resource limitations – most notably, the attacker’s memory – could make the achievable advantage smaller, and thus these proven bounds too pessimistic. Yet, handling memory limitations has eluded existing security proofs.This paper initiates the study of time-memory trade-offs for basic symmetric cryptography. We show that schemes like counter-mode encryption, which are affected by the Birthday Bound, become more secure (in terms of time complexity) as the attacker’s memory is reduced.One key step of this work is a generalization of the Switching Lemma: For adversaries with S bits of memory issuing q distinct queries, we prove an n-to-n bit random function indistinguishable from a permutation as long as $$S \times q \ll 2^n$$S×q≪2n. This result assumes a combinatorial conjecture, which we discuss, and implies right away trade-offs for deterministic, stateful versions of CTR and OFB encryption.We also show an unconditional time-memory trade-off for the security of randomized CTR based on a secure PRF. Via the aforementioned conjecture, we extend the result to assuming a PRP instead, assuming only one-block messages are encrypted.Our results solely rely on standard PRF/PRP security of an underlying block cipher. We frame the core of our proofs within a general framework of indistinguishability for streaming algorithms which may be of independent interest.

The need for high-quality randomness in cryptography makes random-number generation one of its most fundamental tasks.A recent important line of work (initiated by Dodis et al., CCS ’13) focuses on the notion of robustness for pseudorandom number generators (PRNGs) with inputs. These are primitives that use various sources to accumulate sufficient entropy into a state, from which pseudorandom bits are extracted. Robustness ensures that PRNGs remain secure even under state compromise and adversarial control of entropy sources. However, the achievability of robustness inherently depends on a seed, or, alternatively, on an ideal primitive (e.g., a random oracle), independent of the source of entropy. Both assumptions are problematic: seed generation requires randomness to start with, and it is arguable whether the seed or the ideal primitive can be kept independent of the source. This paper resolves this dilemma by putting forward new notions of robustness which enable both (1) seedless PRNGs and (2) primitive-dependent adversarial sources of entropy. To bypass obvious impossibility results, we make a realistic compromise by requiring that the source produce sufficient entropy even given its evaluations of the underlying primitive. We also provide natural, practical, and provably secure constructions based on hash-function designs from compression functions, block ciphers, and permutations. Our constructions can be instantiated with minimal changes to industry-standard hash functions SHA-2 and SHA-3, or key derivation function HKDF, and can be downgraded to (online) seedless randomness extractors, which are of independent interest.On the way we consider both a computational variant of robustness, where attackers only make a bounded number of queries to the ideal primitive, as well as a new information-theoretic variant, which dispenses with this assumption to a certain extent, at the price of requiring a high rate of injected weak randomness (as it is, e.g., plausible on Intel’s on-chip RNG). The latter notion enables applications such as everlasting security. Finally, we show that the CBC extractor, used by Intel’s on-chip RNG, is provably insecure in our model.

Memory-hard functions (MHFs) are moderately-hard functions which enforce evaluation costs both in terms of time and memory (often, in form of a trade-off). They are used e.g. for password protection, password-based key-derivation, and within cryptocurrencies, and have received a considerable amount of theoretical scrutiny over the last few years. However, analyses see MHFs as modes of operation of some underlying hash function $$\mathcal {H}$$, modeled as a monolithic random oracle. This is however a very strong assumption, as such hash functions are built from much simpler primitives, following somewhat ad-hoc design paradigms.This paper initiates the study of how to securely instantiate $$\mathcal {H}$$ within MHF designs using common cryptographic primitives like block ciphers, compression functions, and permutations. Security here will be in a model in which the adversary has parallel access to an idealized version of the underlying primitive. We will provide provably memory-hard constructions from all the aforementioned primitives. Our results are generic, in that we will rely on hard-to-pebble graphs designed in prior works to obtain our constructions.One particular challenge we encounter is that $$\mathcal {H}$$ is usually required to have large outputs (to increase memory hardness without changing the description size of MHFs), whereas the underlying primitives generally have small output sizes.

Format-preserving encryption (FPE) produces ciphertexts which have the same format as the plaintexts. Building secure FPE is very challenging, and recent attacks (Bellare, Hoang, Tessaro, CCS ’16; Durak and Vaudenay, CRYPTO ’17) have highlighted security deficiencies in the recent NIST SP800-38G standard. This has left the question open of whether practical schemes with high security exist.In this paper, we continue the investigation of attacks against FPE schemes. Our first contribution are new known-plaintext message recovery attacks against Feistel-based FPEs (such as FF1/FF3 from the NIST SP800-38G standard) which improve upon previous work in terms of amortized complexity in multi-target scenarios, where multiple ciphertexts are to be decrypted. Our attacks are also qualitatively better in that they make no assumptions on the correlation between the targets to be decrypted and the known plaintexts. We also surface a new vulnerability specific to FF3 and how it handles odd length domains, which leads to a substantial speedup in our attacks.We also show the first attacks against non-Feistel based FPEs. Specifically, we show a strong message-recovery attack for FNR, a construction proposed by Cisco which replaces two rounds in the Feistel construction with a pairwise-independent permutation, following the paradigm by Naor and Reingold (JoC, ’99). We also provide a strong ciphertext-only attack against a variant of the DTP construction by Brightwell and Smith, which is deployed by Protegrity within commercial applications. All of our attacks show that existing constructions fall short of achieving desirable security levels. For Feistel and the FNR schemes, our attacks become feasible on small domains, e.g., 8 bits, for suggested round numbers. Our attack against the DTP construction is practical even for large domains. We provide proof-of-concept implementations of our attacks that verify our theoretical findings.

We initiate the study of symmetric encryption in a regime where the memory of the adversary is bounded. For a block cipher with n-bit blocks, we present modes of operation for encryption and authentication that guarantee security beyond$$2^n$$ encrypted/authenticated messages, as long as (1) the adversary’s memory is restricted to be less than $$2^n$$ bits, and (2) the key of the block cipher is long enough to mitigate memory-less key-search attacks. This is the first proposal of a setting which allows to bypass the $$2^n$$ barrier under a reasonable assumption on the adversarial resources.Motivated by the above, we also discuss the problem of stretching the key of a block cipher in the setting where the memory of the adversary is bounded. We show a tight equivalence between the security of double encryption in the ideal-cipher model and the hardness of a special case of the element distinctness problem, which we call the list-disjointness problem. Our result in particular implies a conditional lower bound on time-memory trade-offs to break PRP security of double encryption, assuming optimality of the worst-case complexity of existing algorithms for list disjointness.

A public random function is a random function that is accessible by all parties, including the adversary. For example, a (public) random oracle is a public random function $\{0,1\}^{*} \to \{0,1\}^n$. The natural problem of constructing a public random oracle from a public random function $\{0,1\}^{m} \to \{0,1\}^n$ (for some $m > n$) was first considered at Crypto 2005 by Coron et al.\ who proved the security of variants of the Merkle-Damg{\aa}rd construction against adversaries issuing up to $O(2^{n/2})$ queries to the construction and to the underlying compression function. This bound is less than the square root of $n2^m$, the number of random bits contained in the underlying random function. In this paper, we investigate domain extenders for public random functions approaching optimal security. In particular, for all $\epsilon \in (0,1)$ and all functions $m$ and $\ell$ (polynomial in $n$), we provide a construction $\mathbf{C}_{\epsilon,m,\ell}(\cdot)$ which extends a public random function $\mathbf{R}: \{0,1\}^{n} \to \{0,1\}^n$ to a function $\mathbf{C}_{\epsilon,m,\ell}(\R): \{0,1\}^{m(n)} \to \{0,1\}^{\ell(n)}$ with time-complexity polynomial in $n$ and $1/\epsilon$ and which is secure against adversaries which make up to $\Theta(2^{n(1-\epsilon)})$ queries. A central tool for achieving high security are special classes of unbalanced bipartite expander graphs with small degree. The achievability of practical (as opposed to complexity-theoretic) efficiency is proved by a non-constructive existence proof. Combined with the iterated constructions of Coron et al., our result leads to the first iterated construction of a hash function $\{0,1\}^{*} \to \{0,1\}^n$ from a component function $\{0,1\}^{n} \to \{0,1\}^n$ that withstands all recently proposed generic attacks against iterated hash functions, like Joux's multi-collision attack, Kelsey and Schneier's second-preimage attack, and Kelsey and Kohno's herding attacks.