On Recent Advances in Key Derivation via the Leftover Hash Lemma

Barak et al. showed how to significantly reduce the entropy loss, which is necessary in general, in the use of the Leftover Hash Lemma (LHL) to derive a secure key for many important cryptographic applications. If one wants this key to be secure against any additional short leakage, then the min-entropy of the source used with the LHL must be big enough. Recently, Berens came up with a notion of collision entropy that is much weaker than min-entropy and allows proving a version of the LHL with leakage robustness but without any entropy saving. We combine both approaches and extend the results of Barak et. al to the collision entropy. Summarizing, we obtain a version of the LHL with optimized entropy loss, leakage robustness and weak entropy requirements

Replacing Probability Distributions in Security Games via Hellinger Distance

Security of cryptographic primitives is usually proved by assuming "ideal" probability distributions. We need to replace them with approximated "real" distributions in the real-world systems without losing the security level. We demonstrate that the Hellinger distance is useful for this problem, while the statistical distance is mainly used in the cryptographic literature. First, we show that for preserving ?-bit security of a given security game, the closeness of 2^{-?/2} to the ideal distribution is sufficient for the Hellinger distance, whereas 2^{-?} is generally required for the statistical distance. The result can be applied to both search and decision primitives through the bit security framework of Micciancio and Walter (Eurocrypt 2018). We also show that the Hellinger distance gives a tighter evaluation of closeness than the max-log distance when the distance is small. Finally, we show that the leftover hash lemma can be strengthened to the Hellinger distance. Namely, a universal family of hash functions gives a strong randomness extractor with optimal entropy loss for the Hellinger distance. Based on the results, a ?-bit entropy loss in randomness extractors is sufficient for preserving ?-bit security. The current understanding based on the statistical distance is that a 2?-bit entropy loss is necessary

On Notions of Security for Deterministic Encryption, and Efficient Constructions Without Random Oracles

The study of deterministic public-key encryption was initiated by Bellare et al. (CRYPTO ’07), who provided the “strongest possible” notion of security for this primitive (called PRIV) and constructions in the random oracle (RO) model. We focus on constructing efficient deterministic encryption schemes without random oracles. To do so, we propose a slightly weaker notion of security, saying that no partial information about encrypted messages should be leaked as long as each message is a-priori hard-to-guess given the others (while PRIV did not have the latter restriction). Nevertheless, we argue that this version seems adequate for many practical applications. We show equivalence of this definition to single-message and indistinguishability-based ones, which are easier to work with. Then we give general constructions of both chosen-plaintext (CPA) and chosen-ciphertext-attack (CCA) secure deterministic encryption schemes, as well as efficient instantiations of them under standard number-theoretic assumptions. Our constructions build on the recently-introduced framework of Peikert and Waters (STOC ’08) for constructing CCA-secure probabilistic encryption schemes, extending it to the deterministic-encryption setting as well

LNCS

We revisit the classical problem of converting an imperfect source of randomness into a usable cryptographic key. Assume that we have some cryptographic application P that expects a uniformly random m-bit key R and ensures that the best attack (in some complexity class) against P(R) has success probability at most δ. Our goal is to design a key-derivation function (KDF) h that converts any random source X of min-entropy k into a sufficiently "good" key h(X), guaranteeing that P(h(X)) has comparable security δ′ which is 'close' to δ. Seeded randomness extractors provide a generic way to solve this problem for all applications P, with resulting security δ′ = O(δ), provided that we start with entropy k ≥ m + 2 log (1/δ) - O(1). By a result of Radhakrishnan and Ta-Shma, this bound on k (called the "RT-bound") is also known to be tight in general. Unfortunately, in many situations the loss of 2 log (1/δ) bits of entropy is unacceptable. This motivates the study KDFs with less entropy waste by placing some restrictions on the source X or the application P. In this work we obtain the following new positive and negative results in this regard: - Efficient samplability of the source X does not help beat the RT-bound for general applications. This resolves the SRT (samplable RT) conjecture of Dachman-Soled et al. [DGKM12] in the affirmative, and also shows that the existence of computationally-secure extractors beating the RT-bound implies the existence of one-way functions. - We continue in the line of work initiated by Barak et al. [BDK+11] and construct new information-theoretic KDFs which beat the RT-bound for large but restricted classes of applications. Specifically, we design efficient KDFs that work for all unpredictability applications P (e.g., signatures, MACs, one-way functions, etc.) and can either: (1) extract all of the entropy k = m with a very modest security loss δ′ = O(δ·log (1/δ)), or alternatively, (2) achieve essentially optimal security δ′ = O(δ) with a very modest entropy loss k ≥ m + loglog (1/δ). In comparison, the best prior results from [BDK+11] for this class of applications would only guarantee δ′ = O(√δ) when k = m, and would need k ≥ m + log (1/δ) to get δ′ = O(δ). - The weaker bounds of [BDK+11] hold for a larger class of so-called "square- friendly" applications (which includes all unpredictability, but also some important indistinguishability, applications). Unfortunately, we show that these weaker bounds are tight for the larger class of applications. - We abstract out a clean, information-theoretic notion of (k,δ,δ′)- unpredictability extractors, which guarantee "induced" security δ′ for any δ-secure unpredictability application P, and characterize the parameters achievable for such unpredictability extractors. Of independent interest, we also relate this notion to the previously-known notion of (min-entropy) condensers, and improve the state-of-the-art parameters for such condensers

Linear Hashing with ℓ∞\ell_\infty guarantees and two-sided Kakeya bounds

We show that a randomly chosen linear map over a finite field gives a good hash function in the $\ell_\infty$ sense. More concretely, consider a set $S \subset \mathbb{F}_q^n$ and a randomly chosen linear map $L : \mathbb{F}_q^n \to \mathbb{F}_q^t$ with $q^t$ taken to be sufficiently smaller than $|S|$. Let $U_S$ denote a random variable distributed uniformly on $S$. Our main theorem shows that, with high probability over the choice of $L$, the random variable $L(U_S)$ is close to uniform in the $\ell_\infty$ norm. In other words, every element in the range $\mathbb{F}_q^t$ has about the same number of elements in $S$ mapped to it. This complements the widely-used Leftover Hash Lemma (LHL) which proves the analog statement under the statistical, or $\ell_1$, distance (for a richer class of functions) as well as prior work on the expected largest 'bucket size' in linear hash functions [ADMPT99]. Our proof leverages a connection between linear hashing and the finite field Kakeya problem and extends some of the tools developed in this area, in particular the polynomial method

The problem with the SURF scheme

There is a serious problem with one of the assumptions made in the security proof of the SURF scheme. This problem turns out to be easy in the regime of parameters needed for the SURF scheme to work. We give afterwards the old version of the paper for the reader's convenience.Comment: Warning : we found a serious problem in the security proof of the SURF scheme. We explain this problem here and give the old version of the paper afterward

Randomness Condensers for Efficiently Samplable, Seed-Dependent Sources

We initiate a study of randomness condensers for sources that are efficiently samplable but may depend on the seed of the con- denser. That is, we seek functions Cond : {0, 1}n ×{0, 1}d → {0, 1}m such that if we choose a random seed S ← {0,1}d, and a source X = A(S) is generated by a randomized circuit A of size t such that X has min- entropy at least k given S, then Cond(X;S) should have min-entropy at least some k′ given S. The distinction from the standard notion of ran- domness condensers is that the source X may be correlated with the seed S (but is restricted to be efficiently samplable). Randomness extractors of this type (corresponding to the special case where k′ = m) have been implicitly studied in the past (by Trevisan and Vadhan, FOCS ‘00). We show that: – Unlike extractors, we can have randomness condensers for samplable, seed-dependent sources whose computational complexity is smaller than the size t of the adversarial sampling algorithm A. Indeed, we show that sufficiently strong collision-resistant hash functions are seed-dependent condensers that produce outputs with min-entropy k′ = m − O(log t), i.e. logarithmic entropy deficiency. – Randomness condensers suffice for key derivation in many crypto- graphic applications: when an adversary has negligible success proba- bility (or negligible “squared advantage” [3]) for a uniformly random key, we can use instead a key generated by a condenser whose output has logarithmic entropy deficiency. – Randomness condensers for seed-dependent samplable sources that are robust to side information generated by the sampling algorithm imply soundness of the Fiat-Shamir Heuristic when applied to any constant-round, public-coin interactive proof system.Engineering and Applied Science

LNCS

HMAC and its variant NMAC are the most popular approaches to deriving a MAC (and more generally, a PRF) from a cryptographic hash function. Despite nearly two decades of research, their exact security still remains far from understood in many different contexts. Indeed, recent works have re-surfaced interest for {\em generic} attacks, i.e., attacks that treat the compression function of the underlying hash function as a black box. Generic security can be proved in a model where the underlying compression function is modeled as a random function -- yet, to date, the question of proving tight, non-trivial bounds on the generic security of HMAC/NMAC even as a PRF remains a challenging open question. In this paper, we ask the question of whether a small modification to HMAC and NMAC can allow us to exactly characterize the security of the resulting constructions, while only incurring little penalty with respect to efficiency. To this end, we present simple variants of NMAC and HMAC, for which we prove tight bounds on the generic PRF security, expressed in terms of numbers of construction and compression function queries necessary to break the construction. All of our constructions are obtained via a (near) {\em black-box} modification of NMAC and HMAC, which can be interpreted as an initial step of key-dependent message pre-processing. While our focus is on PRF security, a further attractive feature of our new constructions is that they clearly defeat all recent generic attacks against properties such as state recovery and universal forgery. These exploit properties of the so-called ``functional graph'' which are not directly accessible in our new constructions