# Cryptography

Haitner, Iftach, Omer Reingold, Salil Vadhan, and Hoeteck Wee. “Inaccessible entropy I: Inaccessible entropy generators and statistically hiding commitments from one-way functions.” arXiv: 2010.05586 [cs.CR] (2020). ArXiv VersionAbstract

Version History: Full version of part of an STOC 2009 paper.

We put forth a new computational notion of entropy, measuring the (in)feasibility of sampling high-entropy strings that are consistent with a given generator. Specifically, the $$i$$'th output block of a generator $$\mathsf{G}$$ has accessible entropy at most $$k$$ if the following holds: when conditioning on its prior coin tosses, no polynomial-time strategy $$\mathsf{\widetilde{G}}$$ can generate valid output for $$\mathsf{G}$$'s $$i$$'th output block with entropy greater than $$k$$. A generator has inaccessible entropy if the total accessible entropy (summed over the blocks) is noticeably smaller than the real entropy of $$\mathsf{G}$$'s output.

As an application of the above notion, we improve upon the result of Haitner, Nguyen, Ong, Reingold, and Vadhan [Sicomp '09], presenting a much simpler and more efficient construction of statistically hiding commitment schemes from arbitrary one-way functions.

Haitner, Iftach, Thomas Holenstein, Omer Reingold, Salil Vadhan, and Hoeteck Wee. “Inaccessible entropy II: IE functions and universal one-way hashing.” Theory of Computing 16, no. 8 (2020): 1-55. Publisher's VersionAbstract

Version History: published earlier in Henri Gilbert, ed., Advances in Cryptology—EUROCRYPT ‘10, Lecture Notes on Computer Science, as "Universal one-way hash functions via inaccessible entropy":

This paper revisits the construction of Universal One-Way Hash Functions (UOWHFs) from any one-way function due to Rompel (STOC 1990). We give a simpler construction of UOWHFs, which also obtains better efficiency and security. The construction exploits a strong connection to the recently introduced notion of inaccessible entropy (Haitner et al. STOC 2009). With this perspective, we observe that a small tweak of any one-way function $$f$$ is already a weak form of a UOWHF: Consider $$F(x', i)$$ that outputs the $$i$$-bit long prefix of $$f(x)$$. If $$F$$ were a UOWHF then given a random $$x$$ and $$i$$ it would be hard to come up with $$x' \neq x$$ such that $$F(x, i) = F(x', i)$$. While this may not be the case, we show (rather easily) that it is hard to sample $$x'$$ with almost full entropy among all the possible such values of $$x'$$. The rest of our construction simply amplifies and exploits this basic property.

With this and other recent works, we have that the constructions of three fundamental cryptographic primitives (Pseudorandom Generators, Statistically Hiding Commitments and UOWHFs) out of one-way functions are to a large extent unified. In particular, all three constructions rely on and manipulate computational notions of entropy in similar ways. Pseudorandom Generators rely on the well-established notion of pseudoentropy, whereas Statistically Hiding Commitments and UOWHFs rely on the newer notion of inaccessible entropy.

Vadhan, Salil. “Computational entropy.” In Providing Sound Foundations for Cryptography: On the Work of Shafi Goldwasser and Silvio Micali (Oded Goldreich, Ed.), 693-726. ACM, 2019. Publisher's VersionAbstract

In this survey, we present several computational analogues of entropy and illustrate how they are useful for constructing cryptographic primitives. Specifically, we focus on constructing pseudorandom generators and statistically hiding commitments from arbitrary one-way functions, and demonstrate that:

1. The security properties of these (and other) cryptographic primitives can be understood in terms of various computational analogues of entropy, and in particular how these computational measures of entropy can be very different from real, information-theoretic entropy.
2. It can be shown that every one-way function directly exhibits some gaps between real entropy and the various computational entropies.
3. Thus we can construct the desired cryptographic primitives by amplifying and manipulating the entropy gaps in a one-way function, through forms of repetition and hashing.

The constructions we present (which are from the past decade) are much simpler and more efficient than the original ones, and are based entirely on natural manipulations of new notions of computational entropy. The two constructions are "dual" to each other, whereby the construction of pseudorandom generators relies on a form of computational entropy ("pseudoentropy") being larger than the real entropy, while the construction of statistically hiding commitments relies on a form of computational entropy ("accessible entropy") being smaller than the real entropy. Beyond that difference, the two constructions share a common structure, using a very similar sequence of manipulations of real and computational entropy. As a warmup, we also "deconstruct" the classic construction of pseudorandom generators from one-way permutations using the modern language of computational entropy.

This survey is written in honor of Shafi Goldwasser and Silvio Micali.

Agrawal, Rohit, Yi-Hsiu Chen, Thibaut Horel, and Salil Vadhan. “Unifying computational entropies via Kullback-Leibler divergence.” In Advances in Cryptology: CRYPTO 2019, A. Boldyreva and D. Micciancio, (Eds), 11693:831-858. Springer Verlag, Lecture Notes in Computer Science, 2019. Publisher's VersionAbstract
Version History:
arXiv, first posted Feb 2019, most recently updated Aug 2019: https://arxiv.org/abs/1902.11202

We introduce hardness in relative entropy, a new notion of hardness for search problems which on the one hand is satisfied by all one-way functions and on the other hand implies both next-block pseudoentropy and inaccessible entropy, two forms of computational entropy used in recent constructions of pseudorandom generators and statistically hiding commitment schemes, respectively. Thus, hardness in relative entropy unifies the latter two notions of computational entropy and sheds light on the apparent “duality” between them. Additionally, it yields a more modular and illuminating proof that one-way functions imply next-block inaccessible entropy, similar in structure to the proof that one-way functions imply next-block pseudoentropy (Vadhan and Zheng, STOC ‘12).
Raghunathan, Ananth, Gil Segev, and Salil P. Vadhan. “Deterministic public-key encryption for adaptively-chosen plaintext distributions.” Journal of Cryptology 31, no. 4 (2018): 1012-1063. Publisher's VersionAbstract

Version History: Preliminary versions in EUROCRYPT ‘13 and Cryptology ePrint report 2013/125.

Bellare, Boldyreva, and O’Neill (CRYPTO '07) initiated the study of deterministic public-key encryption as an alternative in scenarios where randomized encryption has inherent drawbacks. The resulting line of research has so far guaranteed security only for adversarially-chosen plaintext distributions that are independent of the public key used by the scheme. In most scenarios, however, it is typically not realistic to assume that adversaries do not take the public key into account when attacking a scheme.

We show that it is possible to guarantee meaningful security even for plaintext distributions that depend on the public key. We extend the previously proposed notions of security, allowing adversaries to adaptively choose plaintext distributions after seeing the public key, in an interactive manner. The only restrictions we make are that: (1) plaintext distributions are unpredictable (as is essential in deterministic public-key encryption), and (2) the number of plaintext distributions from which each adversary is allowed to adaptively choose is upper bounded by $$2^p$$, where $$p$$ can be any predetermined polynomial in the security parameter. For example, with $$p=0$$ we capture plaintext distributions that are independent of the public key, and with $$p=0(s \log s)$$ we capture, in particular, all plaintext distributions that are samplable by circuits of size $$s$$.

Within our framework we present both constructions in the random-oracle model based on any public-key encryption scheme, and constructions in the standard model based on lossy trapdoor functions (thus, based on a variety of number-theoretic assumptions). Previously known constructions heavily relied on the independence between the plaintext distributions and the public key for the purposes of randomness extraction. In our setting, however, randomness extraction becomes significantly more challenging once the plaintext distributions and the public key are no longer independent. Our approach is inspired by research on randomness extraction from seed-dependent distributions. Underlying our approach is a new generalization of a method for such randomness extraction, originally introduced by Trevisan and Vadhan (FOCS '00) and Dodis (PhD Thesis, MIT, '00).

Bun, Mark, Jonathan Ullman, and Salil Vadhan. “Fingerprinting codes and the price of approximate differential privacy.” SIAM Journal on Computing, Special Issue on STOC '14 47, no. 5 (2018): 1888-1938. Publisher's VersionAbstract

Version HistorySpecial Issue on STOC ‘14. Preliminary versions in STOC ‘14 and arXiv:1311.3158 [cs.CR].

We show new information-theoretic lower bounds on the sample complexity of (ε, δ)- differentially private algorithms that accurately answer large sets of counting queries. A counting query on a database $$D ∈ (\{0, 1\}^d)^n$$ has the form “What fraction of the individual records in the database satisfy the property $$q$$?” We show that in order to answer an arbitrary set $$Q$$ of $$\gg d/ \alpha^2$$ counting queries on $$D$$ to within error $$±α$$ it is necessary that $$n ≥ \tilde{Ω}(\sqrt{d} \log |Q|/α^2ε)$$. This bound is optimal up to polylogarithmic factors, as demonstrated by the private multiplicative weights algorithm (Hardt and Rothblum, FOCS’10). In particular, our lower bound is the first to show that the sample complexity required for accuracy and (ε, δ)-differential privacy is asymptotically larger than what is required merely for accuracy, which is $$O(\log |Q|/α^2 )$$. In addition, we show that our lower bound holds for the specific case of $$k$$-way marginal queries (where $$|Q| = 2^k \binom{d}{k}$$ ) when $$\alpha$$ is not too small compared to d (e.g., when $$\alpha$$ is any fixed constant). Our results rely on the existence of short fingerprinting codes (Boneh and Shaw, CRYPTO’95; Tardos, STOC’03), which we show are closely connected to the sample complexity of differentially private data release. We also give a new method for combining certain types of sample-complexity lower bounds into stronger lower bounds.

Chen, Yi-Hsiu, Mika Goos, Salil P. Vadhan, and Jiapeng Zhang. “A tight lower bound for entropy flattening.” In 33rd Computational Complexity Conference (CCC 2018), 102:23:21-23:28. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik: Leibniz International Proceedings in Informatics (LIPIcs), 2018. Publisher's VersionAbstract

Version History: Preliminary version posted as ECCC TR18-119.

We study entropy flattening: Given a circuit $$C_X$$ implicitly describing an n-bit source $$X$$ (namely, $$X$$ is the output of $$C_X$$  on a uniform random input), construct another circuit $$C_Y$$ describing a source $$Y$$ such that (1) source $$Y$$ is nearly flat (uniform on its support), and (2) the Shannon entropy of $$Y$$ is monotonically related to that of $$X$$. The standard solution is to have $$C_Y$$ evaluate $$C_X$$ altogether $$\Theta(n^2)$$ times on independent inputs and concatenate the results (correctness follows from the asymptotic equipartition property). In this paper, we show that this is optimal among black-box constructions: Any circuit $$C_Y$$ for entropy flattening that repeatedly queries $$C_X$$ as an oracle requires $$\Omega(n^2)$$queries.

Entropy flattening is a component used in the constructions of pseudorandom generators and other cryptographic primitives from one-way functions [12, 22, 13, 6, 11, 10, 7, 24]. It is also used in reductions between problems complete for statistical zero-knowledge [19, 23, 4, 25]. The $$\Theta(n^2)$$ query complexity is often the main efficiency bottleneck. Our lower bound can be viewed as a step towards proving that the current best construction of pseudorandom generator from arbitrary one-way functions by Vadhan and Zheng (STOC 2012) has optimal efficiency.

Haitner, Iftach, and Salil Vadhan. “The Many Entropies in One-way Functions.” In Tutorials on the Foundations of Cryptography, 159-217. Springer, Yehuda Lindell, ed. 2017. Publisher's VersionAbstract

Version History:

Earlier versions: May 2017: ECCC TR 17-084

Dec. 2017: ECCC TR 17-084 (revised)

Computational analogues of information-theoretic notions have given rise to some of the most interesting phenomena in the theory of computation. For example, computational indistinguishability, Goldwasser and Micali [9], which is the computational analogue of statistical distance, enabled the bypassing of Shannon’s impossibility results on perfectly secure encryption, and provided the basis for the computational theory of pseudorandomness. Pseudoentropy, Håstad, Impagliazzo, Levin, and Luby [17], a computational analogue of entropy, was the key to the fundamental result establishing the equivalence of pseudorandom generators and one-way functions, and has become a basic concept in complexity theory and cryptography.

This tutorial discusses two rather recent computational notions of entropy, both of which can be easily found in any one-way function, the most basic cryptographic primitive. The first notion is next-block pseudoentropy, Haitner, Reingold, and Vadhan [14], a refinement of pseudoentropy that enables simpler and more ecient construction of pseudorandom generators. The second is inaccessible entropy, Haitner, Reingold, Vadhan, andWee [11], which relates to unforgeability and is used to construct simpler and more efficient universal one-way hash functions and statistically hiding commitments.

Vadhan, Salil. “The Complexity of Differential Privacy.” In Tutorials on the Foundations of Cryptography, 347-450. Springer, Yehuda Lindell, ed. 2017. Publisher's VersionAbstract

Version History:

August 2016: Manuscript v1 (see files attached)

March 2017: Manuscript v2 (see files attached); Errata

April 2017: Published Version (in Tutorials on the Foundations of Cryptography; see Publisher's Version link and also SPRINGER 2017.PDF, below)

Differential privacy is a theoretical framework for ensuring the privacy of individual-level data when performing statistical analysis of privacy-sensitive datasets. This tutorial provides an introduction to and overview of differential privacy, with the goal of conveying its deep connections to a variety of other topics in computational complexity, cryptography, and theoretical computer science at large. This tutorial is written in celebration of Oded Goldreich’s 60th birthday, starting from notes taken during a minicourse given by the author and Kunal Talwar at the 26th McGill Invitational Workshop on Computational Complexity [1].

Chen, Yi-Hsiu, Kai-Min Chung, Ching-Yi Lai, Salil P. Vadhan, and Xiaodi Wu.Computational notions of quantum min-entropy.” In Poster presention at QIP 2017 and oral presentation at QCrypt 2017, 2017. Publisher's VersionAbstract

Version History

ArXiv v1, 24 April 2017 https://arxiv.org/abs/1704.07309v1
ArXiv v2, 25 April 2017 https://arxiv.org/abs/1704.07309v2
ArXiv v3, 9 September 2017 https://arxiv.org/abs/1704.07309v3
ArXiv v4, 5 October 2017 https://arxiv.org/abs/1704.07309v4

We initiate the study of computational entropy in the quantum setting. We investigate to what extent the classical notions of computational entropy generalize to the quantum setting, and whether quantum analogues of classical theorems hold. Our main results are as follows. (1) The classical Leakage Chain Rule for pseudoentropy can be extended to the case that the leakage information is quantum (while the source remains classical). Specifically, if the source has pseudoentropy at least $$k$$, then it has pseudoentropy at least $$k−ℓ$$ conditioned on an $$ℓ$$-qubit leakage. (2) As an application of the Leakage Chain Rule, we construct the first quantum leakage-resilient stream-cipher in the bounded-quantum-storage model, assuming the existence of a quantum-secure pseudorandom generator. (3) We show that the general form of the classical Dense Model Theorem (interpreted as the equivalence between two definitions of pseudo-relative-min-entropy) does not extend to quantum states. Along the way, we develop quantum analogues of some classical techniques (e.g. the Leakage Simulation Lemma, which is proven by a Non-uniform Min-Max Theorem or Boosting). On the other hand, we also identify some classical techniques (e.g. Gap Amplification) that do not work in the quantum setting. Moreover, we introduce a variety of notions that combine quantum information and quantum complexity, and this raises several directions for future work.

Bun, Mark, Yi-Hsiu Chen, and Salil Vadhan. “Separating computational and statistical differential privacy in the client-server model.” In Martin Hirt and Adam D. Smith, editors, Proceedings of the 14th IACR Theory of Cryptography Conference (TCC `16-B). Lecture Notes in Computer Science. Springer Verlag, 31 October-3 November, 2016. Publisher's VersionAbstract

Version History: Full version posted on Cryptology ePrint Archive, Report 2016/820.

Differential privacy is a mathematical definition of privacy for statistical data analysis. It guarantees that any (possibly adversarial) data analyst is unable to learn too much information that is specific to an individual. Mironov et al. (CRYPTO 2009) proposed several computa- tional relaxations of differential privacy (CDP), which relax this guarantee to hold only against computationally bounded adversaries. Their work and subsequent work showed that CDP can yield substantial accuracy improvements in various multiparty privacy problems. However, these works left open whether such improvements are possible in the traditional client-server model of data analysis. In fact, Groce, Katz and Yerukhimovich (TCC 2011) showed that, in this setting, it is impossible to take advantage of CDP for many natural statistical tasks.

Our main result shows that, assuming the existence of sub-exponentially secure one-way functions and 2-message witness indistinguishable proofs (zaps) for NP, that there is in fact a computational task in the client-server model that can be efficiently performed with CDP, but is infeasible to perform with information-theoretic differential privacy.

Haitner, Iftach, Omer Reingold, and Salil Vadhan. “Efficiency improvements in constructing pseudorandom generators from one-way functions.” SIAM Journal on Computing 42, no. 3 (2013): 1405-1430. Publisher's VersionAbstract

Version HistorySpecial Issue on STOC ‘10.

We give a new construction of pseudorandom generators from any one-way function. The construction achieves better parameters and is simpler than that given in the seminal work of Håstad, Impagliazzo, Levin, and Luby [SICOMP ’99]. The key to our construction is a new notion of next-block pseudoentropy, which is inspired by the notion of “in-accessible entropy” recently introduced in [Haitner, Reingold, Vadhan, and Wee, STOC ’09]. An additional advan- tage over previous constructions is that our pseudorandom generators are parallelizable and invoke the one-way function in a non-adaptive manner. Using [Applebaum, Ishai, and Kushilevitz, SICOMP ’06], this implies the existence of pseudorandom generators in NC$$^0$$ based on the existence of one-way functions in NC$$^1$$.

Mahmoody, Mohammad, Tal Moran, and Salil Vadhan. “Publicly verifiable proofs of sequential work.” In Innovations in Theoretical Computer Science (ITCS ‘13), 373-388. ACM, 2013. Publisher's VersionAbstract

Version HistoryPreliminary version posted as Cryptology ePrint Archive Report 2011/553, under title “Non-Interactive Time-Stamping and Proofs of Work in the Random Oracle Model”.

We construct a publicly verifiable protocol for proving computational work based on collision- resistant hash functions and a new plausible complexity assumption regarding the existence of “inherently sequential” hash functions. Our protocol is based on a novel construction of time-lock puzzles. Given a sampled “puzzle” $$\mathcal{P} \overset{}\gets \mathbf{D}_n$$, where $$n$$ is the security parameter and $$\mathbf{D}_n$$ is the distribution of the puzzles, a corresponding “solution” can be generated using $$N$$ evaluations of the sequential hash function, where $$N > n$$ is another parameter, while any feasible adversarial strategy for generating valid solutions must take at least as much time as $$\Omega(N)$$ sequential evaluations of the hash function after receiving $$\mathcal{P}$$. Thus, valid solutions constitute a “proof” that $$\Omega(N)$$ parallel time elapsed since $$\mathcal{P}$$ was received. Solutions can be publicly and efficiently verified in time $$\mathrm{poly}(n) \cdot \mathrm{polylog}(N)$$. Applications of these “time-lock puzzles” include noninteractive timestamping of documents (when the distribution over the possible documents corresponds to the puzzle distribution $$\mathbf{D}_n$$) and universally verifiable CPU benchmarks.

Our construction is secure in the standard model under complexity assumptions (collision- resistant hash functions and inherently sequential hash functions), and makes black-box use of the underlying primitives. Consequently, the corresponding construction in the random oracle model is secure unconditionally. Moreover, as it is a public-coin protocol, it can be made non- interactive in the random oracle model using the Fiat-Shamir Heuristic.

Our construction makes a novel use of “depth-robust” directed acyclic graphs—ones whose depth remains large even after removing a constant fraction of vertices—which were previously studied for the purpose of complexity lower bounds. The construction bypasses a recent negative result of Mahmoody, Moran, and Vadhan (CRYPTO ‘11) for time-lock puzzles in the random oracle model, which showed that it is impossible to have time-lock puzzles like ours in the random oracle model if the puzzle generator also computes a solution together with the puzzle.

Vadhan, Salil, and Colin Jia Zheng. “A uniform min-max theorem with applications in cryptography.” In Ran Canetti and Juan Garay, editors, Advances in Cryptology—CRYPTO ‘13, Lecture Notes on Computer Science, 8042:93-110. Springer Verlag, Lecture Notes in Computer Science, 2013. Publisher's VersionAbstract
Version History:
Full version published on ECCC2013 and IACR ePrint 2013.

We present a new, more constructive proof of von Neumann’s Min-Max Theorem for two-player zero-sum game — specifically, an algorithm that builds a near-optimal mixed strategy for the second player from several best-responses of the second player to mixed strategies of the first player. The algorithm extends previous work of Freund and Schapire (Games and Economic Behavior ’99) with the advantage that the algorithm runs in poly$$(n)$$ time even when a pure strategy for the first player is a distribution chosen from a set of distributions over $$\{0,1\}^n$$. This extension enables a number of additional applications in cryptography and complexity theory, often yielding uniform security versions of results that were previously only proved for nonuniform security (due to use of the non-constructive Min-Max Theorem).

We describe several applications, including a more modular and improved uniform version of Impagliazzo’s Hardcore Theorem (FOCS ’95), showing impossibility of constructing succinct non-interactive arguments (SNARGs) via black-box reductions under uniform hardness assumptions (using techniques from Gentry and Wichs (STOC ’11) for the nonuniform setting), and efficiently simulating high entropy distributions within any sufficiently nice convex set (extending a result of Trevisan, Tulsiani and Vadhan (CCC ’09)).

Dodis, Yevgeniy, Thomas Ristenpart, and Salil Vadhan. “Randomness condensers for efficiently samplable, seed-dependent sources.” In Ronald Cramer, editor, Proceedings of the 9th IACR Theory of Cryptography Conference (TCC ‘12), Lecture Notes on Computer Science, 7194:618-635. Springer-Verlag, 2012. Publisher's VersionAbstract

We initiate a study of randomness condensers for sources that are efficiently samplable but may depend on the seed of the condenser. That is, we seek functions $$\mathsf{Cond} : \{0,1\}^n \times \{0,1\}^d \to \{0,1\}^m$$such that if we choose a random seed $$S \gets \{0,1\}^d$$, and a source $$X = \mathcal{A}(S)$$ is generated by a randomized circuit $$\mathcal{A}$$ of size $$t$$ such that $$X$$ has min- entropy at least $$k$$ given $$S$$, then $$\mathsf{Cond}(X ; S)$$ should have min-entropy at least some $$k'$$ given $$S$$. The distinction from the standard notion of randomness 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 $$\mathcal{A}$$. Indeed, we show that sufficiently strong collision-resistant hash functions are seed-dependent condensers that produce outputs with min-entropy $$k' = m – \mathcal{O}(\log t)$$, i.e. logarithmic entropy deficiency.

• Randomness condensers suffice for key derivation in many cryptographic applications: when an adversary has negligible success probability (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.