@phdthesis{7896,
abstract = {A search problem lies in the complexity class FNP if a solution to the given instance of the problem can be verified efficiently. The complexity class TFNP consists of all search problems in FNP that are total in the sense that a solution is guaranteed to exist. TFNP contains a host of interesting problems from fields such as algorithmic game theory, computational topology, number theory and combinatorics. Since TFNP is a semantic class, it is unlikely to have a complete problem. Instead, one studies its syntactic subclasses which are defined based on the combinatorial principle used to argue totality. Of particular interest is the subclass PPAD, which contains important problems
like computing Nash equilibrium for bimatrix games and computational counterparts of several fixed-point theorems as complete. In the thesis, we undertake the study of averagecase hardness of TFNP, and in particular its subclass PPAD.
Almost nothing was known about average-case hardness of PPAD before a series of recent results showed how to achieve it using a cryptographic primitive called program obfuscation.
However, it is currently not known how to construct program obfuscation from standard cryptographic assumptions. Therefore, it is desirable to relax the assumption under which average-case hardness of PPAD can be shown. In the thesis we take a step in this direction. First, we show that assuming the (average-case) hardness of a numbertheoretic
problem related to factoring of integers, which we call Iterated-Squaring, PPAD is hard-on-average in the random-oracle model. Then we strengthen this result to show that the average-case hardness of PPAD reduces to the (adaptive) soundness of the Fiat-Shamir Transform, a well-known technique used to compile a public-coin interactive protocol into a non-interactive one. As a corollary, we obtain average-case hardness for PPAD in the random-oracle model assuming the worst-case hardness of #SAT. Moreover, the above results can all be strengthened to obtain average-case hardness for the class CLS ⊆ PPAD.
Our main technical contribution is constructing incrementally-verifiable procedures for computing Iterated-Squaring and #SAT. By incrementally-verifiable, we mean that every intermediate state of the computation includes a proof of its correctness, and the proof can be updated and verified in polynomial time. Previous constructions of such procedures relied on strong, non-standard assumptions. Instead, we introduce a technique called recursive proof-merging to obtain the same from weaker assumptions. },
author = {Kamath Hosdurg, Chethan},
issn = {2663-337X},
pages = {126},
publisher = {IST Austria},
title = {{On the average-case hardness of total search problems}},
doi = {10.15479/AT:ISTA:7896},
year = {2020},
}
@inproceedings{7966,
abstract = {For 1≤m≤n, we consider a natural m-out-of-n multi-instance scenario for a public-key encryption (PKE) scheme. An adversary, given n independent instances of PKE, wins if he breaks at least m out of the n instances. In this work, we are interested in the scaling factor of PKE schemes, SF, which measures how well the difficulty of breaking m out of the n instances scales in m. That is, a scaling factor SF=ℓ indicates that breaking m out of n instances is at least ℓ times more difficult than breaking one single instance. A PKE scheme with small scaling factor hence provides an ideal target for mass surveillance. In fact, the Logjam attack (CCS 2015) implicitly exploited, among other things, an almost constant scaling factor of ElGamal over finite fields (with shared group parameters).
For Hashed ElGamal over elliptic curves, we use the generic group model to argue that the scaling factor depends on the scheme's granularity. In low granularity, meaning each public key contains its independent group parameter, the scheme has optimal scaling factor SF=m; In medium and high granularity, meaning all public keys share the same group parameter, the scheme still has a reasonable scaling factor SF=√m. Our findings underline that instantiating ElGamal over elliptic curves should be preferred to finite fields in a multi-instance scenario.
As our main technical contribution, we derive new generic-group lower bounds of Ω(√(mp)) on the difficulty of solving both the m-out-of-n Gap Discrete Logarithm and the m-out-of-n Gap Computational Diffie-Hellman problem over groups of prime order p, extending a recent result by Yun (EUROCRYPT 2015). We establish the lower bound by studying the hardness of a related computational problem which we call the search-by-hypersurface problem.},
author = {Auerbach, Benedikt and Giacon, Federico and Kiltz, Eike},
booktitle = {Advances in Cryptology – EUROCRYPT 2020},
isbn = {9783030457266},
issn = {0302-9743},
pages = {475--506},
publisher = {Springer Nature},
title = {{Everybody’s a target: Scalability in public-key encryption}},
doi = {10.1007/978-3-030-45727-3_16},
volume = {12107},
year = {2020},
}
@article{5887,
abstract = {Cryptographic security is usually defined as a guarantee that holds except when a bad event with negligible probability occurs, and nothing is guaranteed in that bad case. However, in settings where such failure can happen with substantial probability, one needs to provide guarantees even for the bad case. A typical example is where a (possibly weak) password is used instead of a secure cryptographic key to protect a session, the bad event being that the adversary correctly guesses the password. In a situation with multiple such sessions, a per-session guarantee is desired: any session for which the password has not been guessed remains secure, independently of whether other sessions have been compromised. A new formalism for stating such gracefully degrading security guarantees is introduced and applied to analyze the examples of password-based message authentication and password-based encryption. While a natural per-message guarantee is achieved for authentication, the situation of password-based encryption is more delicate: a per-session confidentiality guarantee only holds against attackers for which the distribution of password-guessing effort over the sessions is known in advance. In contrast, for more general attackers without such a restriction, a strong, composable notion of security cannot be achieved.},
author = {Demay, Gregory and Gazi, Peter and Maurer, Ueli and Tackmann, Bjorn},
issn = {0926227X},
journal = {Journal of Computer Security},
number = {1},
pages = {75--111},
publisher = {IOS Press},
title = {{Per-session security: Password-based cryptography revisited}},
doi = {10.3233/JCS-181131},
volume = {27},
year = {2019},
}
@inproceedings{6430,
abstract = {A proxy re-encryption (PRE) scheme is a public-key encryption scheme that allows the holder of a key pk to derive a re-encryption key for any other key 𝑝𝑘′. This re-encryption key lets anyone transform ciphertexts under pk into ciphertexts under 𝑝𝑘′ without having to know the underlying message, while transformations from 𝑝𝑘′ to pk should not be possible (unidirectional). Security is defined in a multi-user setting against an adversary that gets the users’ public keys and can ask for re-encryption keys and can corrupt users by requesting their secret keys. Any ciphertext that the adversary cannot trivially decrypt given the obtained secret and re-encryption keys should be secure.
All existing security proofs for PRE only show selective security, where the adversary must first declare the users it wants to corrupt. This can be lifted to more meaningful adaptive security by guessing the set of corrupted users among the n users, which loses a factor exponential in Open image in new window , rendering the result meaningless already for moderate Open image in new window .
Jafargholi et al. (CRYPTO’17) proposed a framework that in some cases allows to give adaptive security proofs for schemes which were previously only known to be selectively secure, while avoiding the exponential loss that results from guessing the adaptive choices made by an adversary. We apply their framework to PREs that satisfy some natural additional properties. Concretely, we give a more fine-grained reduction for several unidirectional PREs, proving adaptive security at a much smaller loss. The loss depends on the graph of users whose edges represent the re-encryption keys queried by the adversary. For trees and chains the loss is quasi-polynomial in the size and for general graphs it is exponential in their depth and indegree (instead of their size as for previous reductions). Fortunately, trees and low-depth graphs cover many, if not most, interesting applications.
Our results apply e.g. to the bilinear-map based PRE schemes by Ateniese et al. (NDSS’05 and CT-RSA’09), Gentry’s FHE-based scheme (STOC’09) and the LWE-based scheme by Chandran et al. (PKC’14).},
author = {Fuchsbauer, Georg and Kamath Hosdurg, Chethan and Klein, Karen and Pietrzak, Krzysztof Z},
isbn = {9783030172589},
issn = {16113349},
location = {Beijing, China},
pages = {317--346},
publisher = {Springer Nature},
title = {{Adaptively secure proxy re-encryption}},
doi = {10.1007/978-3-030-17259-6_11},
volume = {11443},
year = {2019},
}
@inproceedings{6528,
abstract = {We construct a verifiable delay function (VDF) by showing how the Rivest-Shamir-Wagner time-lock puzzle can be made publicly verifiable. Concretely, we give a statistically sound public-coin protocol to prove that a tuple (N,x,T,y) satisfies y=x2T (mod N) where the prover doesn’t know the factorization of N and its running time is dominated by solving the puzzle, that is, compute x2T, which is conjectured to require T sequential squarings. To get a VDF we make this protocol non-interactive using the Fiat-Shamir heuristic.The motivation for this work comes from the Chia blockchain design, which uses a VDF as akey ingredient. For typical parameters (T≤2 40, N= 2048), our proofs are of size around 10K B, verification cost around three RSA exponentiations and computing the proof is 8000 times faster than solving the puzzle even without any parallelism.},
author = {Pietrzak, Krzysztof Z},
booktitle = {10th Innovations in Theoretical Computer Science Conference},
isbn = {978-3-95977-095-8},
issn = {1868-8969},
location = {San Diego, CA, United States},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Simple verifiable delay functions}},
doi = {10.4230/LIPICS.ITCS.2019.60},
volume = {124},
year = {2019},
}
@inproceedings{6677,
abstract = {The Fiat-Shamir heuristic transforms a public-coin interactive proof into a non-interactive argument, by replacing the verifier with a cryptographic hash function that is applied to the protocol’s transcript. Constructing hash functions for which this transformation is sound is a central and long-standing open question in cryptography.
We show that solving the END−OF−METERED−LINE problem is no easier than breaking the soundness of the Fiat-Shamir transformation when applied to the sumcheck protocol. In particular, if the transformed protocol is sound, then any hard problem in #P gives rise to a hard distribution in the class CLS, which is contained in PPAD. Our result opens up the possibility of sampling moderately-sized games for which it is hard to find a Nash equilibrium, by reducing the inversion of appropriately chosen one-way functions to #SAT.
Our main technical contribution is a stateful incrementally verifiable procedure that, given a SAT instance over n variables, counts the number of satisfying assignments. This is accomplished via an exponential sequence of small steps, each computable in time poly(n). Incremental verifiability means that each intermediate state includes a sumcheck-based proof of its correctness, and the proof can be updated and verified in time poly(n).},
author = {Choudhuri, Arka Rai and Hubáček, Pavel and Kamath Hosdurg, Chethan and Pietrzak, Krzysztof Z and Rosen, Alon and Rothblum, Guy N.},
booktitle = {Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing - STOC 2019},
isbn = {9781450367059},
location = {Phoenix, AZ, United States},
pages = {1103--1114},
publisher = {ACM Press},
title = {{Finding a Nash equilibrium is no easier than breaking Fiat-Shamir}},
doi = {10.1145/3313276.3316400},
year = {2019},
}
@inbook{6726,
abstract = {Randomness is an essential part of any secure cryptosystem, but many constructions rely on distributions that are not uniform. This is particularly true for lattice based cryptosystems, which more often than not make use of discrete Gaussian distributions over the integers. For practical purposes it is crucial to evaluate the impact that approximation errors have on the security of a scheme to provide the best possible trade-off between security and performance. Recent years have seen surprising results allowing to use relatively low precision while maintaining high levels of security. A key insight in these results is that sampling a distribution with low relative error can provide very strong security guarantees. Since floating point numbers provide guarantees on the relative approximation error, they seem a suitable tool in this setting, but it is not obvious which sampling algorithms can actually profit from them. While previous works have shown that inversion sampling can be adapted to provide a low relative error (Pöppelmann et al., CHES 2014; Prest, ASIACRYPT 2017), other works have called into question if this is possible for other sampling techniques (Zheng et al., Eprint report 2018/309). In this work, we consider all sampling algorithms that are popular in the cryptographic setting and analyze the relationship of floating point precision and the resulting relative error. We show that all of the algorithms either natively achieve a low relative error or can be adapted to do so.},
author = {Walter, Michael},
booktitle = {Progress in Cryptology – AFRICACRYPT 2019},
editor = {Buchmann, J and Nitaj, A and Rachidi, T},
isbn = {9783030236953},
issn = {0302-9743},
location = {Rabat, Morocco},
pages = {157--180},
publisher = {Springer Nature},
title = {{Sampling the integers with low relative error}},
doi = {10.1007/978-3-030-23696-0_9},
volume = {11627},
year = {2019},
}
@inproceedings{7136,
abstract = {It is well established that the notion of min-entropy fails to satisfy the \emph{chain rule} of the form H(X,Y)=H(X|Y)+H(Y), known for Shannon Entropy. Such a property would help to analyze how min-entropy is split among smaller blocks. Problems of this kind arise for example when constructing extractors and dispersers.
We show that any sequence of variables exhibits a very strong strong block-source structure (conditional distributions of blocks are nearly flat) when we \emph{spoil few correlated bits}. This implies, conditioned on the spoiled bits, that \emph{splitting-recombination properties} hold. In particular, we have many nice properties that min-entropy doesn't obey in general, for example strong chain rules, "information can't hurt" inequalities, equivalences of average and worst-case conditional entropy definitions and others. Quantitatively, for any sequence X1,…,Xt of random variables over an alphabet X we prove that, when conditioned on m=t⋅O(loglog|X|+loglog(1/ϵ)+logt) bits of auxiliary information, all conditional distributions of the form Xi|X