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DOI: 10.4230/LIPIcs.ITCS.2026.15

How to Use Nondeterminism in Cryptography

Authors: Marshall Ball and Peter Crawford-Kahrl

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

Nondeterministic reductions have yielded powerful results in the theory of computational complexity, yet are effectively useless in a cryptographic context. The reason for this is simple, a nondeterministic polynomial time adversary can trivially break almost any cryptographic primitive by simply guessing the "key." In order to use this powerful nondeterministic tool kit in the cryptographic context, we initiate the study of cryptography against adversaries with limited nondeterminism: polynomial time nondeterministic algorithms that are restricted to just a few bits of nondeterminism. We demonstrate that limited nondeterministic security is sufficient to prove two foundational results that have eluded our grasp for decades: dream hardness amplification, and extracting ω(log n) hardcore bits.

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Marshall Ball and Peter Crawford-Kahrl. How to Use Nondeterminism in Cryptography. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 15:1-15:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ball_et_al:LIPIcs.ITCS.2026.15,
  author =	{Ball, Marshall and Crawford-Kahrl, Peter},
  title =	{{How to Use Nondeterminism in Cryptography}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{15:1--15:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.15},
  URN =		{urn:nbn:de:0030-drops-253024},
  doi =		{10.4230/LIPIcs.ITCS.2026.15},
  annote =	{Keywords: limited nondeterminism, cryptography, computational complexity, hardness amplification, pseudorandom generators, hardcore bits}
}

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DOI: 10.4230/LIPIcs.CCC.2025.35

How to Construct Random Strings

Authors: Oliver Korten and Rahul Santhanam

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract

We address the following fundamental question: is there an efficient deterministic algorithm that, given 1ⁿ, outputs a string of length n that has polynomial-time bounded Kolmogorov complexity Ω̃(n) or even n - o(n)? Under plausible complexity-theoretic assumptions, stating for example that there is an ε > 0 for which TIME[T(n)] ̸ ⊆ TIME^NP[T(n)^ε]/2^(εn) for appropriately chosen time-constructible T, we show that the answer to this question is positive (answering a question of [Hanlin Ren et al., 2022]), and that the Range Avoidance problem [Robert Kleinberg et al., 2021; Oliver Korten, 2021; Hanlin Ren et al., 2022] is efficiently solvable for uniform sequences of circuits with close to minimal stretch (answering a question of [Rahul Ilango et al., 2023]). We obtain our results by giving efficient constructions of pseudo-random generators with almost optimal seed length against algorithms with small advice, under assumptions of the form mentioned above. We also apply our results to give the first complexity-theoretic evidence for explicit constructions of objects such as rigid matrices (in the sense of Valiant) and Ramsey graphs with near-optimal parameters.

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Oliver Korten and Rahul Santhanam. How to Construct Random Strings. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 35:1-35:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{korten_et_al:LIPIcs.CCC.2025.35,
  author =	{Korten, Oliver and Santhanam, Rahul},
  title =	{{How to Construct Random Strings}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{35:1--35:32},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.35},
  URN =		{urn:nbn:de:0030-drops-237290},
  doi =		{10.4230/LIPIcs.CCC.2025.35},
  annote =	{Keywords: Explicit Constructions, Kolmogorov Complexity, Derandomization}
}

Document

DOI: 10.4230/LIPIcs.CCC.2016.9

Pseudorandomness When the Odds are Against You

Authors: Sergei Artemenko, Russell Impagliazzo, Valentine Kabanets, and Ronen Shaltiel

Published in: LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

Abstract

Impagliazzo and Wigderson (STOC 1997) showed that if E=DTIME(2^O(n)) requires size 2^Omega(n) circuits, then every time T constant-error randomized algorithm can be simulated deterministically in time poly(T). However, such polynomial slowdown is a deal breaker when T=2^(alpha*n), for a constant alpha>0, as is the case for some randomized algorithms for NP-complete problems. Paturi and Pudlak (STOC 2010) observed that many such algorithms are obtained from randomized time T algorithms, for T < 2^o(n), with large one-sided error 1-epsilon, for epsilon=2^(-alpha*n), that are repeated 1/epsilon times to yield a constant-error randomized algorithm running in time T/epsilon=2^((alpha+o(1))*n). We show that if E requires size 2^Omega(n) nondeterministic circuits, then there is a poly(n)-time epsilon-HSG (Hitting-Set Generator) H:{0,1}^(O(log(n)) + log(1/epsilon) -> {0,1}^n, implying that time T randomized algorithms with one-sided error 1-epsilon can be simulated in deterministic time poly(T)/epsilon. In particular, under this hardness assumption, the fastest known constant-error randomized algorithm for k-SAT (for k > 3) by Paturi et al. (J. ACM 2005) can be made deterministic with essentially the same time bound. This is the first hardness versus randomness tradeoff for algorithms for NP-complete problems. We address the necessity of our assumption by showing that HSGs with very low error imply hardness for nondeterministic circuits with "few" nondeterministic bits. Applebaum et al. (CCC 2015) showed that "black-box techniques" cannot achieve poly(n)-time computable epsilon-PRGs (Pseudo-Random Generators) for epsilon=n^-omega(1), even if we assume hardness against circuits with oracle access to an arbitrary language in the polynomial time hierarchy. We introduce weaker variants of PRGs with relative error, that do follow under the latter hardness assumption. Specifically, we say that a function G:{0,1}^r -> {0,1}^n is an (epsilon,delta)-re-PRG for a circuit C if (1-epsilon)*Pr[C(U_n)=1] - delta < Pr[C(G(U_r)=1] < (1+epsilon)*Pr[C(U_n)=1] + delta. We construct poly(n)-time computable (epsilon,delta)-re-PRGs with arbitrary polynomial stretch, epsilon=n^-O(1) and delta=2^(-n^Omega(1)). We also construct PRGs with relative error that fool non-boolean distinguishers (in the sense introduced by Dubrov and Ishai (STOC 2006)). Our techniques use ideas from Paturi and Pudlak (STOC 2010), Trevisan and Vadhan (FOCS 2000), Applebaum et al. (CCC 2015). Common themes in our proofs are "composing" a PRG/HSG with a combinatorial object such as dispersers and extractors, and the use of nondeterministic reductions in the spirit of Feige and Lund (Comp. Complexity 1997).

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Sergei Artemenko, Russell Impagliazzo, Valentine Kabanets, and Ronen Shaltiel. Pseudorandomness When the Odds are Against You. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 9:1-9:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{artemenko_et_al:LIPIcs.CCC.2016.9,
  author =	{Artemenko, Sergei and Impagliazzo, Russell and Kabanets, Valentine and Shaltiel, Ronen},
  title =	{{Pseudorandomness When the Odds are Against You}},
  booktitle =	{31st Conference on Computational Complexity (CCC 2016)},
  pages =	{9:1--9:35},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-008-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{50},
  editor =	{Raz, Ran},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.9},
  URN =		{urn:nbn:de:0030-drops-58375},
  doi =		{10.4230/LIPIcs.CCC.2016.9},
  annote =	{Keywords: Derandomization, pseudorandom generator, hitting-set generator, relative error}
}

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Extended Abstract

DOI: 10.4230/LIPIcs.CCC.2015.582

Incompressible Functions, Relative-Error Extractors, and the Power of Nondeterministic Reductions (Extended Abstract)

Authors: Benny Applebaum, Sergei Artemenko, Ronen Shaltiel, and Guang Yang

Published in: LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

Abstract

A circuit C compresses a function f:{0,1}^n -> {0,1}^m if given an input x in {0,1}^n the circuit C can shrink x to a shorter l-bit string x' such that later, a computationally-unbounded solver D will be able to compute f(x) based on x'. In this paper we study the existence of functions which are incompressible by circuits of some fixed polynomial size s=n^c. Motivated by cryptographic applications, we focus on average-case (l,epsilon) incompressibility, which guarantees that on a random input x in {0,1}^n, for every size s circuit C:{0,1}^n -> {0,1}^l and any unbounded solver D, the success probability Pr_x[D(C(x))=f(x)] is upper-bounded by 2^(-m)+epsilon. While this notion of incompressibility appeared in several works (e.g., Dubrov and Ishai, STOC 06), so far no explicit constructions of efficiently computable incompressible functions were known. In this work we present the following results: 1. Assuming that E is hard for exponential size nondeterministic circuits, we construct a polynomial time computable boolean function f:{0,1}^n -> {0,1} which is incompressible by size n^c circuits with communication l=(1-o(1)) * n and error epsilon=n^(-c). Our technique generalizes to the case of PRGs against nonboolean circuits, improving and simplifying the previous construction of Shaltiel and Artemenko (STOC 14). 2. We show that it is possible to achieve negligible error parameter epsilon=n^(-omega(1)) for nonboolean functions. Specifically, assuming that E is hard for exponential size Sigma_3-circuits, we construct a nonboolean function f:{0,1}^n -> {0,1}^m which is incompressible by size n^c circuits with l=Omega(n) and extremely small epsilon=n^(-c) * 2^(-m). Our construction combines the techniques of Trevisan and Vadhan (FOCS 00) with a new notion of relative error deterministic extractor which may be of independent interest. 3. We show that the task of constructing an incompressible boolean function f:{0,1}^n -> {0,1} with negligible error parameter epsilon cannot be achieved by "existing proof techniques". Namely, nondeterministic reductions (or even Sigma_i reductions) cannot get epsilon=n^(-omega(1)) for boolean incompressible functions. Our results also apply to constructions of standard Nisan-Wigderson type PRGs and (standard) boolean functions that are hard on average, explaining, in retrospective, the limitations of existing constructions. Our impossibility result builds on an approach of Shaltiel and Viola (SIAM J. Comp., 2010).

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Benny Applebaum, Sergei Artemenko, Ronen Shaltiel, and Guang Yang. Incompressible Functions, Relative-Error Extractors, and the Power of Nondeterministic Reductions (Extended Abstract). In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 582-600, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{applebaum_et_al:LIPIcs.CCC.2015.582,
  author =	{Applebaum, Benny and Artemenko, Sergei and Shaltiel, Ronen and Yang, Guang},
  title =	{{Incompressible Functions, Relative-Error Extractors, and the Power of Nondeterministic Reductions}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{582--600},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-81-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{33},
  editor =	{Zuckerman, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.582},
  URN =		{urn:nbn:de:0030-drops-50567},
  doi =		{10.4230/LIPIcs.CCC.2015.582},
  annote =	{Keywords: compression, pseudorandomness, extractors, nondeterministic reductions}
}

4 Search Results for "Artemenko, Sergei"

How to Use Nondeterminism in Cryptography

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How to Construct Random Strings

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Pseudorandomness When the Odds are Against You

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Incompressible Functions, Relative-Error Extractors, and the Power of Nondeterministic Reductions (Extended Abstract)

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