TY - GEN

T1 - Bounded Indistinguishability for Simple Sources

AU - Bogdanov, Andrej

AU - Dinesh, Krishnamoorthy

AU - Filmus, Yuval

AU - Ishai, Yuval

AU - Kaplan, Avi

AU - Srinivasan, Akshayaram

N1 - Publisher Copyright: © Andrej Bogdanov, Krishnamoorthy Dinesh, Yuval Filmus, Yuval Ishai, Avi Kaplan, and Akshayaram Srinivasan; licensed under Creative Commons License CC-BY 4.0

PY - 2022/1/1

Y1 - 2022/1/1

N2 - A pair of sources X, Y over {0, 1}n are k-indistinguishable if their projections to any k coordinates are identically distributed. Can some AC0 function distinguish between two such sources when k is big, say k = n0.1? Braverman's theorem (Commun. ACM 2011) implies a negative answer when X is uniform, whereas Bogdanov et al. (Crypto 2016) observe that this is not the case in general. We initiate a systematic study of this question for natural classes of low-complexity sources, including ones that arise in cryptographic applications, obtaining positive results, negative results, and barriers. In particular: - There exist Ω(√n)-indistinguishable X, Y, samplable by degree-O(log n) polynomial maps (over F2) and by poly(n)-size decision trees, that are Ω(1)-distinguishable by OR. - There exists a function f such that all f(d, ϵ)-indistinguishable X, Y that are samplable by degree-d polynomial maps are ϵ-indistinguishable by OR for all sufficiently large n. Moreover, f(1, ϵ) = ⌈log(1/ϵ)⌉ + 1 and f(2, ϵ) = O(log10(1/ϵ)). - Extending (weaker versions of) the above negative results to AC0 distinguishers would require settling a conjecture of Servedio and Viola (ECCC 2012). Concretely, if every pair of n0.9indistinguishable X, Y that are samplable by linear maps is ϵ-indistinguishable by AC0 circuits, then the binary inner product function can have at most an ϵ-correlation with AC0 ◦ ⨁ circuits. Finally, we motivate the question and our results by presenting applications of positive results to low-complexity secret sharing and applications of negative results to leakage-resilient cryptography.

AB - A pair of sources X, Y over {0, 1}n are k-indistinguishable if their projections to any k coordinates are identically distributed. Can some AC0 function distinguish between two such sources when k is big, say k = n0.1? Braverman's theorem (Commun. ACM 2011) implies a negative answer when X is uniform, whereas Bogdanov et al. (Crypto 2016) observe that this is not the case in general. We initiate a systematic study of this question for natural classes of low-complexity sources, including ones that arise in cryptographic applications, obtaining positive results, negative results, and barriers. In particular: - There exist Ω(√n)-indistinguishable X, Y, samplable by degree-O(log n) polynomial maps (over F2) and by poly(n)-size decision trees, that are Ω(1)-distinguishable by OR. - There exists a function f such that all f(d, ϵ)-indistinguishable X, Y that are samplable by degree-d polynomial maps are ϵ-indistinguishable by OR for all sufficiently large n. Moreover, f(1, ϵ) = ⌈log(1/ϵ)⌉ + 1 and f(2, ϵ) = O(log10(1/ϵ)). - Extending (weaker versions of) the above negative results to AC0 distinguishers would require settling a conjecture of Servedio and Viola (ECCC 2012). Concretely, if every pair of n0.9indistinguishable X, Y that are samplable by linear maps is ϵ-indistinguishable by AC0 circuits, then the binary inner product function can have at most an ϵ-correlation with AC0 ◦ ⨁ circuits. Finally, we motivate the question and our results by presenting applications of positive results to low-complexity secret sharing and applications of negative results to leakage-resilient cryptography.

KW - Bounded indistinguishability

KW - Complexity of sampling

KW - Constant-depth circuits

KW - Leakage-resilient cryptography

KW - Pseudorandomness

KW - Secret sharing

UR - http://www.scopus.com/inward/record.url?scp=85123999985&partnerID=8YFLogxK

U2 - https://doi.org/10.4230/LIPIcs.ITCS.2022.26

DO - https://doi.org/10.4230/LIPIcs.ITCS.2022.26

M3 - منشور من مؤتمر

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 13th Innovations in Theoretical Computer Science Conference, ITCS 2022

A2 - Braverman, Mark

T2 - 13th Innovations in Theoretical Computer Science Conference, ITCS 2022

Y2 - 31 January 2022 through 3 February 2022

ER -