How to prove knowledge of small secrets

Carsten Baum; Ivan Damgård; Kasper Green Larsen; Michael Nielsen

doi:10.1007/978-3-662-53015-3_17

How to prove knowledge of small secrets

Carsten Baum, Ivan Damgård, Kasper Green Larsen, Michael Nielsen

Aarhus University

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

23 Scopus citations

Abstract

We propose a new zero-knowledge protocol applicable to additively homomorphic functions that map integer vectors to an Abelian group. The protocol demonstrates knowledge of a short preimage and achieves amortised efficiency comparable to the approach of Cramer and Damgård from Crypto 2010, but gives a much tighter bound on what we can extract from a dishonest prover. Towards achieving this result, we develop an analysis for bins-and-balls games that might be of independent interest. We also provide a general analysis of rewinding of a cut-and-choose protocol as well as a method to use Lyubachevsky’s rejection sampling technique efficiently in an interactive protocol when many proofs are given simultaneously. Our new protocol yields improved proofs of plaintext knowledge for (Ring-)LWE-based cryptosystems, where such general techniques were not known before. Moreover, they can be extended to prove preimages of homomorphic hash functions as well.

Original language	English
Title of host publication	Advances in Cryptology - 36th Annual International Cryptology Conference, CRYPTO 2016, Proceedings
Editors	Matthew Robshaw, Jonathan Katz
Publisher	Springer Verlag
Pages	478-498
Number of pages	21
ISBN (Print)	9783662530146
DOIs	https://doi.org/10.1007/978-3-662-53015-3_17
State	Published - 2016
Externally published	Yes
Event	36th Annual International Cryptology Conference, CRYPTO 2016 - Santa Barbara, United States Duration: 14 Aug 2016 → 18 Aug 2016

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	9816
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	36th Annual International Cryptology Conference, CRYPTO 2016
Country/Territory	United States
City	Santa Barbara
Period	14/08/16 → 18/08/16

Bibliographical note

Publisher Copyright:
© International Association for Cryptologic Research 2016.

Funding

K.G. Larsen—Supported by the Center for Massive Data Algorithmics, a Center of the Danish National Research Foundation, grant DNRF84, a Villum Young Investigator Grant and an AUFF Starting Grant. C. Baum, I. Damgård and M. Nielsen—Supported by The Danish National Research Foundation and The National Science Foundation of China (under the grant 61061130540) for the Sino-Danish Center for the Theory of Interactive Computation, within which part of this work was performed; by the CFEM research center (supported by the Danish Strategic Research Council) within which part of this work was performed; and by the Advanced ERC grant MPCPRO.

Funders	Funder number
AUFF
Advanced ERC
CFEM
Sino-Danish Center for the Theory of Interactive Computation
Strategiske Forskningsråd
Horizon 2020 Framework Programme	669255
Danmarks Grundforskningsfond	DNRF84
National Natural Science Foundation of China	61061130540

Keywords

Homomorphic hashing
Integer commitments
Lattice-based encryption
Proofs of plaintext knowledge

Access to Document

10.1007/978-3-662-53015-3_17

Cite this

Baum, C., Damgård, I., Larsen, K. G., & Nielsen, M. (2016). How to prove knowledge of small secrets. In M. Robshaw, & J. Katz (Eds.), Advances in Cryptology - 36th Annual International Cryptology Conference, CRYPTO 2016, Proceedings (pp. 478-498). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9816). Springer Verlag. https://doi.org/10.1007/978-3-662-53015-3_17

Baum, Carsten ; Damgård, Ivan ; Larsen, Kasper Green et al. / How to prove knowledge of small secrets. Advances in Cryptology - 36th Annual International Cryptology Conference, CRYPTO 2016, Proceedings. editor / Matthew Robshaw ; Jonathan Katz. Springer Verlag, 2016. pp. 478-498 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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Baum, C, Damgård, I, Larsen, KG & Nielsen, M 2016, How to prove knowledge of small secrets. in M Robshaw & J Katz (eds), Advances in Cryptology - 36th Annual International Cryptology Conference, CRYPTO 2016, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9816, Springer Verlag, pp. 478-498, 36th Annual International Cryptology Conference, CRYPTO 2016, Santa Barbara, United States, 14/08/16. https://doi.org/10.1007/978-3-662-53015-3_17

How to prove knowledge of small secrets. / Baum, Carsten; Damgård, Ivan; Larsen, Kasper Green et al.
Advances in Cryptology - 36th Annual International Cryptology Conference, CRYPTO 2016, Proceedings. ed. / Matthew Robshaw; Jonathan Katz. Springer Verlag, 2016. p. 478-498 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9816).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AB - We propose a new zero-knowledge protocol applicable to additively homomorphic functions that map integer vectors to an Abelian group. The protocol demonstrates knowledge of a short preimage and achieves amortised efficiency comparable to the approach of Cramer and Damgård from Crypto 2010, but gives a much tighter bound on what we can extract from a dishonest prover. Towards achieving this result, we develop an analysis for bins-and-balls games that might be of independent interest. We also provide a general analysis of rewinding of a cut-and-choose protocol as well as a method to use Lyubachevsky’s rejection sampling technique efficiently in an interactive protocol when many proofs are given simultaneously. Our new protocol yields improved proofs of plaintext knowledge for (Ring-)LWE-based cryptosystems, where such general techniques were not known before. Moreover, they can be extended to prove preimages of homomorphic hash functions as well.

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Baum C, Damgård I, Larsen KG, Nielsen M. How to prove knowledge of small secrets. In Robshaw M, Katz J, editors, Advances in Cryptology - 36th Annual International Cryptology Conference, CRYPTO 2016, Proceedings. Springer Verlag. 2016. p. 478-498. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-662-53015-3_17

How to prove knowledge of small secrets

Abstract

Publication series

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Bibliographical note

Funding

Keywords

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