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Erschienen in:
On Removing Graded Encodings from Functional Encryption Kapitel lesen Erstes Kapitel lesen

2017 | OriginalPaper | Buchkapitel

Removing the Strong RSA Assumption from Arguments over the Integers

verfasst von : Geoffroy Couteau, Thomas Peters, David Pointcheval

Erschienen in: Advances in Cryptology – EUROCRYPT 2017

Verlag: Springer International Publishing

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Abstract

Committing integers and proving relations between them is an essential ingredient in many cryptographic protocols. Among them, range proofs have been shown to be fundamental. They consist in proving that a committed integer lies in a public interval, which can be seen as a particular case of the more general Diophantine relations: for the committed vector of integers \(\varvec{x}\), there exists a vector of integers \(\varvec{w}\) such that \(P(\varvec{x},\varvec{w})=0\), where P is a polynomial.
In this paper, we revisit the security strength of the statistically hiding commitment scheme over the integers due to Damgård-Fujisaki, and the zero-knowledge proofs of knowledge of openings. Our first main contribution shows how to remove the Strong RSA assumption and replace it by the standard RSA assumption in the security proofs. This improvement naturally extends to generalized commitments and more complex proofs without modifying the original protocols.
As a second contribution, we design an interactive technique turning commitment scheme over the integers into commitment scheme modulo a prime p. Still under the RSA assumption, this results in more efficient proofs of relations between committed values. Our methods thus improve upon existing proof systems for Diophantine relations both in terms of performance and security. We illustrate that with more efficient range proofs under the sole RSA assumption.

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Vorheriges Kapitel A New Structural-Differential Property of 5-Round AES
Nächstes Kapitel Magic Adversaries Versus Individual Reduction: Science Wins Either Way
Fußnoten
1
It should be noted that in our proof, the bound a will depend on the success probability of the adversary.
 
2
The classical trick that consists of using \(\lambda _i = \lambda ^i\) is not efficient here since we are in the integers, and so no reduction can be applied.
 
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Metadaten
Titel
Removing the Strong RSA Assumption from Arguments over the Integers
verfasst von
Geoffroy Couteau
Thomas Peters
David Pointcheval
Copyright-Jahr
2017
Buch
Advances in Cryptology – EUROCRYPT 2017

Print ISBN: 978-3-319-56613-9

Electronic ISBN: 978-3-319-56614-6

Copyright-Jahr: 2017

DOI
https://doi.org/10.1007/978-3-319-56614-6_11