Cryptology ePrint Archive: Report 2015/712

Adaptive Proofs have Straightline Extractors (in the Random Oracle Model)

David Bernhard and Bogdan Warinschi and Ngoc Khanh Nguyen

Abstract: Abstract. The concept of adaptive security for proofs of knowledge was recently studied by Bernhard et al. They formalised adaptive security in the ROM and showed that the non-interactive version of the Schnorr protocol obtained using the Fiat-Shamir transformation is not adaptively secure unless the one-more discrete logarithm problem is easy. Their only construction for adaptively secure protocols used the Fischlin transformation [3] which yields protocols with straight-line extractors. In this paper we provide two further key insights. Our main result shows that any adaptively secure protocol must have a straight-line extractor: even the most clever rewinding strategies cannot oer any benets against adaptive provers.

Then, we show that any Fiat-Shamir transformed SIGMA-protocol is not adaptively secure unless a related problem which we call the SIGMA-one-wayness problem is easy. This assumption concerns not just Schnorr but applies to a whole class of SIGMA-protocols including e.g. Chaum-Pedersen and representation proofs. We also prove that SIGMA-one-wayness is hard in the generic group model. Taken together, these results suggest that Fiat-Shamir transformed SIGMA-protocols should not be used in settings where adaptive security is important.

Category / Keywords: foundations / zero-knowledge, sigma protocol, adaptive security, metareduction, discrete logarithm

Date: received 16 Jul 2015, last revised 18 Oct 2016

Contact author: bernhard at cs bris ac uk

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Note: major revision - new author added

Version: 20161018:152920 (All versions of this report)

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