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Abstract: A family $(S_t)$ of sets is $p$-bounded Diophantine if $S_t$ has a representing $p$-bounded polynomial $R_{S,t}$, s.t.~$x\in S_t \iff (\exists y)[R_{S}(x;y)=0]$. We say that $(S_t)$ is unbounded Diophantine if additionally, $R_{S,t}$ is a fixed $t$-independent polynomial. We show that $p$-bounded (resp., unbounded) Diophantine set has a polynomial-size (resp., constant-size) statistical zero-knowledge proof system that a committed tuple $x$ belongs to $S$. We describe efficient SZK proof... (Update)
Context of citations to this paper: More
...with an efficient (statistical) zero knowledge correctness proof. By using recently proposed cryptographic range proofs [Bou00,Lip01], both the bidder seller and the seller authority communication complexity of the LAN scheme are of order (V log 2 B) bits, where V...
...and Yuri Matiyasevich for fruitful discussions on the Hilbert s tenth problem. This paper obsolotes an earlier technical report [Lip01]. Efficient Diophantine membership arguments can be given for many interesting sets S ZZ. We did certainly not mention all...
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BibTeX entry: (Update)
Helger Lipmaa. Statistical Zero-Knowledge Proofs from Diophantine Equations. Preliminary version, 25 October 2001. http://citeseer.ist.psu.edu/lipmaa01statistical.html More
@misc{ lipmaa01statistical,
author = "H. Lipmaa",
title = "Statistical Zero-Knowledge Proofs from Diophantine Equations",
text = "Helger Lipmaa. Statistical Zero-Knowledge Proofs from Diophantine Equations.
Preliminary version, 25 October 2001.",
year = "2001",
url = "citeseer.ist.psu.edu/lipmaa01statistical.html" }
Citations (may not include all citations):162 Public-Key Cryptosystems Based on Composite Degree Residuosi.. - Paillier - 1999
119 Proofs of Partial Knowledge and Simplified Design of Witness.. - Cramer, Damgard et al. - 1994
103 Statistical Zero Knowledge Protocols to Prove Modular Polyno.. (context) - Fujisaki, Okamoto - 1997
68 Prime Number and Computer Methods for Factorization (context) - Riesel - 1994
54 Easy Come - Easy Go Divisible Cash - Chan, Frankel et al. - 1998
45 a Simplification and Some Applications of Paillier's Probabi.. (context) - Damgard, Jurik et al. - 2001
17 An Integer Commitment Scheme Based on Groups with Hidden Ord.. - Damgard, Fujisaki - 2001
13 Efficient Proofs that a Committed Number Lies in an Interval (context) - Boudot - 2000
9 Communications in Pure and Applied Mathematics (context) - Rabin, Shallit et al. - 1986
7 Personal communication (context) - Shallit - 2001
4 On Transformation of Interactive Proofs that Preserve the Pr.. - Vadhan - 2000
3 Efficient Convertible Undeniable Signature Schemes - Michels, Stadler - 1997
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