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Paper: LatticeBased ZeroKnowledge Arguments for Integer Relations
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Conference:  CRYPTO 2018 
Abstract:  We provide latticebased protocols allowing to prove relations among committed integers. While the most general zeroknowledge proof techniques can handle arithmetic circuits in the lattice setting, adapting them to prove statements over the integers is nontrivial, at least if we want to handle exponentially large integers while working with a polynomialsize modulus q. For a polynomial L, we provide zeroknowledge arguments allowing a prover to convince a verifier that committed Lbit bitstrings x, y and z are the binary representations of integers X, Y and Z satisfying $$Z=X+Y$$ over $$\mathbb {Z}$$. The complexity of our arguments is only linear in L. Using them, we construct arguments allowing to prove inequalities $$X<Z$$ among committed integers, as well as arguments showing that a committed X belongs to a public interval $$[\alpha ,\beta ]$$, where $$\alpha $$ and $$\beta $$ can be arbitrarily large. Our range arguments have logarithmic cost (i.e., linear in L) in the maximal range magnitude. Using these tools, we obtain zeroknowledge arguments showing that a committed element X does not belong to a public set S using $$\widetilde{\mathcal {O}}(n \cdot \log S)$$ bits of communication, where n is the security parameter. We finally give a protocol allowing to argue that committed Lbit integers X, Y and Z satisfy multiplicative relations $$Z=XY$$ over the integers, with communication cost subquadratic in L. To this end, we use our protocol for integer addition to prove the correct recursive execution of Karatsuba’s multiplication algorithm. The security of our protocols relies on standard lattice assumptions with polynomial modulus and polynomial approximation factor. 
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BibTeX
@inproceedings{crypto201828830, title={LatticeBased ZeroKnowledge Arguments for Integer Relations}, booktitle={Advances in Cryptology – CRYPTO 2018}, series={Lecture Notes in Computer Science}, publisher={Springer}, volume={10992}, pages={700732}, doi={10.1007/9783319968810_24}, author={Benoît Libert and San Ling and Khoa Nguyen and Huaxiong Wang}, year=2018 }