Paper 2018/716

Lattice-Based Zero-Knowledge Arguments for Integer Relations

Benoît Libert, San Ling, Khoa Nguyen, and Huaxiong Wang


We provide lattice-based protocols allowing to prove relations among committed integers. While the most general zero-knowledge proof techniques can handle arithmetic circuits in the lattice setting, adapting them to prove statements over the integers is non-trivial, at least if we want to handle exponentially large integers while working with a polynomial-size modulus $q$. For a polynomial $L$, we provide zero-knowledge arguments allowing a prover to convince a verifier that committed $L$-bit 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 zero-knowledge 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 $L$-bit 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.

Available format(s)
Cryptographic protocols
Publication info
A minor revision of an IACR publication in CRYPTO 2018
Lattice-based cryptographyzero-knowledge argumentsinteger relationsrange proofsnon-membership proofs
Contact author(s)
khoantt @ ntu edu sg
2018-08-01: received
Short URL
Creative Commons Attribution


      author = {Benoît Libert and San Ling and Khoa Nguyen and Huaxiong Wang},
      title = {Lattice-Based Zero-Knowledge Arguments for Integer Relations},
      howpublished = {Cryptology ePrint Archive, Paper 2018/716},
      year = {2018},
      note = {\url{}},
      url = {}
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