# Lattice-Based SNARGs and Their Application to More Efficient Obfuscation

**Authors:**

*D. Boneh, Y. Ishai, A. Sahai, and D. Wu*

** Abstract: **

Succinct non-interactive arguments (SNARGs) enable verifying NP computations with substantially
lower complexity than that required for classical NP verification. In this work, we
first construct a lattice-based SNARG candidate with quasi-optimal succinctness (where the
argument size is quasilinear in the security parameter). Further extension of our methods yields
the first SNARG (from any assumption) that is quasi-optimal in terms of both prover overhead
(polylogarithmic in the security parameter) as well as succinctness. Moreover, because our constructions
are lattice-based, they plausibly resist quantum attacks. Central to our construction is
a new notion of linear-only vector encryption which is a generalization of the notion of linear-only
encryption introduced by Bitansky et al. (TCC 2013). We conjecture that variants of Regev
encryption satisfy our new linear-only definition. Then, together with new information-theoretic
approaches for building statistically-sound linear PCPs over small finite fields, we obtain the
first quasi-optimal SNARGs.
We then show a surprising connection between our new lattice-based SNARGs and the
concrete efficiency of program obfuscation. All existing obfuscation candidates currently rely
on multilinear maps. Among the constructions that make black-box use of the multilinear
map, obfuscating a circuit of even moderate depth (say, 100) requires a multilinear map with
multilinearity degree in excess of 2^{100}. In this work, we show that an ideal obfuscation of
both the decryption function in a fully homomorphic encryption scheme and a variant of the
verification algorithm of our new lattice-based SNARG yields a general-purpose obfuscator for all
circuits. Finally, we give some concrete estimates needed to obfuscate this “obfuscation-complete”
primitive. We estimate that at 80-bits of security, a (black-box) multilinear map with ≈ 2^{12}
levels of multilinearity suffices. This is over 2^{80} times more efficient than existing candidates,
and thus, represents an important milestone towards implementable program obfuscation for all
circuits.

** Reference:**

In proceedings of Eurocrypt 2017, pp. 247-277.

**Full paper:**
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