Paper 2022/1313

Bounded Surjective Quadratic Functions over $\mathbb F_p^n$ for MPC-/ZK-/FHE-Friendly Symmetric Primitives

Abstract

Motivated by new applications such as secure Multi-Party Computation (MPC), Fully Homomorphic Encryption (FHE), and Zero-Knowledge proofs (ZK), many MPC-, FHE- and ZK-friendly symmetric-key primitives that minimize the number of multiplications over $\mathbb F_p$ for a large prime $p$ have been recently proposed in the literature. These symmetric primitives are usually defined via invertible functions, including (i) Feistel and Lai-Massey schemes and (ii) SPN constructions instantiated with invertible non-linear S-Boxes. However, the ``invertibility'' property is actually never required in any of the mentioned applications. In this paper, we discuss the possibility to set up MPC-/FHE-/ZK-friendly symmetric primitives instantiated with non-invertible bounded surjective functions. With respect to one-to-one functions, any output of a $l$-bounded surjective function admits at most $l$ pre-images. The simplest example is the square map $x\mapsto x^2$ over $\mathbb F_p$ for a prime $p\ge 3$, which is (obviously) $2$-bounded surjective. When working over $\mathbb F_p^n$ for $n\ge 2$, we set up bounded surjective functions by re-considering the recent results proposed by Grassi, Onofri, Pedicini and Sozzi at FSE/ToSC 2022 as starting points. Given a quadratic local map $F:\mathbb F_p^m \rightarrow \mathbb F_p$ for $m\in\{1,2,3\}$, they proved that the shift-invariant non-linear function over $\mathbb F_p^n$ defined as $\mathcal S_F(x_0, x_1, \ldots, x_{n-1}) = y_0\| y_1\| \ldots \| y_{n-1}$ where $y_i := F(x_i, x_{i+1})$ is never invertible for any $n\ge 2\cdot m-1$. Here, we prove that - the quadratic function $F:\mathbb F_p^m \rightarrow \mathbb F_p$ for $m\in\{1,2\}$ that minimizes the probability of having a collision for $\mathcal S_F$ over $\mathbb F_p^n$ is of the form $F(x_0, x_1) = x_0^2 + x_1$ (or equivalent); - the function $\mathcal S_F$ over $\mathbb F_p^n$ defined as before via $F(x_0, x_1) = x_0^2 + x_1$ (or equivalent) is $2^n$-bounded surjective. As concrete applications, we propose modified versions of the MPC-friendly schemes MiMC, HadesMiMC, and (partially of) Hydra, and of the FHE-friendly schemes Masta, Pasta, and Rubato. By instantiating them with the bounded surjective quadratic functions proposed in this paper, we are able to improve the security and/or the performances in the target applications/protocols.