Paper 2026/1940

Efficient Pseudorandom Correlation Generators over Binary Extension Fields and More

Abstract

In this work, we construct highly efficient pseudorandom correlation generators (PCGs) over binary extension fields. PCGs allow for the distribution of a great amount of correlated randomness by having each party locally expand a short seed, enabling efficient multi-party computation (MPC) protocols with silent and sublinear preprocessing. Our PCGs achieve nearly linear time, both asymptotically and concretely, where expanding $N$ VOLEs (vector oblivious linear evaluations) over $\mathbb{F}_{2^k}$ consists of $O(N)$ PRG calls, $O(N)$ multiplications in $\mathbb{F}_{2^k}$ and $O(kN\log{N})$ binary XORs. In fact, the $O(kN\log{N})$ XORs are insignificant in practical parameter regimes where $N\approx 2^{20}$. Concretely, we achieve a throughput of $\approx 5,000,000$ VOLEs per second over $\mathbb{F}_{2^{128}}$, which is at least $4.7$x, $1.9$x and $1.6$x faster than the state-of-the-art PCGs of Expand-Accumulate (EA), Block-Accumulate (BA) and Expand-Convolute (EC) codes, respectively. For oblivious linear evaluations (OLEs), expanding $N$ instances requires $O(\lambda N)$ PRG calls, $O(N)$ multiplications in $\mathbb{F}_{2^k}$ and $O(kN\log{N})$ binary XORs, where $\lambda$ is the security parameter. Based on the $\mathsf{libOTe}$ library, we also give the first full implementation of PCGs for OLEs over extension fields, which shows a throughput of $\approx 42,000$ OLEs per second over $\mathbb{F}_{2^{128}}$. At the heart of our technical contribution is an efficient multivariate FFT algorithm. It evaluates any $n$-variate polynomial $f(x_1,\ldots,x_n)$ over $\mathbb{F}_{2^k}$ of individual degree$\le2$ using exactly $\frac{5}{6}kN\log_3{N}$ additions and $\frac{1}{6}kN\log_3{N}$ multiplications in $\mathbb{F}_4$, where $N=3^n$ and $k$ is an arbitrary even integer. Note that this is considerably faster than normal FFT algorithms that require $O(N\log{N})$ multiplications in the underlying field $\mathbb{F}_{2^k}$. Along the way, we generalize our approach to work over a degree-$k$ Galois extension of $\mathbb{Z}_{2^{\ell}}$, denoted by $\mathbb{GR}(2^{\ell},k)$. Similarly, our algorithm takes only $\frac{7}{6}kN\log_3{N}$ additions and $\frac{1}{6}kN\log_3{N}$ multiplications in $\mathbb{GR}(2^{\ell},2)$ to evaluate $f(x_1,\ldots,x_n)\in\mathbb{GR}(2^{\ell},k)[x_1,\ldots,x_n]_{\leq 2}$. This implies practically efficient PCGs for (V)OLE over Galois rings. As a side contribution of independent interest, we derive fast encoding algorithms for a family of Quasi-Abelian codes, which are even concretely $1.66$x faster than Repeat-Accumulate-Accumulate (RAA) codes over $\mathbb{F}_{2^{128}}$, whose encoding only involves a linear number of additions and two random permutations. As Quasi-Abelian codes have been proven to attain concretely large minimum distance in a recent work of Li et al. (CRYPTO 2026), this opens a door to building more efficient succinct arguments over extension fields.