Asymptotically-Good Arithmetic Secret Sharing over Z/(p^\ell Z) with Strong Multiplication and Its Applications to Efficient MPC

This paper deals with (1) asymptotics of strongly-multiplicative arithmetic secret sharing over an arbitrary xed ring R` := Z/pZ (p > 0 prime, ` > 0 an integer) and supporting an unbounded number of players n, and with (2) its applications to communication complexity of arithmetic MPC over this ring. For each integer r > 0, let R`(r) be the degree-r Galois-ring extension of R`, with maximal ideal p, residue eld (R`(r))/p = Fpr , and |R`(r)| = p. Using the theory of AG-codes over nite elds and over rings, combined with nontrivial algebraic-geometric lifting techniques, we show that, for arbitrary xed ring R` = Z/pZ, there is a xed integer r̂ = r̂(p) > 0 and a (dense) family of R`(r̂)-linear codes C of unbounded length such that: Denoting the reduction of C modulo p (an Fpr̂ -linear code) by C, each of C, (C)⊥ (dual), (C)∗2 ( square under Schur-product ) is asymptotically good. Each of C, C⊥, C∗2 is free over R`(r̂), with the same dimension as its reduction. Therefore, each has the same minimum distance as its reduction. Particularly, each is asymptotically good. All constructions are e cient. This implies arithmetic secret sharing over the xed ring Z/pZ (rather, the constant-degree extension) with unbounded (dense) n, secret-space dimension Ω(n), share-space dimension O(1), t-privacy Ω(n) with t-wise share-uniformity and 1/3 − t/n > 0 a constant arbitrarily close to 0, and, last-but-not-least, multiplicativity-locality n− t. This extends Chen-Cramer (CRYPTO 2006), which only works over any (large enough) nite elds, signi cantly. Concrete parameters we show here are at least as large. We also show a similar lifting result for asymptotically-good reverse multiplication-friendly embeddings (RFME) and we show how to get an asymptoticallygood alternative for the functionality of hyper-invertible matrices (essential for e cient active-security MPC), as the latter are inherently asymptoticallybad. Finally, we give two applications to general arithmetic MPC over Z/pZ (in the BGW-model with active, perfect security) with communication complexity signi cantly better than the obvious approach based on combining MPC over Fp with added circuitry for emulation of the basic Z/pZ-operations over Fp. Concretely, recent results by Cascudo-Cramer-Xing-Yuan on amortized complexity of MPC (CRYPTO 2018) are now achievable over these rings instead of nite elds, with the same asymptotic complexity and adversary rates.