Efficient Extended GCD and Class Groups from Secure Integer Arithmetic

Abstract

In this paper we first present an efficient protocol for the secure computation of the extended greatest common divisor, assuming basic secure integer arithmetic common to many MPC frameworks. The protocol is based on Bernstein and Yang’s constant-time 2-adic algorithm, which we adapt to work purely over the integers. This yields a much better approach for the MPC setting, but raises a new concern about the growth of the Bézout coefficients. By a careful analysis we are able to prove that the Bézout coefficients in our protocol will never exceed $3\max (a,b)$ in absolute value for inputs a and b. Next, we present efficient protocols for implementing class groups of imaginary quadratic number fields in the MPC setting. We start from Shanks’ original algorithms for the efficient composition of binary quadratic forms and combine these with our particular adaptation of a forms reduction algorithm due to Agarwal and Frandsen. We will formulate this result in terms of secure groups, which are introduced as oblivious data structures implementing finite groups in a privacy-preserving manner. Our results show how class group operations can be run efficiently between multiple parties operating jointly on secret-shared group elements. We have integrated secure class groups in MPyC along with other instances of secure groups such as Schnorr groups and elliptic curves.