Universal Equivalence and Majority of Probabilistic Programs over Finite Fields

Abstract

We study decidability problems for equivalence of probabilistic programs for a core probabilistic programming language over finite fields of fixed characteristic. The programming language supports uniform sampling, addition, multiplication, and conditionals and thus is sufficiently expressive to encode Boolean and arithmetic circuits. We consider two variants of equivalence: The first one considers an interpretation over the finite field F q , while the second one, which we call universal equivalence, verifies equivalence over all extensions F q k of F q . The universal variant typically arises in provable cryptography when one wishes to prove equivalence for any length of bitstrings, i.e., elements of F 2 k for any k . While the first problem is obviously decidable, we establish its exact complexity, which lies in the counting hierarchy. To show decidability and a doubly exponential upper bound of the universal variant, we rely on results from algorithmic number theory and the possibility to compare local zeta functions associated to given polynomials. We then devise a general way to draw links between the universal probabilistic problems and widely studied problems on linear recurrence sequences. Finally, we study several variants of the equivalence problem, including a problem we call majority, motivated by differential privacy. We also define and provide some insights about program indistinguishability, proving that it is decidable for programs always returning 0 or 1.

Skolem Meets Schanuel

“…In the remainder of this section we give an alternative formula for the coefficients of f as p-adically convergent sums of rational numbers. This provides a simple method to compute v p (a j ) that avoids computing p-adic approximations of the characteristic roots, as would be needed if we were to directly use (5).…”

“…Tests were run for 48 hours using a 60-second timeout 4 , generating 82367 test instances. 5 The results are presented in Tables 1 and 2. In particular, from order 7 onwards the tool is unable to handle more than half of the instances within one minute, with the timeout percentage jumping significantly from order 6.…”

“…Testing was restricted to 16 parallel threads (50% of the computer's resources) for institutional reasons. 5 7 instances were discarded: 6 happened to be the zero sequence, one resulted in an exception (outside of the main tool code) which was later fixed. 2 Table listing statistical information for successful runs, by order.…”

“…The Skolem Problem, along with closely related questions such as the Positivity Problem, is intimately connected to various fundamental topics in program analysis and automated verification, such as the termination and model checking of simple while loops [3,18,27] or the algorithmic analysis of stochastic systems [2,1,5,13,28]. It also appears in a variety of other contexts, such as formal power series [29,33] and control theory [9,16].…”

Computing zeta functions of large polynomial systems over finite fields

“…They proved that program equivalence in their setting is decidable, and gave an algorithm with doubly exponential complexity. We now briefly review their terminology (from [BJK,Sec. 2.2]), and how their algorithm requires a non-trivial use of zeta functions.…”