SMT Solving for Finite Field

Overview

Finite filed arithmetic is a fundamental component of many cryptographic protocols.

Recent works have significantly advanced SMT solving for finite fields, including:

  • MCSat-based techniques in Yices2

  • Split Gröbner Bases approaches

  • Non-linear reasoning methods

Implementation

Arlib provides a flexible implementation for SMT solving over finite fields in the arlib/smt/ff module. It supports multiple encoding strategies:

  • Bit-vector encoding (ff_bv_solver.py): Translates finite field operations to bit-vector constraints

  • Integer encoding (ff_int_solver.py): Uses modular arithmetic via integer constraints

The core components include:

  • Formula parsing and representation (ff_parser.py)

  • Translation to bit-vector formulas using Z3

  • Translation to integer arithmetic with modular operations

  • Support for standard finite field operations (addition, multiplication, equality)

Bit-vector Encoding

The bit-vector approach represents finite field elements as fixed-width bit-vectors:

  • Field elements are encoded as bit-vectors with width log₂(field_size)

  • Field operations are translated to bit-vector operations with modular constraints

  • Range constraints ensure values stay within the field bounds

Integer Encoding

The integer encoding represents field elements as integers with modular arithmetic:

  • Field elements are constrained to the range [0, field_size-1]

  • Field operations are translated to integer operations with explicit modulo operations

  • This approach leverages integers and non-linear arithmetic reasoning in SMT solvers

Usage Example

Here’s a simple example of using the finite field solver:

from arlib.smt.ff.ff_bv_solver import solve_qfff

# Example formula in SMT-LIB format
smt_input = """
(set-info :smt-lib-version 2.6)
(set-logic QF_FF)
(declare-fun x () (_ FiniteField 17))
(declare-fun y () (_ FiniteField 17))
(assert (= (ff.add x y) #f3m17))
(assert (= (ff.mul x y) #f5m17))
(check-sat)
"""

# Solve the formula
solve_qfff(smt_input)

This will solve the system of equations x + y = 3 and x * y = 5 in the finite field GF(17).

References

  • Hader, T., Kaufmann, D., Irfan, A., Graham-Lengrand, S., & Kovács, L. (2024).

    MCSat-based Finite Field Reasoning in the Yices2 SMT Solver. ArXiv:2402.17927.

  • Ozdemir, A., Pailoor, S., Bassa, A., Ferles, K., Barrett, C., & Dillig, I. (2024).

    Split Gröbner Bases for Satisfiability Modulo Finite Fields. In Computer Aided Verification (CAV).

  • Ozdemir, A., Kremer, G., Tinelli, C., & Barrett, C. (2023).

    Satisfiability modulo finite fields. In Computer Aided Verification (CAV).

  • Hader, T., Kaufmann, D., & Kovács, L. (2023).

    SMT solving over finite field arithmetic. In Logic for Programming, Artificial Intelligence and Reasoning (LPAR).

  • Hader, T. (2022).

    Non-linear SMT-reasoning over finite fields. Master’s thesis, TU Wien.

  • Hader, T., & Kovács, L. (2022).

    Non-linear SMT-reasoning over finite fields. In SMT Workshop. Extended Abstract.