  The sumcheck protocol, introduced in 1992, is an interactive proof which is a
key component of many probabilistic proof systems in computational complexity
theory and cryptography, some of which have been deployed. However, none of
these proof systems based on the sumcheck protocol enjoy a formally-verified
security analysis. In this paper, we make progress in this direction by
providing a formally verified security analysis of the sumcheck protocol using
the interactive theorem prover Isabelle/HOL. We follow a general and modular
approach.
  First, we give a general formalization of public-coin interactive proofs. We
then define a generalized sumcheck protocol for which we axiomatize the
underlying mathematical structure and we establish its soundness and
completeness. Finally, we prove that these axioms hold for multivariate
polynomials, the original setting of the sumcheck protocol. Our modular
analysis facilitates formal verification of sumcheck instances based on
different mathematical structures with little effort, by simply proving that
these structures satisfy the axioms. Moreover, the analysis supports the
development and formal verification of future cryptographic protocols using the
sumcheck protocol as a building block.
