Alignment with DF goals (BGI, Platform growth, community)

BGI (Building Generative Intelligence): Contributes substantial advancements to AI research by linking logical reasoning with computation. Platform Growth: Novel cryptographic tools will attract researchers. Community: Fosters collaboration across disciplines.

Problem description

Current homomorphic encryption schemes primarily focus on arithmetic and boolean operations, limiting their applicability to more complex logic. Our proposal overcomes these constraints. We present a conceptual framework for extending homomorphic encryption beyond arithmetic or Boolean operations into the domain of intuitionistic logic proofs and, by the Curry-Howard correspondence, into the domain of typed functional programs. We begin by reviewing well-known homomorphic encryption schemes for arithmetic operations, and then discuss the adaptation of similar concepts to support logical inference steps in intuitionistic logic. Key to our construction are polynomial functors and Bounded Natural Functors (BNFs), which serve as a categorical substrate on which logic formulas and proofs are represented and manipulated. We outline a complexity-theoretic hardness assumption – the BNF Distinguishing Problem, constructed via a reduction from Subgraph Isomorphism, providing a foundation for cryptographic security. Finally, we describe how these methods can homomorphically encode the execution of total, dependently typed functional programs, and outline strategies for making the approach potentially efficient, including software optimizations and hardware acceleration.

Proposed Solutions

Our framework extends homomorphic encryption using: Representation via Polynomial Functors and BNFs: Logical formulas and proofs are represented as objects leveraging polynomial functors and BNFs. Cryptographic Security Foundation: Anchored in the BNF Distinguishing Problem, reducing from Subgraph Isomorphism. Homomorphic Encoding of Functional Programs: Secure execution of total, dependently typed functional programs. Efficiency Enhancements: Algorithmic and hardware acceleration strategies. A Simple Example: Addition Modulo 4 Plaintext Setup: Integers modulo 4 are represented as "shift functions" on set ( A = {a_0, a_1, a_2, a_3} ). Integer ( k ) maps to function ( f_k(a_i) = a_{(i+k) mod 4} ). Universal Functor: ( F(X) = X^4 ) represents all functions from A to A. Encryption via Quotienting: A secret BNF induces equivalence classes, obscuring the shift functions. Homomorphic Computation: Function composition on equivalence classes computes addition modulo 4: ( [f_1] star [f_3] = [f_1 circ f_3] = [f_{(1+3) mod 4}] = [f_0] ). The evaluator operates on equivalence classes without knowing the underlying integers, ensuring privacy.