Verified Pythagorean Composition for Adaptive Cryptographic Games: Noise Flooding in Homomorphic Encryption

2026-08-13

AI Summary

This study addresses the excessive security bound degradation caused by noise padding in homomorphic encryption under adaptive composition. Leveraging Rocq and SSProve, we construct a novel relational program logic featuring Pythagorean judgment logic and a verified trace compiler. This approach enables lossless composition of conditional KL-divergence, effectively eliminating intermediate conversion overheads. Consequently, we achieve a machine-checked tight security proof that optimizes the security bound under adaptive queries to incur only square-root loss. These contributions significantly enhance both the precision and practicality of formal verification for cryptographic primitives, providing rigorous guarantees where previous methods suffered from loose bounds due to compositional penalties.

Abstract

Noise flooding is a standard defense against decryption attacks on approximate homomorphic encryption, but its security proof is unusually sensitive to composition. Replacing each of $q$ adaptive decryption answers with a statistically close simulation and applying an ordinary hybrid argument loses linearly in $q$. The cryptographic proof instead accumulates conditional Kullback-Leibler (KL) costs and converts to statistical distance once, giving the parameter-critical square-root loss. We machine-check this argument using Rocq and SSProve. Given any fully homomorphic encryption scheme that is approximately correct and IND-CPA secure, we formalize a reduction for every $q$-query IND-CPAD adversary and prove \[ \Pr[\mathsf{IND\text{-}CPAD}_{\mathsf{NF}}^{\mathcal A}=1] \leq β_{\mathsf{CPA}}(\mathcal B_{\mathcal A,q}) + \frac{\sqrt{qn}}{2γ}. \] where $n$ is the plaintext dimension and $γ$ is the flooding-width multiplier. Our proof constructs a new relational program logic over SSProve semantics. Its Pythagorean judgment composes conditional KL budgets without converting them to statistical distance, and a verified trace compiler lifts a local oracle rule to arbitrary adaptive programs with a single final conversion.

Problem

• Noise Flooding
• Homomorphic Encryption
• Adaptive Composition
• Machine-Checked Proof
• IND-CPAD

Innovation

• Pythagorean Composition
• Noise Flooding
• Relational Program Logic
• Machine-Checked Proof
• Homomorphic Encryption