Haolun Tang | Searcharxiv

arXiv2026

Quaternary-Squeeze Quantum Identity Authentication: Direction-Scaling Security via Single-Mode Squeezed Light

Quantum identity authentication (QIA) has emerged as a crucial technology for secure communication systems, particularly in the burgeoning era of quantum communications. This paper proposes a novel QIA protocol based on non-classical characteristics of squeezed light fields. By exploiting quantum noise reduction properties of quadrature squeezed coherent states, the protocol fundamentally thwarts eavesdropping attempts by Heisenberg-limited uncertainty constraints. The fidelity parameter for decoy states is utilized to detect spoofing attacks, and the dynamic key update mechanism fundamentally eliminates vulnerabilities caused by key reuse. Security information ratio analysis shows that the protocol is able to resist Gaussian-cloner attacks and detect eavesdropping. Moreover, the security threshold can be further enhanced with higher squeezing, allowing tunable protection levels adaptable to different threat scenarios. Compared with binary-squeezed protocols, our proposed four-direction (quaternary-dimensional) squeezing halves the eavesdropper's guessing probability and enlarges the fidelity gap by 29% tightening the discrimination threshold without relying on extra hardware, thus facilitating practical implementation.

quant-ph↗

arXiv2026

Formalization of Amicable Numbers Theory

This paper presents a formalization of the theory of amicable numbers in the Lean~4 proof assistant. Two positive integers $m$ and $n$ are called an amicable pair if the sum of proper divisors of $m$ equals $n$ and the sum of proper divisors of $n$ equals $m$. Our formalization introduces the proper divisor sum function $\propersum(n) = \sigma(n) - n$, defines the concepts of amicable pairs and amicable numbers, and computationally verifies historically famous amicable pairs. Furthermore, we formalize basic structural theorems, including symmetry, non-triviality, and connections to abundant/deficient numbers. A key contribution is the complete formal proof of the classical Th\={a}bit formula (9th century), using index-shifting and the \texttt{zify} tactic. Additionally, we provide complete formal proofs of both Th\={a}bit's rule and Euler's generalized rule (1747), two fundamental theorems for generating amicable pairs. A major achievement is the first complete formalization of the Borho-Hoffmann breeding method (1986), comprising 540 lines with 33 theorems and leveraging automated algebra tactics (\texttt{zify} and \texttt{ring}) to verify complex polynomial identities. We also formalize extensions including sociable numbers (aliquot cycles), betrothed numbers (quasi-amicable pairs), parity constraint theorems, and computational search bounds for coprime pairs ($>10^{65}$). We verify the smallest sociable cycle of length 5 (Poulet's cycle) and computationally verify specific instances. The formalization comprises 2076 lines of Lean code organized into Mathlib-candidate and paper-specific modules, with 139 theorems and all necessary infrastructure for divisor sum multiplicativity and coprimality reasoning.

cs.LO↗