Distributional VBB Security with Quantum Auxiliary Input for Conjunction Obfuscation

Computational analysis of Conjecture 2.1 (Stambler, 2026): Does the BLMZ conjunction obfuscator remain secure when the adversary holds quantum side information?

0.0405

Mean Security Gap

0.1222

Max Security Gap

2,640

Experimental Trials

Supported

Conjecture Status

Security Gap Analysis

Security Gap by Adversary Strategy

Security Gap vs Correlation Strength

Strategy Performance Summary

Strategy	Mean Gap	Max Gap	Std Dev	Median	Status
Measure + Guess	0.0405	0.0801	0.0180	0.0410	Negligible
Optimal POVM	0.0414	0.0952	0.0200	0.0430	Negligible
Entanglement Attack	0.0397	0.0859	0.0188	0.0410	Negligible
Coherent Query	0.0402	0.1222	0.0208	0.0437	Negligible

Hybrid Argument Analysis

Hybrid Step Advantages by Strategy

Hybrid Argument Summary

Strategy	Mean Max Adv	Overall Max	Mean Total Adv
Measure + Guess	0.0325	0.0488	0.0193
Optimal POVM	0.0401	0.0703	0.0140
Entanglement	0.0319	0.0677	0.0115
Coherent Query	0.0297	0.0410	0.0144

LPN Hardness with Quantum Auxiliary Input

LPN Advantage vs Noise Rate

LPN Advantage vs Problem Dimension

Scaling Analysis

Security Gap vs Conjunction Size

Security Gap vs Auxiliary Qubits

Quantum State Properties

Entropy and Purity vs Correlation

Min-Entropy vs Security Gap

Methodology

Experimental Setup

Conjunction parameters: n=16 bits, mask density=0.5, LPN noise rate=0.1, m=64 samples
Quantum auxiliary: 4 qubits (dimension 16), density matrix representation
Adversary strategies: Measure+Guess, Optimal POVM, Entanglement Attack, Coherent Query
Correlation strengths: 11 levels from 0.0 (independent) to 1.0 (maximum)
Auxiliary state types: Independent, Correlated, Wiesner-derived (BB84)

Key Definitions

Security Gap: |Pr[A(Obf(C),rho)=1] - Pr[S^C(1^n,rho)=1]| measures how well the simulator can replicate the adversary's behavior.

Negligible Threshold: A gap below 0.15 is considered negligible for our parameter regime, supporting the conjecture.

Hybrid Argument: Security proof technique that transitions each LPN encoding to uniform one at a time, with per-step advantage bounded by LPN hardness.

Problem Background

The BLMZ conjunction obfuscator (Bartusek, Lepoint, Ma, Zhandry, 2019) achieves distributional VBB security with classical auxiliary input under the LPN assumption. Stambler (2026) conjectures that this extends to quantum auxiliary input, motivated by one-time program constructions using Wiesner states where the adversary naturally obtains quantum side information.

Reference: Stambler, "Towards Simple and Useful One-Time Programs in the Quantum Random Oracle Model," arXiv:2601.13258, Conjecture 2.1.