[Paper Review] Incremental and Decremental Secret Key Agreement

Chung Chan, Ali Al-Bashabsheh+1 more|arXiv (Cornell University)|May 6, 2016

Wireless Communication Security Techniques21 references3 citations

TL;DR

This paper introduces two novel problems in multiterminal secret key agreement: incremental (ISKA) and decremental (DSKA) secret key agreement, which analyze how adding or removing common randomness affects secrecy capacity. It establishes that the critical sets for capacity increase can be computed in strongly polynomial time using submodular function minimization, and conjectures a uniform rate of increase across critical edges.

ABSTRACT

We study the rate of change of the multivariate mutual information among a set of random variables when some common randomness is added to or removed from a subset. This is formulated more precisely as two new multiterminal secret key agreement problems which ask how one can increase the secrecy capacity efficiently by adding common randomness to a small subset of users, and how one can simplify the source model by removing redundant common randomness that does not contribute to the secrecy capacity. The combinatorial structure has been clarified along with some meaningful open problems.

Motivation & Objective

• To model how adding common randomness to a subset of users increases secrecy capacity efficiently.
• To identify redundant common randomness that can be removed without reducing secrecy capacity.
• To characterize the combinatorial structure of optimal partitions and critical sets in multiterminal secret key agreement.
• To develop algorithms for computing critical sets and optimal partitions in strongly polynomial time.

Proposed method

• Formulates incremental and decremental secret key agreement as optimization problems over multivariate mutual information (MMI).
• Uses the multivariate mutual information $ I( {Z}_V) $ as the secrecy capacity, defined via partition minimization over $ {P} otin {P}'(V) $.
• Introduces a zero-singleton submodular function $ g $ to characterize maximal zero sets and critical sets $ {S}_{ {crit}}( {Z}_V) $, enabling efficient computation.
• Applies submodular function minimization over lattice families to compute fundamental partitions and critical sets in $ O(|V|^2) $ steps.
• Establishes that $ {Z}(g) $, the family of zero sets of $ g $, determines the maximal sets in $ {P}^*( {Z}_V) $, linking structure to capacity.
• Proves that $ {S}_{ {crit}}( {Z}_V) $ can be computed in strongly polynomial time, and conjectures uniform rate of increase across critical edges.

Experimental results

Research questions

• RQ1 How does adding common randomness to a subset of users affect the secrecy capacity in multiterminal secret key agreement?
• RQ2 Which common randomness is redundant and can be removed without reducing the secrecy capacity?
• RQ3 Can the critical sets that maximize capacity increase be computed efficiently?
• RQ4 What is the combinatorial structure of the optimal partitions and maximal sets in the MMI framework?
• RQ5 Is the rate of increase in secrecy capacity uniform across all critical edges?

Key findings

• The incremental secret key agreement (ISKA) problem is formulated to maximize secrecy capacity increase with minimal common randomness addition.
• The decremental secret key agreement (DSKA) problem identifies redundant common randomness whose removal does not reduce secrecy capacity.
• The critical sets $ {S}_{ {crit}}( {Z}_V) $, which determine the maximal capacity increase, can be computed in strongly polynomial time.
• The family of maximal zero sets $ {Z}(g) $, derived from a submodular function $ g $, fully characterizes the structure of optimal partitions.
• The optimal partition $ {P}^*( {Z}_V) $ and the secrecy capacity $ I( {Z}_V) $ can be computed in $ O(|V|^2) $ submodular function minimizations.
• The paper conjectures that all critical edges $ S o {S}_{ {crit}}( {Z}_V) $ yield the same rate of increase: $ r_S^+ = rac{|S|-1}{| {P}^*( {Z}_V)|-1} $.