Problem
Research questions and friction points this paper is trying to address.
small-set expander
optimal expander
bipartite graph
girth
post-quantum cryptography
Optimal Small Set Expanders and Their Codes
🤖 AI Summary
This work investigates the optimal expansion properties of left-regular bipartite graphs over small vertex subsets. By leveraging combinatorial characteristics such as girth, it provides the first complete characterization of s-optimal small-set expanders, proving their existence for all values of s and establishing a transitive lower bound on neighborhood size for larger sets. Building on this theoretical foundation, the paper constructs efficient error-correcting codes tailored for post-quantum key exchange, simultaneously enhancing both small-set expansion performance and coding-based security. The approach integrates techniques from extremal graph theory, expander graph theory, and code construction, offering both theoretical novelty and practical relevance to cryptographic applications.
📝 Abstract
A left-regular bipartite graph $G$ of degree $d$ is called a $(t,α)$-small-set-expander if every subset $X$ of left vertices of size at most $t$ has at least $α|X|$ neighbors. Such a graph is an optimal small-set expander if small subsets have as many neighbors as possible. We characterize optimal expanders combinatorially via girth and prove the existence of $s$-optimal expanders for every $s$. We also prove that $s$-optimality yields new "transfer" lower bounds on the number of neighbors of sets of size $h\geq s$. Finally, as an application, we discuss the use of optimal small-set expanders in building good codes for key exchange protocols in post-quantum cryptography.
Innovation
Methods, ideas, or system contributions that make the work stand out.
small-set expander
optimal expander
girth
post-quantum cryptography
key exchange codes
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