Sarah Miracle

Reverse Cycle Walking and Its Applications

We study the problem of constructing a block-cipher on a “possibly-strange” set \(\mathcal{S}\) using a block-cipher on a larger set \(\mathcal{T}\). Such constructions are useful in format-preserving encryption, where for example the set \(\mathcal{S}\) might contain “valid 9-digit social security numbers” while \(\mathcal{T}\) might be the set of 30-bit strings. Previous work has solved this problem using a technique called cycle walking, first formally analyzed by Black and Rogaway. Assuming the size of \(\mathcal{S}\) is a constant fraction of the size of \(\mathcal{T}\), cycle walking allows one to encipher a point \(x \in \mathcal{S}\) by applying the block-cipher on \(\mathcal{T}\) a small expected number of times and O(N) times in the worst case, where \(N = |\mathcal{T}|\), without any degradation in security. We introduce an alternative to cycle walking that we call reverse cycle walking, which lowers the worst-case number of times we must apply the block-cipher on \(\mathcal{T}\) from O(N) to \(O(\log N)\). Additionally, when the underlying block-cipher on \(\mathcal{T}\) is secure against \(q = (1-\epsilon )N\) adversarial queries, we show that applying reverse cycle walking gives us a cipher on \(\mathcal{S}\) secure even if the adversary is allowed to query all of the domain points. Such fully secure ciphers have been the the target of numerous recent papers.

Sampling and Counting 3-Orientations of Planar Triangulations

Given a planar triangulation, a 3-orientation is an orientation of the internal edges so all internal vertices have out-degree three. Each 3-orientation gives rise to a unique edge coloring known as a Schnyder wood that has proven powerful for various computing and combinatorics applications. We consider natural Markov chains for sampling uniformly from the set of 3-orientations. First, we study a “triangle-reversing” chain on the space of 3-orientations of a fixed triangulation that reverses the orientation of the edges around a triangle in each move. We show that, when restricted to planar triangulations of maximum degree six, this Markov chain is rapidly mixing and we can approximately count 3-orientations. Next, we construct a triangulation with high degree on which this Markov chain mixes slowly. Finally, we consider an “edge-flipping” chain on the larger state space consisting of 3-orientations of all planar triangulations on a fixed number of vertices. We prove that this chain is always rapidly mixing.

Cycle Slicer: An Algorithm for Building Permutations on Special Domains

We introduce an algorithm called Cycle Slicer that gives new solutions to two important problems in format-preserving encryption: domain targeting and domain completion. In domain targeting, where we wish to use a cipher on domain \(\mathcal {X}\) to construct a cipher on a smaller domain \(\mathcal{S}\subseteq \mathcal {X}\), using Cycle Slicer leads to a significantly more efficient solution than Miracle and Yilek’s Reverse Cycle Walking (ASIACRYPT 2016) in the common setting where the size of \(\mathcal{S}\) is large relative to the size of \(\mathcal {X}\). In domain completion, a problem recently studied by Grubbs, Ristenpart, and Yarom (EUROCRYPT 2017) in which we wish to construct a cipher on domain \(\mathcal {X}\) while staying consistent with existing mappings in a lazily-sampled table, Cycle Slicer provides an alternative construction with better worst-case running time than the Zig-Zag construction of Grubbs et al. Our analysis of Cycle Slicer uses a refinement of the Markov chain techniques for analyzing matching exchange processes, which were originally developed by Czumaj and Kutylowski (Rand. Struct. & Alg. 2000).

Algorithms to approximately count and sample conforming colorings of graphs

Given a multigraphź G and a functionź F that assigns a forbidden ordered pair of colors to each edge e , we say a coloringź C of the vertices is conforming to ź F if for all e = ( u , v ) , ( C ( u ) , C ( v ) ) ź F ( e ) . Conforming colorings generalize many natural graph theoretic concepts, including independent sets, vertex colorings, list colorings, H -colorings and adapted colorings and consequently there are known complexity barriers to sampling and counting. We introduce natural Markov chains on the set of conforming colorings and provide general conditions for when they can be used to design efficient Monte Carlo algorithms for sampling and approximate counting.

Algorithms to Approximately Count and Sample Conforming Colorings of Graphs

Abstract Conforming colorings naturally generalize many graph theory structures, including independent sets, vertex colorings, list colorings, H-colorings and adapted colorings. Given a multigraph G and a function F that assigns a forbidden ordered pair of colors to each edge e, we say a coloring C of the vertices is conforming to F if, for all e = ( u , v ) , F ( e ) ≠ ( C ( u ) , C ( v ) ) . We consider Markov chains on the set of conforming colorings and provide some general conditions for when they can be used to construct efficient Monte Carlo algorithms for sampling and counting.