[Paper Review] Optimization of Tree Modes for Parallel Hash Functions

Kévin Atighehchi, Robert Rolland|arXiv (Cornell University)|Dec 21, 2015

Cryptographic Implementations and Security20 references3 citations

TL;DR

This paper proposes an optimized tree mode for parallel hash functions that minimizes running time first, then reduces the number of required processors without compromising performance. It achieves this by strategically reconfiguring the tree topology to balance efficiency and resource usage, offering a dual-optimization framework applicable to non-associative and non-commutative operations.

ABSTRACT

This paper focuses on parallel hash functions based on tree modes of operation for a compression function. We discuss the various forms of optimality that can be obtained when designing such parallel hash functions. The first result is a scheme which optimizes the tree topology in order to decrease at best the running time. Then, without affecting the optimal running time we show that we can slightly change the corresponding tree topology so as to decrease at best the number of required processors as well. Consequently, the resulting scheme optimizes in the first place the running time and in the second place the number of required processors. The present work is of independent interest if we consider the problem of parallelizing the evaluation of an expression where the operator used is neither associative nor commutative.

Motivation & Objective

• To design a parallel hash function that minimizes running time through optimal tree topology.
• To reduce the number of required processors without increasing the running time.
• To address the challenge of parallelizing expressions with non-associative and non-commutative operators.
• To provide a framework that optimizes both time and resource usage in a hierarchical, tree-based computation model.

Proposed method

• The paper models the hash computation as a tree structure where internal nodes represent compression function applications.
• It formulates an optimization problem to minimize the depth of the tree, thereby minimizing running time.
• It then modifies the tree topology under the constraint of fixed minimal depth to reduce the number of processors.
• The method leverages structural properties of the tree to identify and replace high-processor nodes with lower-usage alternatives.
• It applies combinatorial optimization techniques to balance load and reduce parallelism overhead.
• The approach is general enough to be applied to any expression evaluation where operators are neither associative nor commutative.

Experimental results

Research questions

• RQ1 How can the running time of a parallel hash function be minimized through tree topology design?
• RQ2 Can the number of required processors be reduced without increasing the running time?
• RQ3 What structural changes to the tree topology allow for reduced processor usage while preserving optimal performance?
• RQ4 To what extent can the optimization be generalized to non-associative and non-commutative operations?

Key findings

• The proposed scheme achieves optimal running time by minimizing the depth of the tree structure used in the hash computation.
• The same optimal running time is preserved while reducing the number of required processors through strategic topology reconfiguration.
• The method enables efficient parallelization of expressions involving non-associative and non-commutative operators.
• The dual optimization—first time, then space—results in a practical and efficient parallel hash function design.
• The framework is applicable beyond hashing, extending to any computation with similar algebraic constraints.