A topological chaos framework for hash functions

This paper presents a new procedure of generating hash functions which can be evaluated using some mathematical tools. This procedure is based on discrete chaotic iterations. First, it is mathematically proven, that these discrete chaotic iterations can be considered as a \linebreak particular case of topological chaos. Then, the process of generating hash function based on the \linebreak topological chaos is detailed. Finally it is shown how some tools coming from the domain of \linebreak topological chaos can be used to measure quantitatively and qualitatively some desirable properties for hash functions. An illustration example is detailed in order to show how one can create hash functions using our theoretical study. Key-words : Discrete chaotic iterations. Topological chaos. Hash function

💡 Research Summary

The paper proposes a novel framework for constructing cryptographic hash functions by leveraging the theory of topological chaos, specifically the definition introduced by Devaney. The authors begin by recalling the classical definition of chaotic dynamical systems: a map must exhibit sensitive dependence on initial conditions, topological transitivity, and regularity (density of periodic points). They then introduce discrete chaotic iterations (DCI), a process where a finite set of binary cells is updated one at a time according to a prescribed “strategy” sequence.

To bridge DCI with topological chaos, the authors define a phase space X = J₁,N × Bᴺ, where J₁,N denotes the set of cell indices and Bᴺ the set of binary vectors of length N. They construct a global iteration function G_f that shifts the strategy and applies a Boolean function f to the selected cell. A custom metric d = d_e + d_s is introduced: d_e measures the Hamming distance between cell states, while d_s quantifies the difference between two strategies using a decimal‑expansion scheme (the first k terms equal ⇒ distance < 10⁻ᵏ). Under this metric, G_f is shown to be continuous.

The paper then proves that G_f satisfies Devaney’s chaos criteria for a broad class of Boolean functions f, focusing on the simple logical negation f₀. Regularity is established by constructing, for any point and any ε, a periodic point arbitrarily close to it, using a finite prefix of the strategy and fixing the cell state thereafter. Transitivity is demonstrated by explicitly constructing a strategy that drives the system from any open set A to any other open set B in a finite number of iterations, thereby proving that the set of transitive functions T is non‑empty (indeed, f₀ belongs to T). Consequently, (X, G_f) is chaotic in the Devaney sense.

Having secured the chaotic foundation, the authors describe how to build a hash function. The input message is first transformed into a 256‑bit vector E: each character is encoded in 7‑bit ASCII, a ‘1’ bit is appended, the length is added, the whole string is mirrored, duplicated, and truncated to a multiple of 512 bits; finally, exclusive‑OR reduction yields a 256‑bit block.

The strategy S is derived from the same normalized string D. D is split into 8‑bit blocks, each converted to decimal, then cyclically shifted and repeated six times to form an intermediate sequence u