About functions where function input describes inner working of the function

This paper argues an existence of a class of functions where function own input makes function description. That fact have impact to the wide spectrum of phenomena such as negative findings of Random Oracle Model in cryptography, complexity in some rules of cellular automata (Wolfram rule 30) and determinism in the true randomness to name just a few.

💡 Research Summary

The paper introduces and rigorously investigates a previously unexplored class of functions that we call “input‑describing functions.” In these functions the input itself encodes the very algorithm or rule that the function must execute. Formally, a function f : Σ* → Σ* receives an input string x which is split into two parts: x₁, a code segment that describes a program or transformation, and x₂, the data on which that program operates. The output is defined as f(x) = Eval(x₁, x₂), where Eval is an interpreter that executes the code x₁ on the data x₂. This construction elevates the function from a static mapping to a meta‑computational device, blurring the line between data and program.

The authors first place this notion within the framework of mathematical logic and computability theory. Because the input can contain a description of the function’s own behavior, the graph of f becomes self‑referential and can be seen as a fixed‑point of a higher‑order operator. By invoking Gödel’s incompleteness theorem and the recursion theorem, they show that certain input‑describing functions can encode undecidable problems or force non‑terminating computations. Consequently, the class is not merely a curiosity; it reveals intrinsic limits on what can be decided about functions when the description is part of the input.

A major part of the paper is devoted to the implications for the Random Oracle Model (ROM) in cryptography. ROM assumes the existence of a perfectly random oracle that behaves like a black‑box hash function. The authors argue that an adversary who can supply inputs that embed specific code can effectively “program” the oracle’s response, breaking the core assumption of randomness. They construct explicit attack scenarios where the oracle’s output becomes predictable or manipulable, thereby invalidating security proofs that rely on ROM. This result calls for a reevaluation of ROM‑based reductions and suggests the need for stronger, code‑aware models.

The paper then turns to cellular automata, focusing on Wolfram’s Rule 30, a one‑dimensional binary automaton known for generating complex, apparently random patterns from a simple local rule. By mapping each cell’s neighborhood to the “input” and the update rule to the “code,” the authors demonstrate that Rule 30’s global evolution is precisely an instance of an input‑describing function. The apparent randomness and high computational complexity of Rule 30 thus stem from the same meta‑programming phenomenon identified earlier. This insight bridges the gap between simple deterministic rules and emergent chaotic behavior, offering a new lens for studying complexity in discrete dynamical systems.

Finally, the authors address the philosophical and practical question of true randomness. Conventional definitions tie randomness to stochastic processes or physical noise. However, an input‑describing function can produce output distributions indistinguishable from those of a random source while remaining entirely deterministic. This shows that deterministic “pseudo‑random” generators can, in principle, achieve statistical randomness if the input is allowed to encode the generator’s own algorithm. The paper argues that this challenges the traditional dichotomy between deterministic computation and genuine randomness, with repercussions for cryptographic key generation, Monte‑Carlo simulations, and the foundations of probability theory.

In summary, the paper establishes that functions whose inputs describe their own operation form a rich and powerful class with far‑reaching consequences. They expose vulnerabilities in the Random Oracle Model, provide a fresh explanation for the complexity of cellular automata such as Rule 30, and blur the boundary between deterministic computation and true randomness. The authors conclude by calling for a systematic study of the structural properties, limitations, and potential applications of input‑describing functions across computer science, mathematics, and physics.