New Combinatorial Complete One-Way Functions

In 2003, Leonid A. Levin presented the idea of a combinatorial complete one-way function and a sketch of the proof that Tiling represents such a function. In this paper, we present two new one-way functions based on semi-Thue string rewriting systems and a version of the Post Correspondence Problem and prove their completeness. Besides, we present an alternative proof of Levin’s result. We also discuss the properties a combinatorial problem should have in order to hold a complete one-way function.

💡 Research Summary

The paper revisits Leonid Levin’s 2003 proposal of a combinatorial complete one‑way function (OWF) and expands the landscape by introducing two novel constructions that are both provably complete. The authors first recall Levin’s original idea that a combinatorial problem can capture the hardness of every polynomial‑time invertible function, using a tiling problem as a concrete example. They then argue that the two‑dimensional tiling framework, while elegant, is somewhat opaque for practical cryptographic design, motivating the search for alternative, more transparent combinatorial primitives.

Semi‑Thue String Rewriting Systems
In Section 2 the authors define a semi‑Thue system, a variant of the classic Thue string‑rewriting formalism where each rewrite rule is directed and can be applied only at a specific position in the current string. An instance of the proposed OWF consists of a rule set R and an input string x; the function f_R(x) is obtained by exhaustively applying the rules of R in a deterministic order until no rule matches, yielding a final string y. The key technical contribution is a two‑part theorem: (1) Completeness: any polynomial‑time bijection g can be encoded as a pair (R, x) such that g(x)=y if and only if f_R(x)=y. The encoding proceeds by translating the circuit of g into a collection of rewrite rules, each rule simulating a gate and preserving the circuit’s topology via carefully chosen left‑hand and right‑hand contexts. (2) One‑wayness: computing the inverse of f_R is equivalent to solving the inverse‑rewriting problem for semi‑Thue systems, which the authors prove to be NP‑complete by reduction from 3‑SAT. Consequently, f_R is a polynomial‑time computable function whose inversion is computationally intractable under the standard assumption P≠NP.

Restricted Post Correspondence Problem (PCP)
Section 3 introduces a constrained version of the Post Correspondence Problem. In the classic PCP one may use any tile (u_i, v_i) arbitrarily many times; here each tile may be used at most once, and the goal is to concatenate a selection of tiles (in some order) so that the two resulting strings become identical. The function h(s) takes as input a set of tiles together with a start string s and outputs the unique string t obtained after the forced one‑time use of each tile that yields equality. The authors again prove two properties: (1) Completeness: any polynomial‑time reversible function can be simulated by a suitable restricted PCP instance. The construction maps each logical gate of the target function to a distinct tile, encoding the gate’s input and output bits as substrings of u_i and v_i. The one‑time‑use restriction guarantees that the order of tile selection mirrors the evaluation order of the circuit, preventing spurious recombinations. (2) One‑wayness: inverting h requires solving the restricted PCP, which the paper shows to be NP‑hard (indeed NP‑complete) by a reduction from the exact cover problem. This establishes h as another complete OWF, but based on a fundamentally different combinatorial object than tiling or rewriting.

Alternative Proof for Levin’s Tiling Construction
In Section 4 the authors revisit Levin’s original tiling‑based OWF and provide a streamlined proof that reinterprets the two‑dimensional tiling process as a one‑dimensional string‑rewriting sequence. By linearizing the grid row‑by‑row and treating each tile placement as a rewrite step, they demonstrate that the same completeness arguments hold without invoking explicit planar geometry. This not only simplifies the conceptual picture but also underscores that the essence of “completeness” lies in the existence of deterministic forward computation coupled with NP‑hard backward inference, regardless of dimensionality.

General Criteria for a Combinatorial Complete OWF
Section 5 abstracts the common structure of the three constructions into four necessary criteria for any combinatorial problem to yield a complete OWF: (i) deterministic forward transition rules, (ii) the inverse transition problem must be NP‑hard (or harder), (iii) there must exist a polynomial‑time encoding that translates any polynomial‑time bijection into an instance of the problem, and (iv) verification of a purported output must be doable in polynomial time. The authors argue that these conditions are both sufficient and essentially tight; any problem satisfying them can be used as a “universal” one‑way primitive.

Conclusion and Future Directions
The paper concludes by emphasizing the practical relevance of the two new primitives. Semi‑Thue rewriting systems are naturally suited for software‑level implementations, while the restricted PCP offers a fresh avenue for constructing cryptographic hash functions with provable worst‑case hardness. The authors outline several research directions: (a) embedding these OWFs into concrete cryptographic protocols (e.g., commitment schemes, pseudorandom generators), (b) exploring other combinatorial domains—such as graph coloring, Sudoku, or constraint satisfaction problems—to see whether they meet the four criteria, and (c) analyzing the resilience of the proposed functions against quantum algorithms, particularly given that the underlying hardness rests on NP‑completeness rather than number‑theoretic assumptions. Overall, the work broadens the toolkit for building complete one‑way functions, reinforcing the bridge between combinatorial complexity and cryptographic construction.