One-Way Functions Linked to Computational Limits and State Generation.

The bedrock of secure communication relies upon the existence of one-way functions – mathematical problems easily computed in one direction, but computationally intractable to reverse. Despite decades of research, proving their existence remains an open challenge in theoretical computer science. New work explores this fundamental question by connecting it to the concept of ‘lossy reductions’ – transformations that simplify problems while preserving key characteristics. Researchers from Sorbonne Université (Pouria Fallahpour and Alex B. Grilo), alongside Garazi Muguruza of QuSoft at the University of Amsterdam, and Mahshid Riahinia from DIENS at École Normale Supérieure, detail these findings in a paper entitled ‘Cryptography from Lossy Reductions: Towards OWFs from ETH, and Beyond’. Their analysis establishes a relationship between the existence of one-way functions and the efficiency of these lossy reductions, with implications for the limits of computation under the widely held Exponential Time Hypothesis (ETH).

Recent research establishes a direct relationship between the existence of one-way functions (OWFs) and the properties of lossy reductions, offering new constraints on computational complexity. The findings demonstrate that either OWFs exist, or the runtime of any lossy reduction for a ‘promise problem’ is bounded by the infimum of the runtime of worst-case solvers for that problem.

A promise problem is a computational task where a ‘yes’ or ‘no’ answer is guaranteed; this allows for a precise formulation of the computational challenge and its implications for reduction complexity. Lossy reductions are mappings from one computational problem to another where solutions to the original problem do not uniquely determine solutions to the reduced problem. The research focuses on ‘mildly-lossy’ reductions, requiring only effective function on sparse, uniform input distributions, simplifying analysis and broadening applicability.

Researchers demonstrate that the existence of a mildly-lossy reduction implies the existence of one-way state generators – functions that produce unpredictable sequences. This partial extension suggests potential for further exploration within cryptographic contexts.

The study also investigates weak fine-grained OWFs – a more restricted class of one-way functions – and shows that considering these functions leads to improved runtime bounds. By framing the problem as a specific instance, the results provide sufficient conditions for the existence of (fine-grained) OWFs, contingent upon the validity of the Exponential Time Hypothesis (ETH). The ETH posits that solving certain problems (like 3-SAT) requires exponential time, even with optimal algorithms.

Conversely, the absence of (fine-grained) OWFs implies limitations on instance compression and randomization, again under the ETH framework. This bidirectional relationship strengthens the connection between OWFs and established complexity assumptions, providing a more robust foundation for cryptographic security.

Furthermore, the research demonstrates that worst-case to average-case Karp reductions and randomized encodings fall under the category of mildly-lossy reductions. This connection strengthens the understanding of reduction types and computational hardness, allowing for a tightening of previously established runtime bounds. These findings provide more precise limits on the efficiency of computational reductions, with implications for the design and analysis of cryptographic primitives.