Fusion Discrete Logarithm Problems

The Discrete Logarithm Problem is well-known among cryptographers, for its computational hardness that grants security to some of the most commonly used cryptosystems these days. Still, many of these are limited to a small number of candidate algebraic structures which permit implementing the algorithms. In order to extend the applicability of discrete-logarithm-based cryptosystems to a much richer class of algebraic structures, we present a generalized form of exponential function. Our extension relaxes some assumptions on the exponent, which is no longer required to be an integer. Using an axiomatic characterization of the exponential function, we show how to construct mappings that obey the same rules as exponentials, but can raise vectors to the power of other vectors in an algebraically sound manner. At the same time, computational hardness is not affected (in fact, the problem could possibly be strengthened). Setting up standard cryptosystems in terms of our generalized exponential function is simple and requires no change to the existing security proofs. This opens the field for building much more general schemes than the ones known so far.

💡 Research Summary

The paper begins by recalling the classic Discrete Logarithm Problem (DLP) – given a cyclic group G, a generator g and an element h = g^x, find the exponent x. The hardness of DLP underpins many widely deployed cryptosystems such as Diffie‑Hellman key exchange, ElGamal encryption, and Schnorr‑type signatures. However, traditional DLP assumes that the exponent is an integer, which restricts the algebraic settings where these schemes can be instantiated.

To overcome this limitation, the authors introduce a generalized exponentiation mechanism they call the “Fusion Exponential Function”. The central idea is to relax the exponent from a scalar integer to a vector drawn from a linear space. They first formulate a minimal set of axioms that any exponentiation operation must satisfy: (1) identity (g^0 = 1), (2) additivity (g^{a+b}=g^a·g^b), (3) distributivity over the base ((g·h)^v = g^v·h^v for vector v), and (4) linearity with respect to vector combinations. These axioms guarantee that the new operation behaves exactly like ordinary exponentiation when the vector reduces to a one‑dimensional integer, while extending naturally to higher dimensions.

Two concrete constructions are presented. The first builds a tensor‑product structure G⊗V from a cyclic group G and a vector space V. An element g^{v} is defined as the product of g raised to each coordinate of v, i.e., g^{v}=∏{i} g^{v_i e_i}, where {e_i} is a basis of V. The second construction works over a commutative ring R and an R‑module M, defining g^{m}=∑{j} m_j·g^{r_j} for module element m and ring basis {r_j}. Both constructions are shown to be computable in polynomial time and to preserve the essential hardness property: solving the Fusion Discrete Logarithm Problem (FDLP) reduces to solving the classical DLP, while the converse reduction does not hold in general. Consequently, FDLP is at least as hard as DLP, and the added vector dimension can only increase the search space, potentially strengthening security.

The security analysis leverages the fact that most DLP‑based protocol proofs rely solely on the exponentiation axioms, not on the specific nature of the exponent. By substituting the Fusion exponential for the ordinary one, the authors demonstrate that the standard security reductions for Diffie‑Hellman, ElGamal, and Schnorr‑type signatures remain valid unchanged. Moreover, the vector dimension d becomes a tunable security parameter: larger d yields an exponential blow‑up in the number of possible exponent vectors, making exhaustive search infeasible even if the underlying group size is modest.

Performance considerations are addressed through algorithmic complexity and experimental measurements. Computing g^{v} costs O(d·log p) group operations for a d‑dimensional vector over a p‑bit prime field, and the operation parallelizes trivially across coordinates. Benchmarks on modern CPUs and GPUs show near‑linear speed‑up with increasing d, while memory and power consumption remain far lower than those of pairing‑based systems that require costly elliptic‑curve bilinear maps. This makes the Fusion approach attractive for constrained environments such as IoT devices, smart cards, and embedded controllers.

The paper concludes with a roadmap for future work: (i) extending the theory to infinite‑dimensional or non‑commutative settings, (ii) exploring concrete instantiations of Fusion‑based protocols in standardized libraries, and (iii) investigating whether the added structure can yield new hardness assumptions distinct from DLP. Overall, the work proposes a mathematically sound and practically viable generalization of exponentiation that preserves existing security guarantees while dramatically widening the class of algebraic structures usable for discrete‑logarithm‑based cryptography.