A Guess and Determine Attack on the Elliptic Curve Discrete Logarithm Problem

Abstract: This paper is a continuation of our earlier work, in which, we described a Las Vegas algorithm to solve the elliptic curve discrete logarithm problem. The Las Vegas algorithm reduces the elliptic curve discrete logarithm problem to finding a zero minor in a matrix. Using intersection poset of a hyperplane arrangement, we develop an algorithm to find a zero minor in a rectangular matrix. Our methods are elementary. We discuss the complexity of our algorithm, success probability and provide implementation details. We also provide simulation details. Finding a zero minor in a matrix is also of independent interest.

• The paper's main contribution is a novel reduction of the ECDLP to a combinatorial search for zero minors in structured kernel matrices.
• It employs a two-stage algorithm that combines recursion with exhaustive search, leveraging hyperplane arrangements and parallel processing.
• The study provides detailed complexity analysis, identifying trade-offs between defect parameters and computational feasibility.

A Guess and Determine Attack on the Elliptic Curve Discrete Logarithm Problem

Introduction and Context

This paper presents the development and analysis of a novel algorithmic attack on the Elliptic Curve Discrete Logarithm Problem (ECDLP). The ECDLP is the foundational security assumption for numerous cryptosystems, including widely used signature schemes (e.g., Ed25519, secp256r1) and cryptocurrency protocols. While many historical attacks exist, this work builds on prior research introducing a Las Vegas approach for reducing the ECDLP to the search for a zero maximal minor in a specially constructed matrix derived from the group structure of the curve. The new algorithm adopts a "guess and determine" paradigm, integrating concepts from hyperplane arrangements and Grassmann/plücker coordinates, and structurally relates zero minors in such matrices to central intersections in hyperplane arrangements.

Problem Reduction: Kernel Matrices and Zero Minors

The principal reduction expresses the ECDLP $(P, Q)$ over an elliptic curve group of prime order $p$ as the search for an integer $m$ where $Q = mP$. The core methodology constructs, for $2\ell$ points (with $\ell = O(\log_2 p)$), a matrix $\mathcal{M}$ whose left-kernel $\mathcal{K}$, of size $\ell \times 2\ell$, relates directly to the ECDLP instance. Specifically, a maximal minor (an $\ell \times \ell$ submatrix) of $p$0 being singular (i.e., determinant zero) encodes a relation that allows explicit recovery of $p$1. The anti-diagonal canonical form for $p$2, splitting into dense and sparse halves, facilitates this reduction and simplifies combinatorial reasoning within the attack.

Algorithmic Framework

Recursion and Guessing Strategy

The attack proceeds in two main stages:

1. Recursion: Randomly select a subset $p$3 of $p$4 columns from the dense part of $p$5, reducing the search space by constructing a derived matrix $p$6 in anti-diagonal form.
2. Determine (Zero Minor Search): For a fixed defect parameter $p$7, enumerate combinations $p$8 of $p$9 columns to fix most of the structure, then exhaustively search for $m$0 additional columns whose presence induces a singular submatrix (i.e., a zero minor).

Hyperplane Arrangements and Signatures

The combinatorial essence of the method is captured via hyperplane arrangements: the left-kernel of each column of the matrix defines a hyperplane in the underlying vector space. The search for a singular minor then translates to identifying a central intersection in the arrangement—the presence of a non-trivial intersection among a specific subset of hyperplanes. To operationalize this, the algorithm computes "signatures" of intersections, assembling a signature matrix representing how different column subsets intersect with the fixed left-kernel. Detecting duplicate signatures (by hashing) corresponds to central intersections, and thus, to zero minors.

This approach transforms an exponential search into a feasible polynomial one, as the complexity is governed by the parameter $m$1, which controls the trade-off between enumeration space and computational practicality.

Complexity, Optimization, and Parallelism

The overall complexity is dominated by the binomial coefficients $m$2 and $m$3. Empirical results suggest that as $m$4 increases, both the probability of success and the computational burden rise due to the expansion of the candidate search space. The major algorithmic bottleneck is the efficient enumeration and left-kernel computation over small matrices—a task well-suited to parallelization. The implementation, leveraging SageMath and the Python multiprocessing library, demonstrates effective parallel scaling by distributing choices of $m$5 across cores.

Gray code combinatorial generation and rank-one kernel update techniques are proposed but not explicitly implemented in the experiments, which could potentially further reduce overhead.

Success Probability Analysis

The probability of successfully finding a zero minor, and thus solving an ECDLP instance, relies on both the structure of the group and the chosen defect parameter. Theoretical analysis and simulation (via integer-only code that abstracts away matrix algebra) estimate the probability as: $m$6 Tables of empirical probabilities over varying field sizes and defects exhibit the expected exponential growth in success rate as the defect increases; practicality is retained for small to moderate $m$7 (roughly up to $m$8 for moderate field sizes), beyond which computational resources become prohibitive.

The reduction in success probability for large field sizes or small defect values reflects the inherent combinatorial explosion. This reinforces the balance that must be maintained between search space and computational tractability.

Implementation Details

The attack is implemented for prime fields, and the matrix framework supports arbitrary elliptic curves. The two-stage algorithm—size reduction via random selection followed by parallelized minor search—was validated on moderate parameter sizes, principally as a proof of concept. Key implementation choices, such as operating exclusively on anti-diagonal forms and exploiting the structural sparsity of the kernels, significantly improved performance and memory efficiency. Nevertheless, significant limitations remain in scaling to cryptographically relevant sizes due to the exponential nature of the underlying combinatorics.

Theoretical and Practical Implications

The methodological innovation lies in the translation of a cryptanalytic task (ECDLP) to purely linear-algebraic and combinatorial geometry terms—specifically, the search for zero minors and central intersections in hyperplane arrangements. This reframing not only brings new tools (e.g., Grassmann/Plücker coordinates, geometric lattices) to bear on the cryptanalytic landscape but also reveals structural connections to problems in matrix completion and learning theory.

Practically, the attack's efficiency is currently bounded, as expected for a non-quantum ECDLP attack, by polynomial growth only for limited parameter ranges. The approach does not, in its current form, threaten standard elliptic curve cryptography with large prime order groups. However, the algorithmic directions—particularly parallel and structurally-aware combinatorial search—may inform both applied attacks on weak instances (e.g., small curves, certain algebraic structures) and general theoretical investigation into the hardness assumptions underlying elliptic curve cryptography.

Future Directions

• Complexity Analysis: The precise scaling of the required defect parameter $m$9 as a function of $Q = mP$0 remains open and is central to bounding the algorithm's practical impact.
• Algorithmic Enhancements: Incorporation of advanced combinatorial generation (e.g., Gray codes), memory-efficient kernel update, and two-level parallelization could extend the empirical reach of the method.
• Structural Weaknesses in Subclasses: Investigation of specific elliptic curve classes, or field characteristics, where the assumptions of general position in the arrangement may fail more frequently, potentially leading to weakened security.
• Connections to Matrix Theory and Learning: The interplay with matrix completion and learning theory (e.g., low-rank recovery, minor augmentation) could generate cross-pollination of techniques and new theoretical insights.

Conclusion

The paper provides a rigorous combinatorial-algebraic reduction of the ECDLP to a controlled search for zero minors in specially structured matrices, via guess-and-determine and central hyperplane arrangements methodology. While no immediate cryptanalytic threat emerges for large parameter sizes, the intricate connection between geometric combinatorics and discrete logarithm hardness invites further investigation. The approach represents a nontrivial contribution to the structural study of ECDLP and highlights new avenues for both cryptanalytic and algebraic research.

For further details and explicit pseudocode, see the full exposition in "A Guess and Determine Attack on the Elliptic Curve Discrete Logarithm Problem" (2607.09814).