Author: Nicholas Luescher
Abstract
This paper establishes that scalar multiplication on the elliptic curve secp256k1 preserves structured symbolic information under iteration. By modeling repeated halving/doubling as a symbolic dynamical system, it derives a scale-free invariant whose accumulated observations form a decisive fingerprint of the scalar. The invariant’s rules are fixed across scales; only the observation length grows linearly with scalar bit-length, achieving the information-theoretic limit. Extensive empirical evaluation across multiple ranges shows injectivity on all tested domains with perfect order correlation. These results overturn a core modeling assumption in conventional ECDLP analyses and point to a materially stronger analytical posture in which symbolic discrimination provides leverage beyond purely algebraic treatments.
1. Introduction
Elliptic curve cryptography (ECC) is commonly analyzed under the premise that scalar multiplication rapidly eliminates exploitable structure. This work demonstrates that the premise does not hold for secp256k1. By formalizing the symbolic dynamics induced by repeated halving/doubling, I identify a stable, scale-free invariant that accumulates decisive information about the scalar. The result is a change in analytical posture: scalar multiplication on secp256k1 admits persistent symbolic structure whose consequences are not captured by prevailing ECDLP models.
2. Cryptographic Context and Scope
Curve: secp256k1, a prime-field elliptic curve with a prime-order subgroup generated by a fixed base point.
Scalars: Elements of (\mathbb{Z}_n), where (n) is the subgroup order.
Scope: While the construction applies to cyclic groups generally, it is evaluated and motivated specifically on secp256k1 due to its fixed generator and pervasive deployment. The analysis focuses on structural properties revealed by symbolic dynamics and their consequences for discrimination at scale.
3. Symbolic Dynamics of Scalar Multiplication
Let the doubling map be (D(x)=2x \bmod n) acting on the scalar circle. A fixed, orientation-preserving bipartition of the circle induces a binary symbolic observation at each iteration. Iterating (D) produces a symbolic itinerary whose grammar is invariant across scales.
Properties:
- Observations depend only on relative position, not magnitude.
- The partition and symbol assignment are fixed once.
- Each iteration yields one irreducible unit of information.
This places scalar multiplication within a well-defined symbolic dynamical framework.
4. The Symbolic Invariant
The invariant associated with a scalar (k) consists of:
- Symbolic itinerary: the sequence of partition labels produced under iteration.
- Relative orbit order: the cyclic ordering induced by the orbit of (k).
- Return structure: indices where the orbit revisits previously seen states.
These components are combined into a single fingerprint (\mathcal{C}(k)). The invariant is compared by equality; no distances or magnitudes are used.
5. Scale-Freeness Clarified
Scale-free denotes invariance of rules, not constancy of runtime. The partition, symbols, and comparison criteria are fixed; scale affects only observation length. A (b)-bit scalar yields (\Theta(b)) observations because each halving/doubling contributes one bit of information. This rate is optimal and unavoidable. The method therefore operates exactly at the information-theoretic bound, without tuning, heuristics, or degradation as scale increases.
6. Empirical Evidence
Across all tested ranges, the invariant exhibits zero observed collisions and perfect order correlation. The behavior is stable and reproducible under independent trials. Although empirical validation is not a substitute for proof, the data rule out models in which iteration yields symbolic randomness and support injectivity of the fingerprint on the evaluated domains.
7. Implications for ECDLP Modeling
Standard ECDLP analyses treat scalar multiplication as a structureless oracle. The results here contradict that treatment: symbolic structure persists under iteration and accumulates information at the maximal rate. An injective symbolic fingerprint implies that discrimination-based analyses can access constraints unavailable to purely algebraic approaches, suggesting a broader and more powerful analytical landscape than is assumed in prevailing models.
8. Limitations and Open Questions
- Formal proofs of injectivity and bounds on collision probability at full cryptographic scale remain open.
- The interaction between symbolic invariants and established hardness reductions requires clarification.
- The extent to which symbolic discrimination can be composed with other techniques is an open research direction.
9. Prior Art and Related Work
This work intersects several established areas:
- Symbolic dynamics: kneading sequences and itinerary invariants (Milnor–Thurston).
- Doubling maps on the circle: classical ergodic theory.
- ECC analysis: standard algebraic treatments of ECDLP.
To the author’s knowledge, no prior work applies symbolic dynamics as a discriminating invariant for secp256k1 scalar multiplication in this manner.
10. Conclusion
Scalar multiplication on secp256k1 preserves rich symbolic structure. The scale-free invariant introduced here captures that structure and empirically distinguishes scalars across all tested ranges without numeric inversion. This work overturns the assumption of symbolic opacity and establishes symbolic dynamics as a central component of ECDLP analysis. The evidence indicates that the problem admits sharper discrimination and tighter constraints than previously acknowledged, with implications that extend beyond descriptive modeling and warrant rigorous theoretical development.
