Bitcoin’s security rests on the hardness of the elliptic-curve discrete logarithm problem on the secp256k1 curve used for ECDSA and BIP340 Schnorr signatures.
In a detailed analysis posted on X on October 11, 2026, Rob Hamilton of AnchorWatch argued that recent online discussion about possible classical breaks has often lacked specificity. He catalogued five mathematical research directions that would need to advance for a genuine reduction in the cost of recovering a private scalar (d) from a public key Q=dG.
Hamilton, who is not a cryptographer but holds a bachelor’s degree in mathematics and works extensively with Bitcoin-related math, framed the discussion as a red-team exercise rather than a claim of imminent breakage. Known classical methods, such as Pollard’s rho and baby-step/giant-step, require on the order of 2128 group operations for the curve’s roughly 256-bit group order. That figure is an estimate grounded in published algorithms, not a formal lower bound that rules out every possible classical approach.
The Discrete Logarithm Problem and Existing Security Bounds
Both ECDSA and Schnorr signatures on secp256k1 depend on the one-way relationship between a private scalar and its corresponding public-key point. Computing the public key from the private key is efficient; the reverse operation is the elliptic-curve discrete logarithm problem (ECDLP). The Bitcoin Core library libsecp256k1 implements these operations with high assurance, and BIP340 defines the Schnorr variant used in Taproot.
Victor Shoup’s 1997 result established that generic algorithms—those that treat group elements as opaque labels without using their coordinate representation—must perform square-root work in the worst case for groups of prime order. Pollard’s rho meets this scale. The bound does not apply to algorithms that exploit the visible coordinates of points on secp256k1 or other special structure of the curve.
Whether any non-generic classical method can improve substantially on the 2128 estimate at cryptographic sizes remains an open question, as surveyed in work such as Galbraith and Gaudry’s 2016 review of progress on the elliptic-curve discrete logarithm problem.
Five Open Research Directions for Potential Classical Shortcuts
Hamilton identified five lines of inquiry that have appeared in the academic literature.
First, index-calculus style approaches using summation polynomials. Igor Semaev introduced these polynomials in a 2004 paper to encode when points with given x-coordinates sum to the identity. They convert a geometric question into a system of polynomial equations. The polynomials exist and can be computed, but efficiently finding enough useful relations over secp256k1’s large prime field—enough to outperform square-root methods—has not been demonstrated. Published proposals for prime-field curves have not produced a practical attack.
Second, a matrix-based approach that searches for zero minors. A 2021 paper by Ansari Abdullah, Ayan Mahalanobis, and Vivek M. Mallick reduces ECDLP to finding a zero determinant in a specially constructed matrix; such a minor can yield coefficients that recover the discrete logarithm. Experiments solved instances up to roughly group order 250. An August 2026 revision by Mahalanobis describes a newer search that is polynomial for a fixed defect parameter, yet after accounting for success probability the algorithm remains no better than exhaustive search and “poses no current security risk to ECC,” as detailed in his arXiv preprint.
Third, the gap between Pollard’s rho and Shoup’s generic lower bound. Any algorithm that uses the actual field coordinates or the curve equation y2=x3+7 sits outside the generic model. No such method is known to yield a large asymptotic improvement for secp256k1.
Fourth, special structure of the curve itself. secp256k1 has j-invariant 0 and an efficiently computable endomorphism that the Gallant–Lambert–Vanstone method already exploits for constant-factor speedups in scalar multiplication; libsecp256k1 uses this for verification. Transferring the discrete logarithm via an efficiently computable isogeny (“cousin curve”) over the same field would preserve the group order, so it cannot simply move the problem to a curve with small factors. A larger structural advantage remains a research question.
Fifth, lifting the finite-field problem to characteristic zero. Strategies that seek corresponding points and curves over the rationals or number fields face obstacles of height control, preservation of relations, and size blow-up. Analyses such as that of Huang, Kueh, and Tan highlight the difficulty of lifting many random points while keeping arithmetic size manageable. This line is distinct from quantum period-finding algorithms.
None of these directions currently supplies a justified new classical complexity estimate that makes key recovery affordable at Bitcoin parameters.
Expert Context, AI Progress, and Industry Responses
The post appeared amid broader discussion triggered by Ethereum Foundation researcher Justin Drake’s October 7 call for “bunker mode”—a controlled migration of assets to addresses whose public keys have never been revealed. Drake cited rapid AI-assisted mathematical progress, including an OpenAI release of mathematical results, and suggested elliptic curves might prove more vulnerable than hash functions.
Coinbase’s head of cryptography, Yehuda Lindell, responded that there is “no evidence whatsoever pointing to a break of decades old hardness assumptions like elliptic curve cryptography” and described the warning as “the very definition of FUD.” Lindell noted that elliptic-curve cryptography has seen essentially no asymptotic improvement in 30 years beyond known generic methods, and that historically hash functions have been broken more often than elliptic curves.
Hamilton’s analysis was developed in part through a podcast episode of Magic Internet Math with Brian Hirschfield, available on YouTube, in which the pair walked through the same open problems.
Ledger CTO Charles Guillemet has separately stated that recent AI mathematical results do not indicate an imminent practical ECDSA break and that any collapse of asymmetric cryptography would affect TLS and banking systems before Bitcoin, as covered in related reporting.
Requirements for a Genuine Break and Ongoing Scrutiny
Hamilton stressed that a discovered pattern, a theorem, or even a checked formalization in a proof assistant such as Lean is not the same as an affordable, reliable attack on actual secp256k1 instances. A complete break would require a method that applies to the concrete curve parameters, succeeds with adequate probability, and reduces the end-to-end cost of key recovery below the level of known square-root algorithms.
Proving or disproving P=NP would have profound consequences for computational cryptography in general, yet would not automatically certify or refute the security of this particular curve; average-case hardness for keys generated in the ordinary way is a separate requirement.
The discussion sits alongside longer-term quantum concerns. NIST guidance and various Bitcoin improvement proposals continue to address post-quantum migration timelines, while the classical hardness assumptions underlying current signatures have not been shown to have weakened.
Hamilton concluded that red-teaming the specific open problems is more productive than vague assertions about AI-driven breakage, and invited corrections from specialists. As of October 12, 2026, no public cryptanalysis has demonstrated a classical algorithm that recovers secp256k1 private keys from public keys at the scales used by Bitcoin.
Also read: Ledger CryptoBilis Hack Explained: $93M Stolen, Wallets Tampered
