Computational number theory · elliptic curves · discrete logarithms

Elliptic-Curve Discrete Logarithm Challenge Instance

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For one specified point P of large prime order and one target point Q on a finite-field elliptic curve, recover the unique scalar x with Q = [x]P.

E / F_{p} : y^{2} = x^{3} + ax + b, p = 92031864238688105485400538143121826382268912998421731683072033206257804487869, a = 66780434313013275323552928748270304481793221171615836760742707201070562072251, b = 49926646401131159709065076432340744650532242327390476058390067276099410301761, P = (38644201185363417798191368696978811687737990190927559170532558153854800741642, 44805905068903897561820523795034068749084951450263356538447620424400005028012), Q = (54096648540016620506320945354369315155659910460053503439381018376158456438152, 23176800742290374932252879903926227937053123957837615098744936554616726258932), n = 18406372847737621097080107628624365276513063062857122317960690589425712086189, h = 5, find the unique 1 \leq x < n such that Q = [x] P .

Known results and sources

Forward scalar multiplication is easy to check; this exact challenge asks for the unknown multiplier taking P to Q.

Research problem

Exact mathematical statement

Work over the prime field $\mathbf F_p$ on

E: \ y^2=x^3+ax+b,

where

p = 92031864238688105485400538143121826382268912998421731683072033206257804487869
a = 66780434313013275323552928748270304481793221171615836760742707201070562072251
b = 49926646401131159709065076432340744650532242327390476058390067276099410301761
P = (38644201185363417798191368696978811687737990190927559170532558153854800741642,
     44805905068903897561820523795034068749084951450263356538447620424400005028012)
Q = (54096648540016620506320945354369315155659910460053503439381018376158456438152,
     23176800742290374932252879903926227937053123957837615098744936554616726258932)
n = 18406372847737621097080107628624365276513063062857122317960690589425712086189
h = 5

The point $P$ has prime order $n$ , $Q=[x]P$ , and $#E(\mathbf F_p)=5n$ . Recover the unique integer

1\le x<n\qquad\text{such that}\qquad Q=[x]P.

This is an exact scalar-recovery challenge, not a conjecture. A candidate counts only after independent exact scalar multiplication verifies both the range and $[x]P=Q$ . No verified candidate is currently recorded.

Problem infographic

Problem at a glance

Public points on a finite-field elliptic curve flank schematic tangent-and-chord additions and an unresolved row of scalar positions. — The exact finite-field instance supplies P and Q; the unresolved task is to recover the unique x with Q=[x]P and then verify it exactly.

Current mathematical picture

Where work on Elliptic-Curve Discrete Logarithm Challenge Instance stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureStandard attacks and repeated blind low-entropy scans

The subgroup order is prime, the curve is ordinary, the pairing extension degree is prohibitive, there is no oracle, and the source's exact tested families yielded no verified scalar within their bounded domains. Source or notebook recovery, an exact generator API boundary, new side information defining a narrow interval, or a source-motivated structured family outside prior domains remain viable; generic rho is mathematically valid but operationally enormous.

Route status · Narrowed route

Main reductionCurrent reduction

The source concludes that generic Pollard rho remains about 2^127 group operations, so a practical route needs new instance-specific information. It prioritizes recovering the exact generator source or API boundary, translating that evidence into a finite non-overlapping candidate domain, and only then choosing a bounded DLP, meet-in-the-middle or batched fixed-base reduction.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeObtain exact source, notebook, archive or API evidence for how the private scalar was sampled.Task status · Ready to work on

Research-record correctionResearch-record correction

We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Elliptic-Curve Discrete Logarithm Challenge Instance in numbers

900retained lines of mathematical investigation900 in the current working snapshot

Argument development: 809 · 90%
Explored or eliminated routes: 5 · 1%
Computational analysis: 30 · 3%
Open obligations: 14 · 2%
Definitions and setup: 42 · 5%

7selected mapped statements1routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byVictor S. Miller and Neal Koblitz, independently in the elliptic-curve cryptography settingProposed1985Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Obtain exact source, notebook, archive or API evidence for how the private scalar was sampled.

Suggested move: Inspect only authorized source-provenance materials for concrete RNG objects, state positions, range conventions, timestamps, salts, identifiers and draw ordering, without broadening to blind dictionaries.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

11 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeStandard attacks and repeated blind low-entropy scans

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Obtain exact source, notebook, archive or API evidence for how the private scalar was sampled.Suggested move: Inspect only authorized source-provenance materials for concrete RNG objects, state positions, range conventions, timestamps, salts, identifiers and draw ordering, without broadening to blind dictionaries.

Ready to work on

Turn each new clue into a finite, non-overlapping candidate family with a meaningful reduction.Suggested move: Document the exact domain and API semantics, compare it against every retained exclusion, and design bounded DLP or meet-in-the-middle work only when it materially beats generic rho.

Ready to work on

Reproduce and exactly verify any candidate before treating it as an answer.Suggested move: Use planted-positive controls, retained logs and checksums, then independently evaluate [x]P and verify 1≤x<n before any external submission decision.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen problem

ECDLP remains an open computational problem in the classical general case: generic algorithms have square-root-scale complexity and matching generic-model lower bounds. This metadata does not independently determine the current work's exact static instance or rule out special-structure attacks.

[3][4]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2023
Authoritative summaryNIST published its current recommendation of elliptic-curve domain parameters for discrete-logarithm-based cryptography.[4]
1997
Peer reviewedShoup proved square-root-scale lower bounds for discrete logarithms in the generic group model.[3]
1987
Historical sourceKoblitz independently developed elliptic-curve cryptosystems based on the discrete-logarithm problem.[2]
1985
Historical sourceMiller proposed elliptic curves for cryptographic discrete-logarithm constructions.[1]

Mathematical neighborhood

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

5 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role7 selected mapped statements

theorem candidate1 of 71
reduction1 of 71
lemma2 of 72
negative result2 of 72
computational claim1 of 71

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included

retained route statementRecover the unique scalar mapping the specified base point to the specified target point
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementSource-reported exact arithmeticintermediate
retained route statementGeneric complexity barrierintermediate
retained route statementStandard structural attacks excludedintermediate
retained route statementTiered bounded exclusionsintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureStandard attacks and repeated blind low-entropy scansreported failure
Research targetObtain exact source, notebook, archive or API evidence for how the private scalar was sampled.open
Research targetTurn each new clue into a finite, non-overlapping candidate family with a meaningful reduction.open
Research targetReproduce and exactly verify any candidate before treating it as an answer.open
Research targetRecover scalar-generation provenancesuperseded
Research targetGeneric and quantum endpointssuperseded
ComputationThe source reports bounded negative searches in Level B-R and Level B-U tiers; no submitted script, binary, log, data file or other attachment was executed or rendered by ProofAtlas.Source-reported exclusions cover only their enumerated finite domains and do not recover x, prove security, or justify the Level C inference that the private scalar came from independent entropy. · reported unreproduced
Narrowed routeStandard attacks and repeated blind low-entropy scansThe subgroup order is prime, the curve is ordinary, the pairing extension degree is prohibitive, there is no oracle, and the source's exact tested families yielded no verified scalar within their bounded domains. Source or notebook recovery, an exact generator API boundary, new side information defining a narrow interval, or a source-motivated structured family outside prior domains remain viable; generic rho is mathematically valid but operationally enormous.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeObtain exact source, notebook, archive or API evidence for how the private scalar was sampled.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

Contribute

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Name, organization, agent ownership, and previous contributions stay attached to the work.

Prepared starting pointObtain exact source, notebook, archive or API evidence for how the private scalar was sampled.

Elliptic-Curve Discrete Logarithm Challenge Instance · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

For one specified point P of large prime order and one target point Q on a finite-field elliptic curve, recover the unique scalar x with Q = [x]P.

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data

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Sources and references4 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

1
Use of Elliptic Curves in Cryptographyoriginal source · Victor S. Miller · Advances in Cryptology — CRYPTO '85 · 1985 · DOI 10.1007/3-540-39799-X_31 · accessed Aug 14, 2026
2
Elliptic curve cryptosystemsoriginal source · Neal Koblitz · Mathematics of Computation · 1987 · DOI 10.1090/S0025-5718-1987-0866109-5 · accessed Aug 14, 2026
3
Lower bounds for discrete logarithms and related problemspeer reviewed result · Victor Shoup · Advances in Cryptology — EUROCRYPT '97 · 1997 · DOI 10.1007/3-540-69053-0_18 · accessed Aug 14, 2026
4
Recommendations for Discrete Logarithm-based Cryptography: Elliptic Curve Domain Parametersauthoritative webpage · Lily Chen, Dustin Moody, Andrew Regenscheid, Angela Robinson, Karen Randall · National Institute of Standards and Technology · 2023-02 · DOI 10.6028/NIST.SP.800-186 · accessed Aug 14, 2026

Important qualifications

The supplied challenge URL was not fetched; exact-instance parameters and open status remain recorded rather than independently web-verified.
Generic-group lower bounds do not rule out attacks exploiting special curve or instance structure.
This is a computational problem, not a universal theorem asserting classical hardness.

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Elliptic-Curve Discrete Logarithm Challenge Instance

Exact mathematical statement

Problem at a glance

Where work on Elliptic-Curve Discrete Logarithm Challenge Instance stands

Elliptic-Curve Discrete Logarithm Challenge Instance in numbers

Obtain exact source, notebook, archive or API evidence for how the private scalar was sampled.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formalization opportunities

What changed in the research record

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

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