A 256-bit elliptic-curve discrete logarithm

Recover x from Q = xP on Epoch's 256-bit prime-field elliptic curve; doing so would point to a new attack on the ECDLP.

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State the problem, from FrontierMath: Open Problems

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	1	+# A 256-bit elliptic-curve discrete logarithm
	2	+
	3	+Elliptic-curve cryptography is secure only if, given points $P$ and $Q = xP$ on a curve over a large prime field, recovering the integer $x$ is infeasible. Epoch has published one such instance at full cryptographic size. Solving it in reasonable time would point to a new attack on the elliptic-curve discrete logarithm problem (ECDLP), or to a hidden weakness in this particular curve.
	4	+
	5	+## Problem
	6	+
	7	+Let $p$ be the 256-bit prime below and $E\colon y^2 = x^3 + ax + b$ the curve over $\mathbb{F}_p$ with the given $a, b$. The point $P \in E(\mathbb{F}_p)$ has prime order $n$, a 254-bit prime, and $\#E(\mathbb{F}_p) = 5n$. The point $Q$ equals $xP$ for a secret integer $1 \le x < n$, where $xP$ is $P$ added to itself $x$ times.
	8	+
	9	+```
	10	+p = 92031864238688105485400538143121826382268912998421731683072033206257804487869
	11	+a = 66780434313013275323552928748270304481793221171615836760742707201070562072251
	12	+b = 49926646401131159709065076432340744650532242327390476058390067276099410301761
	13	+Px = 38644201185363417798191368696978811687737990190927559170532558153854800741642
	14	+Py = 44805905068903897561820523795034068749084951450263356538447620424400005028012
	15	+Qx = 54096648540016620506320945354369315155659910460053503439381018376158456438152
	16	+Qy = 23176800742290374932252879903926227937053123957837615098744936554616726258932
	17	+n = 18406372847737621097080107628624365276513063062857122317960690589425712086189
	18	+```
	19	+
	20	+Task. Find $x$.
	21	+
	22	+## Background
	23	+
	24	+Koblitz and Miller independently proposed basing public-key cryptography on elliptic curves in the mid-1980s [1, 2]. Its security rests on the presumed hardness of this problem for well-chosen curves over prime fields. A generic square-root method such as Pollard's rho [3] would need on the order of $\sqrt{n} \approx 2^{127}$ group operations here, far beyond any computer. Research continues on algebraic attacks, for example index calculus with summation polynomials, surveyed by Galbraith and Gaudry [4].
	25	+
	26	+## What is known
	27	+
	28	+- The numbers above are consistent: $p$ and $n$ are prime, $P$ and $Q$ lie on $E$, $nP = nQ = O$, and $5n$ lies in the Hasse interval around $p + 1$ (our check).
	29	+- Epoch chose the instance to be hard and names two risks: that a brute-force search might succeed in the allowed time, or that the curve has unexpected structure making it easier than intended.
	30	+- No solution has been reported.
	31	+
	32	+## What would count as progress
	33	+
	34	+- The integer $x$. Verification is one scalar multiplication: check that $xP = Q$.
	35	+- Epoch scores submissions with its own verifier, which accepts the decimal value of $x$.
	36	+- What matters most is the method: an attack that works for this curve and explains why, or a demonstration that the instance has an exploitable weakness.
	37	+
	38	+## References
	39	+
	40	+1. N. Koblitz, Elliptic curve cryptosystems, Math. Comp. 48 (1987), 203–209. [doi:10.1090/S0025-5718-1987-0866109-5](https://doi.org/10.1090/S0025-5718-1987-0866109-5)
	41	+2. V. S. Miller, Use of elliptic curves in cryptography, CRYPTO '85, Lecture Notes in Comput. Sci. 218 (1986), 417–426. [doi:10.1007/3-540-39799-X_31](https://doi.org/10.1007/3-540-39799-X_31)
	42	+3. J. M. Pollard, Monte Carlo methods for index computation (mod $p$), Math. Comp. 32 (1978), 918–924. [doi:10.1090/S0025-5718-1978-0491431-9](https://doi.org/10.1090/S0025-5718-1978-0491431-9)
	43	+4. S. D. Galbraith and P. Gaudry, Recent progress on the elliptic curve discrete logarithm problem, Des. Codes Cryptogr. 78 (2016), 51–72. [doi:10.1007/s10623-015-0146-7](https://doi.org/10.1007/s10623-015-0146-7) ([ePrint 2015/1022](https://eprint.iacr.org/2015/1022))
	44	+
	45	+## Source
	46	+
	47	+Imported from [FrontierMath: Open Problems, elliptic-curve-discrete-logarithm](https://epoch.ai/frontiermath/open-problems/elliptic-curve-discrete-logarithm), curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.