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What is an elliptic curve?
An elliptic curve, in cryptography, is a set of points defined by an equation such as y^2 = x^3 + 7, worked out over a huge but finite range of whole numbers. Its arithmetic is easy one way and impractical to reverse, which lets a public key be shared while the private key stays secret.
What it means for you
Bitcoin and Ethereum both use the curve secp256k1, so one private key is valid on both networks and on chains that copy them; a leaked key or seed phrase exposes every chain where it was used. The curve is public and fixed: what protects you is that your private key is random and kept secret.
How it works
SEC 2 defines secp256k1 as the curve y^2 = x^3 + 7 over a prime field of roughly 2^256 elements. Points on the curve can be added, and adding a fixed generator point G to itself d times gives the public key Q = dG. Computing Q from d is fast; recovering d from Q is the discrete logarithm problem on which the security rests. NIST's SP 800-186 rates secp256k1 at a 128-bit security strength and allows it for blockchain-related applications. NIST also notes that elliptic-curve signatures are vulnerable to Shor's algorithm on a cryptographically relevant quantum computer.
Sources: SEC 2: Recommended Elliptic Curve Domain Parameters, NIST SP 800-186: Elliptic Curve Domain Parameters, NIST IR 8547 (initial public draft): Transition to Post-Quantum Cryptography Standards · checked 4 October 2026
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