Every Bitcoin transaction and every Signal message leans on an elliptic curve to prove who signed what. They are not the same curve, and the difference is not cosmetic. Bitcoin and Ethereum run on secp256k1, a curve Certicom published in 1999 with almost no marketing behind it. Signal, WireGuard, modern SSH keys, and the X25519 key exchange in TLS 1.3 run on Curve25519, a curve Daniel J. Bernstein built in 2005 specifically to be fast and hard to implement wrong. Both give you roughly 128-bit security. Neither is going away. This comparison pulls apart the math, the benchmarks, and the deployment reality so you can pick the right one for your project instead of copying whatever the last tutorial used. What Curve25519 and secp256k1 Actually Are Curve25519 is a specific elliptic curve defined over the prime field 2^255 – 19, described by Bernstein as a Diffie-Hellman function built for speed and safety at the same time. Its equation, y^2 = x^3 + 486662x^2 + x, is written in Montgomery form, a shape chosen for one purpose: fast, constant-time scalar multiplication (Wikipedia). The signature scheme built on the same field, Ed25519, uses a birationally equivalent twisted Edwards curve so the same underlying arithmetic powers both key exchange (X25519) and signing (Ed25519). secp256k1 is a different animal. It is a short Weierstrass curve, y^2 = x^3 + 7, specified in Certicom’s SEC 2 standard alongside a family of NIST-style curves (SEC 2: Recommended Elliptic Curve Domain Parameters). Satoshi Nakamoto picked it for Bitcoin’s ECDSA signatures in 2008 or 2009, and Ethereum inherited the choice. secp256k1 was not designed around a specific implementation strategy the way Curve25519 was. Its constants are simple (a = 0, b = 7), which some developers read as easier to audit than the more opaque parameter generation used