secp256k1

The elliptic curve Bitcoin uses for all signatures (ECDSA + Schnorr). Defined by SECG (SEC 2: Recommended Elliptic Curve Domain Parameters).

Parameters

p = 2^256 - 2^32 - 977   (the field prime)
a = 0
b = 7
G = (0x79BE667E_F9DCBBAC_55A06295_CE870B07_029BFCDB_2DCE28D9_59F2815B_16F81798,
     0x483ADA77_26A3C465_5DA4FBFC_0E1108A8_FD17B448_A6855419_9C47D08F_FB10D4B8)
n = 0xFFFFFFFF_FFFFFFFF_FFFFFFFF_FFFFFFFE_BAAEDCE6_AF48A03B_BFD25E8C_D0364141
h = 1   (cofactor)

The curve equation: y^2 = x^3 + 7 (mod p).

Point operations

Point addition: P + Q with chord-and-tangent rule mod p.
Scalar multiplication: k*G is the public key for private scalar k.
Point doubling: 2P via tangent line.
Point at infinity: identity for the group.

Key encoding

Public key (full)

Uncompressed (65 bytes): 0x04 || x || y (32-byte each).
Compressed (33 bytes): 0x02 || x if y even, 0x03 || x if y odd.

x-only (BIP340, Taproot)

32 bytes: just x. Implicit even-y convention.
Used in Schnorr signatures and Taproot output keys.

Why secp256k1 (not P-256)?

No NSA-derived seed in the parameter generation (P-256 has the controversial Dual_EC seeds and the ad-hoc base point).
Endomorphism λ = ... allows fast multiplication: any point P has a known (λ·P) derivable cheaply, used in optimized scalar mult.
Sufficient security: ~128-bit classical security against discrete-log attacks, parity to symmetric-128. See the quantum caveat below.

Quantum outlook (as of September 2026)

The ~128-bit figure above is classical security. Shor's algorithm solves the discrete log on an elliptic curve in polynomial time, so no secp256k1 key size helps against a cryptographically relevant quantum computer; the exposed surface is any published public key (a Taproot x-only key sits in the scriptPubKey, a reused-address ECDSA key sits in a spent input). None of this changes how you sign or verify today.

Two consensus-track drafts frame the response. Both are Draft in the BIPs repo as of September 2026, with no activation parameters:

BIP 360, "Pay-to-Merkle-Root (P2MR)" (Beast, Heilman, Foxen Duke; Consensus soft fork, assigned 2024-12-18, spec version 0.12.1) — P2TR with the key path removed, so the output publishes no EC point. It targets long exposure attacks only and deliberately picks no post-quantum signature algorithm, deferring that to a later proposal.
BIP 361, "Post Quantum Migration and Legacy Signature Sunset" (Lopp et al.; Consensus soft fork, Type: Informational, assigned 2026-02-11) — a phased sunset that first disallows sending to quantum-vulnerable outputs, then encumbers ECDSA/Schnorr spends with a quantum-safe rescue protocol.

Detail lives in schnorr/SKILL.md (BIP340 key exposure) and ../../protocol/proposals/SKILL.md (output types, migration policy, rescue mechanics). libsecp256k1 ships no post-quantum module.

Critical operations to never mess up

Modular inverse: Use Fermat's little theorem a^(p-2) mod p or extended Euclidean. Constant-time required for security.
Scalar generation: 32 bytes of CSPRNG, reduce mod n. Reject zero. On a signing device, show the raw device-generated entropy and mix in independent user entropy (dice/coin) — both practices were reinforced by the July 2026 Coldcard advisory below.
Range check pubkey: x in [0, p), y satisfies curve equation.
Co-factor: cofactor h = 1, so any non-zero point on curve is in the prime-order subgroup. No subgroup check needed.

Recommended library: libsecp256k1

The C library maintained at github.com/bitcoin-core/secp256k1:

Constant-time everywhere by default.
Optimized scalar mult using endomorphism.
Modular APIs: secp256k1_ec_*, secp256k1_ecdsa_*, secp256k1_xonly_* / secp256k1_keypair_* (extrakeys), secp256k1_schnorrsig_* (BIP340), secp256k1_ecdh_*, secp256k1_recovery_*, secp256k1_musig_* (MuSig2, BIP327 — the C prefix has no 2), secp256k1_ellswift_* (BIP324), secp256k1_silentpayments_* (BIP352). All modules are configure-time options; as of master (September 2026) every one defaults to enabled except recovery, which is off by default.
There is no FROST module upstream (checked against master, September 2026). secp256k1-frost is a third-party fork, not a libsecp256k1 module — see frost/SKILL.md.
Bindings: rust-secp256k1, python-bitcoinlib (via cffi), secp256k1-node, etc.

Avoid rolling your own EC math in production. Subtle errors here are catastrophic.

Current release (as of September 2026)

v0.8.0 (2026-08-03). Modules are exactly: ecdh, ellswift, extrakeys, musig, recovery, schnorrsig, silentpayments.

v0.8.0 added the silentpayments module (BIP352 send/receive; needs full transaction data, no light-client scanning yet) and secp256k1_context_set_sha256_compression. secp256k1_ellswift_xdh now rejects secret keys >= the curve order instead of silently reducing them.
v0.8.0 removed secp256k1_context_no_precomp (use secp256k1_context_static), the secp256k1_schnorrsig_sign alias (use secp256k1_schnorrsig_sign32) and the SECP256K1_GNUC_PREREQ macro. The two symbols break old callers at link time; the macro breaks them at compile time.
v0.7.0 (2025-07-21) made the static context pointers const and removed the deprecated secp256k1_ec_privkey_* aliases (use secp256k1_ec_seckey_*); CMake minimum rose to 3.22 and CMake builds stopped being experimental. v0.7.1 (2026-01-26) is maintenance: wider stack-secret clearing, a parallel unit-test framework, and a fix for an x86_64 assembly feature check that could silently fall back to the slower C implementation.
rust-secp256k1 (crate secp256k1) is at 0.33.1 on crates.io (2026-08-29). The 0.32 series added MuSig2 support and removed the Secp256k1 context from the public API in favour of a global context — a breaking change for every caller.

Common bugs

Using a non-CT mod-inverse → side-channel key leak.
Accepting public keys without curve-equation check → invalid-curve attack on (P + Q) operations.
Confusing n (group order) with p (field prime) — they differ.
Generating private keys from a buggy RNG (e.g., Math.random() in JS) → catastrophic. Shipped-hardware case: Coldcard's July 2026 advisory — a build/link error routed rng_get() to MicroPython's Yasmarang PRNG from March 2021 on (~40 bits of effective search space on Mk2/Mk3, ~72 on Mk4/Mk5/Q, both preliminary). Fixed in 5.6.0 (Mk4/Mk5), 1.5.0Q (Q), 4.2.0 (Mk2/Mk3), Edge 6.6.0X/6.6.0QX; updating does not repair an already-generated seed. Details in ../../hardware/coldcard/SKILL.md.

bitcoin-secp256k1