ECDSA on secp256k1

ECDSA is the original Bitcoin signature scheme. Used for legacy P2PKH, P2SH, and SegWit v0 (P2WPKH/P2WSH). Replaced by Schnorr in Taproot.

Sign (private key d, message hash h)

1. Generate nonce k ∈ [1, n-1].
2. R = k * G
3. r = R.x mod n.    if r == 0, retry.
4. s = k^(-1) * (h + r*d)  mod n.   if s == 0, retry.
5. Apply low-s normalization: if s > n/2, s = n - s.   (BIP146)
6. Output (r, s).

The signature is (r, s) ∈ Z_n × Z_n.

Verify (public key Q, message hash h, signature (r, s))

1. Check r, s ∈ [1, n-1].
2. w = s^(-1) mod n.
3. u1 = h * w mod n,  u2 = r * w mod n.
4. P = u1*G + u2*Q.
5. Verify P.x mod n == r.

Deterministic k (RFC6979)

CRITICAL: never reuse k. If you sign two different messages with the same k, the private key leaks.

d * (s1 - s2) = k * (h2 - h1)  (mod n)
→ d = (s1 - s2)^(-1) * (h2 - h1) * k  (mod n)

RFC6979 derives k deterministically from (d, h) via HMAC-SHA256. Bitcoin libraries default to RFC6979 — never accept "user-provided" k unless you implement extra randomness mixing (RFC6979 + extra entropy pattern, used by some hardware wallets to defeat fault attacks).

DER encoding (for tx witness)

Bitcoin Script uses ASN.1 DER:

30 LL 02 LR <r> 02 LS <s>

Where:

30 = SEQUENCE tag.
LL = total length following.
02 = INTEGER tag.
LR / LS = length of r / s.
<r> / <s> = big-endian, with leading 0x00 byte if high bit set (DER strict).

Followed by 1-byte sighash flag in tx witness:

<DER signature> <sighash byte>

Total length typically 71-72 bytes.

Strict DER (BIP66)

Bitcoin Core enforces strict DER since 2015 to defeat malleability:

No extra leading bytes.
Length fields exactly minimal.
Integers must be DER-canonical.

Non-DER signatures are rejected at script execution.

Low-s (BIP146)

Both (r, s) and (r, n-s) verify against the same message and key. This is malleability. BIP146 enforces s ≤ n/2 as standard policy (strictly enforced in Tapscript and as standard in segwit; mempool rejects but consensus historically accepts both forms).

Recoverable signatures

A 65-byte ECDSA encoding <header byte> r s lets verifier recover the public key from the signature alone:

header byte = 27 + recovery_id + (4 if compressed pubkey).
recovery_id ∈ [0, 3]: encodes which of (R, -R, R+n, -R+n) to use.

Used by:

BIP137 message signing.
Lightning gossip messages (signed announcements).
Some Ethereum-bridge tools (ecrecover).

Not used in raw tx witness (regular DER is more compact for that).

Common errors

Reusing k → key leak. Never accept an external nonce.
Skipping low-s normalization → mempool rejection.
Including a high-s signature in standardness-strict mempools → tx goes nowhere.
Off-by-one in DER length encoding (the most common DER bug).
Confusing nonce k with private key d in code reviews.

bitcoin-ecdsa