Verification of a Bitcoin signature from https://github.com/bitcoinjs/bitcoinjs-lib/blob/master/src/ecdsa.js

verifysig.js

function verify (hash, signature, Q) {
  typeforce(types.tuple(
    types.Hash256bit,
    types.ECSignature,
    types.ECPoint
  ), arguments)

  var n = secp256k1.n
  var G = secp256k1.G

  var r = signature.r
  var s = signature.s

  // 1.4.1 Enforce r and s are both integers in the interval [1, n − 1]
  if (r.signum() <= 0 || r.compareTo(n) >= 0) return false
  if (s.signum() <= 0 || s.compareTo(n) >= 0) return false

  // 1.4.2 H = Hash(M), already done by the user
  // 1.4.3 e = H
  var e = BigInteger.fromBuffer(hash)

  // Compute s^-1
  var sInv = s.modInverse(n)

  // 1.4.4 Compute u1 = es^−1 mod n
  //               u2 = rs^−1 mod n
  var u1 = e.multiply(sInv).mod(n)
  var u2 = r.multiply(sInv).mod(n)

  // 1.4.5 Compute R = (xR, yR)
  //               R = u1G + u2Q
  var R = G.multiplyTwo(u1, Q, u2)

  // 1.4.5 (cont.) Enforce R is not at infinity
  if (secp256k1.isInfinity(R)) return false

  // 1.4.6 Convert the field element R.x to an integer
  var xR = R.affineX

  // 1.4.7 Set v = xR mod n
  var v = xR.mod(n)

  // 1.4.8 If v = r, output "valid", and if v != r, output "invalid"
  return v.equals(r)
}