research/dkg-ecdsa: Key generation

script/research/dkg-ecdsa/dkg_ecdsa.sage

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+# https://eprint.iacr.org/2021/060.pdf
+from hashlib import sha256
+
+t = 3  # Threshold
+n = 5  # Participants
+
+# secp256k1
+p = 2^256-2^32-977
+q = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
+Fp = GF(p)
+Fq = GF(q)
+Ep = EllipticCurve(Fp, [0, 7])
+Ep.set_order(q)
+
+# Base point
+g = Ep([
+    0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798,
+    0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8,
+])
+
+# Key Generation
+# ==============
+xi = []
+Xi = []
+Xi_proofs = []
+for i in range(n):
+    x_i = Fq.random_element()
+    X_i = g * x_i
+    xi.append(x_i)
+    Xi.append(X_i)
+
+    # Schnorr proof of knowledge
+    k = Fq.random_element()
+    R = k * g
+    hasher = sha256()
+    hasher.update(str(X_i.xy()))
+    hasher.update(str(R.xy()))
+    e = Fq(int(hasher.hexdigest(), 16))
+    s = k + e * x_i
+    Xi_proofs.append((R, s))
+    # Verifier computes: e = hash(X || r),  s*g == R+e*X