Merge #15826: Pure python EC

b67978529a Add comments to Python ECDSA implementation (John Newbery)
8c7b9324ca Pure python EC (Pieter Wuille)

Pull request description:

  This removes the dependency on OpenSSL for the interaction tests, by providing a pure-Python
  toy implementation of secp256k1.

ACKs for commit b67978:
  jnewbery:
    utACK b67978529ad02fc2665f2362418dc53db2e25e17

Tree-SHA512: 181445eb08b316c46937b80dc10aa50d103ab1fdddaf834896c0ea22204889f7b13fd33cbcbd00ddba15f7e4686fe0d9f8e8bb4c0ad0e9587490c90be83966dc

test/functional/feature_assumevalid.py

@@ -32,7 +32,7 @@ Start three nodes:
 import time
 
 from test_framework.blocktools import (create_block, create_coinbase)
-from test_framework.key import CECKey
+from test_framework.key import ECKey
 from test_framework.messages import (
     CBlockHeader,
     COutPoint,
@@ -105,9 +105,9 @@ class AssumeValidTest(BitcoinTestFramework):
         self.blocks = []
 
         # Get a pubkey for the coinbase TXO
-        coinbase_key = CECKey()
+        coinbase_key = ECKey()
-        coinbase_key.set_secretbytes(b"horsebattery")
+        coinbase_key.generate()
-        coinbase_pubkey = coinbase_key.get_pubkey()
+        coinbase_pubkey = coinbase_key.get_pubkey().get_bytes()
 
         # Create the first block with a coinbase output to our key
         height = 1

test/functional/feature_block.py

@@ -7,7 +7,7 @@ import copy
 import struct
 
 from test_framework.blocktools import create_block, create_coinbase, create_tx_with_script, get_legacy_sigopcount_block
-from test_framework.key import CECKey
+from test_framework.key import ECKey
 from test_framework.messages import (
     CBlock,
     COIN,
@@ -85,9 +85,9 @@ class FullBlockTest(BitcoinTestFramework):
         self.bootstrap_p2p()  # Add one p2p connection to the node
 
         self.block_heights = {}
-        self.coinbase_key = CECKey()
+        self.coinbase_key = ECKey()
-        self.coinbase_key.set_secretbytes(b"horsebattery")
+        self.coinbase_key.generate()
-        self.coinbase_pubkey = self.coinbase_key.get_pubkey()
+        self.coinbase_pubkey = self.coinbase_key.get_pubkey().get_bytes()
         self.tip = None
         self.blocks = {}
         self.genesis_hash = int(self.nodes[0].getbestblockhash(), 16)
@@ -481,7 +481,7 @@ class FullBlockTest(BitcoinTestFramework):
             tx.vin.append(CTxIn(COutPoint(b39.vtx[i].sha256, 0), b''))
             # Note: must pass the redeem_script (not p2sh_script) to the signature hash function
             (sighash, err) = SignatureHash(redeem_script, tx, 1, SIGHASH_ALL)
-            sig = self.coinbase_key.sign(sighash) + bytes(bytearray([SIGHASH_ALL]))
+            sig = self.coinbase_key.sign_ecdsa(sighash) + bytes(bytearray([SIGHASH_ALL]))
             scriptSig = CScript([sig, redeem_script])
 
             tx.vin[1].scriptSig = scriptSig
@@ -1225,7 +1225,7 @@ class FullBlockTest(BitcoinTestFramework):
             tx.vin[0].scriptSig = CScript()
             return
         (sighash, err) = SignatureHash(spend_tx.vout[0].scriptPubKey, tx, 0, SIGHASH_ALL)
-        tx.vin[0].scriptSig = CScript([self.coinbase_key.sign(sighash) + bytes(bytearray([SIGHASH_ALL]))])
+        tx.vin[0].scriptSig = CScript([self.coinbase_key.sign_ecdsa(sighash) + bytes(bytearray([SIGHASH_ALL]))])
 
     def create_and_sign_transaction(self, spend_tx, value, script=CScript([OP_TRUE])):
         tx = self.create_tx(spend_tx, 0, value, script)

test/functional/test_framework/key.py

@@ -1,216 +1,386 @@
-# Copyright (c) 2011 Sam Rushing
+# Copyright (c) 2019 Pieter Wuille
-"""ECC secp256k1 OpenSSL wrapper.
+# Distributed under the MIT software license, see the accompanying
+# file COPYING or http://www.opensource.org/licenses/mit-license.php.
+"""Test-only secp256k1 elliptic curve implementation
 
-WARNING: This module does not mlock() secrets; your private keys may end up on
+WARNING: This code is slow, uses bad randomness, does not properly protect
-disk in swap! Use with caution!
+keys, and is trivially vulnerable to side channel attacks. Do not use for
+anything but tests."""
+import random
 
-This file is modified from python-bitcoinlib.
+def modinv(a, n):
-"""
+    """Compute the modular inverse of a modulo n
 
-import ctypes
+    See https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Modular_integers.
-import ctypes.util
+    """
-import hashlib
+    t1, t2 = 0, 1
+    r1, r2 = n, a
+    while r2 != 0:
+        q = r1 // r2
+        t1, t2 = t2, t1 - q * t2
+        r1, r2 = r2, r1 - q * r2
+    if r1 > 1:
+        return None
+    if t1 < 0:
+        t1 += n
+    return t1
 
-ssl = ctypes.cdll.LoadLibrary(ctypes.util.find_library ('ssl') or 'libeay32')
+def jacobi_symbol(n, k):
+    """Compute the Jacobi symbol of n modulo k
 
-ssl.BN_new.restype = ctypes.c_void_p
+    See http://en.wikipedia.org/wiki/Jacobi_symbol
-ssl.BN_new.argtypes = []
 
-ssl.BN_bin2bn.restype = ctypes.c_void_p
+    For our application k is always prime, so this is the same as the Legendre symbol."""
-ssl.BN_bin2bn.argtypes = [ctypes.c_char_p, ctypes.c_int, ctypes.c_void_p]
+    assert k > 0 and k & 1, "jacobi symbol is only defined for positive odd k"
+    n %= k
+    t = 0
+    while n != 0:
+        while n & 1 == 0:
+            n >>= 1
+            r = k & 7
+            t ^= (r == 3 or r == 5)
+        n, k = k, n
+        t ^= (n & k & 3 == 3)
+        n = n % k
+    if k == 1:
+        return -1 if t else 1
+    return 0
 
-ssl.BN_CTX_free.restype = None
+def modsqrt(a, p):
-ssl.BN_CTX_free.argtypes = [ctypes.c_void_p]
+    """Compute the square root of a modulo p when p % 4 = 3.
 
-ssl.BN_CTX_new.restype = ctypes.c_void_p
+    The Tonelli-Shanks algorithm can be used. See https://en.wikipedia.org/wiki/Tonelli-Shanks_algorithm
-ssl.BN_CTX_new.argtypes = []
 
-ssl.ECDH_compute_key.restype = ctypes.c_int
+    Limiting this function to only work for p % 4 = 3 means we don't need to
-ssl.ECDH_compute_key.argtypes = [ctypes.c_void_p, ctypes.c_int, ctypes.c_void_p, ctypes.c_void_p]
+    iterate through the loop. The highest n such that p - 1 = 2^n Q with Q odd
+    is n = 1. Therefore Q = (p-1)/2 and sqrt = a^((Q+1)/2) = a^((p+1)/4)
 
-ssl.ECDSA_sign.restype = ctypes.c_int
+    secp256k1's is defined over field of size 2**256 - 2**32 - 977, which is 3 mod 4.
-ssl.ECDSA_sign.argtypes = [ctypes.c_int, ctypes.c_void_p, ctypes.c_int, ctypes.c_void_p, ctypes.c_void_p, ctypes.c_void_p]
+    """
+    if p % 4 != 3:
+        raise NotImplementedError("modsqrt only implemented for p % 4 = 3")
+    sqrt = pow(a, (p + 1)//4, p)
+    if pow(sqrt, 2, p) == a % p:
+        return sqrt
+    return None
 
-ssl.ECDSA_verify.restype = ctypes.c_int
+class EllipticCurve:
-ssl.ECDSA_verify.argtypes = [ctypes.c_int, ctypes.c_void_p, ctypes.c_int, ctypes.c_void_p, ctypes.c_int, ctypes.c_void_p]
+    def __init__(self, p, a, b):
+        """Initialize elliptic curve y^2 = x^3 + a*x + b over GF(p)."""
+        self.p = p
+        self.a = a % p
+        self.b = b % p
 
-ssl.EC_KEY_free.restype = None
+    def affine(self, p1):
-ssl.EC_KEY_free.argtypes = [ctypes.c_void_p]
+        """Convert a Jacobian point tuple p1 to affine form, or None if at infinity.
 
-ssl.EC_KEY_new_by_curve_name.restype = ctypes.c_void_p
+        An affine point is represented as the Jacobian (x, y, 1)"""
-ssl.EC_KEY_new_by_curve_name.argtypes = [ctypes.c_int]
+        x1, y1, z1 = p1
+        if z1 == 0:
+            return None
+        inv = modinv(z1, self.p)
+        inv_2 = (inv**2) % self.p
+        inv_3 = (inv_2 * inv) % self.p
+        return ((inv_2 * x1) % self.p, (inv_3 * y1) % self.p, 1)
 
-ssl.EC_KEY_get0_group.restype = ctypes.c_void_p
+    def negate(self, p1):
-ssl.EC_KEY_get0_group.argtypes = [ctypes.c_void_p]
+        """Negate a Jacobian point tuple p1."""
+        x1, y1, z1 = p1
+        return (x1, (self.p - y1) % self.p, z1)
 
-ssl.EC_KEY_get0_public_key.restype = ctypes.c_void_p
+    def on_curve(self, p1):
-ssl.EC_KEY_get0_public_key.argtypes = [ctypes.c_void_p]
+        """Determine whether a Jacobian tuple p is on the curve (and not infinity)"""
+        x1, y1, z1 = p1
+        z2 = pow(z1, 2, self.p)
+        z4 = pow(z2, 2, self.p)
+        return z1 != 0 and (pow(x1, 3, self.p) + self.a * x1 * z4 + self.b * z2 * z4 - pow(y1, 2, self.p)) % self.p == 0
 
-ssl.EC_KEY_set_private_key.restype = ctypes.c_int
+    def is_x_coord(self, x):
-ssl.EC_KEY_set_private_key.argtypes = [ctypes.c_void_p, ctypes.c_void_p]
+        """Test whether x is a valid X coordinate on the curve."""
+        x_3 = pow(x, 3, self.p)
+        return jacobi_symbol(x_3 + self.a * x + self.b, self.p) != -1
 
-ssl.EC_KEY_set_conv_form.restype = None
+    def lift_x(self, x):
-ssl.EC_KEY_set_conv_form.argtypes = [ctypes.c_void_p, ctypes.c_int]
+        """Given an X coordinate on the curve, return a corresponding affine point."""
+        x_3 = pow(x, 3, self.p)
+        v = x_3 + self.a * x + self.b
+        y = modsqrt(v, self.p)
+        if y is None:
+            return None
+        return (x, y, 1)
 
-ssl.EC_KEY_set_public_key.restype = ctypes.c_int
+    def double(self, p1):
-ssl.EC_KEY_set_public_key.argtypes = [ctypes.c_void_p, ctypes.c_void_p]
+        """Double a Jacobian tuple p1
 
-ssl.i2o_ECPublicKey.restype = ctypes.c_void_p
+        See https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates - Point Doubling"""
-ssl.i2o_ECPublicKey.argtypes = [ctypes.c_void_p, ctypes.c_void_p]
+        x1, y1, z1 = p1
+        if z1 == 0:
+            return (0, 1, 0)
+        y1_2 = (y1**2) % self.p
+        y1_4 = (y1_2**2) % self.p
+        x1_2 = (x1**2) % self.p
+        s = (4*x1*y1_2) % self.p
+        m = 3*x1_2
+        if self.a:
+            m += self.a * pow(z1, 4, self.p)
+        m = m % self.p
+        x2 = (m**2 - 2*s) % self.p
+        y2 = (m*(s - x2) - 8*y1_4) % self.p
+        z2 = (2*y1*z1) % self.p
+        return (x2, y2, z2)
 
-ssl.EC_POINT_new.restype = ctypes.c_void_p
+    def add_mixed(self, p1, p2):
-ssl.EC_POINT_new.argtypes = [ctypes.c_void_p]
+        """Add a Jacobian tuple p1 and an affine tuple p2
 
-ssl.EC_POINT_free.restype = None
+        See https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates - Point Addition (with affine point)"""
-ssl.EC_POINT_free.argtypes = [ctypes.c_void_p]
+        x1, y1, z1 = p1
+        x2, y2, z2 = p2
+        assert(z2 == 1)
+        # Adding to the point at infinity is a no-op
+        if z1 == 0:
+            return p2
+        z1_2 = (z1**2) % self.p
+        z1_3 = (z1_2 * z1) % self.p
+        u2 = (x2 * z1_2) % self.p
+        s2 = (y2 * z1_3) % self.p
+        if x1 == u2:
+            if (y1 != s2):
+                # p1 and p2 are inverses. Return the point at infinity.
+                return (0, 1, 0)
+            # p1 == p2. The formulas below fail when the two points are equal.
+            return self.double(p1)
+        h = u2 - x1
+        r = s2 - y1
+        h_2 = (h**2) % self.p
+        h_3 = (h_2 * h) % self.p
+        u1_h_2 = (x1 * h_2) % self.p
+        x3 = (r**2 - h_3 - 2*u1_h_2) % self.p
+        y3 = (r*(u1_h_2 - x3) - y1*h_3) % self.p
+        z3 = (h*z1) % self.p
+        return (x3, y3, z3)
 
-ssl.EC_POINT_mul.restype = ctypes.c_int
+    def add(self, p1, p2):
-ssl.EC_POINT_mul.argtypes = [ctypes.c_void_p, ctypes.c_void_p, ctypes.c_void_p, ctypes.c_void_p, ctypes.c_void_p, ctypes.c_void_p]
+        """Add two Jacobian tuples p1 and p2
 
-# this specifies the curve used with ECDSA.
+        See https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates - Point Addition"""
-NID_secp256k1 = 714 # from openssl/obj_mac.h
+        x1, y1, z1 = p1
+        x2, y2, z2 = p2
+        # Adding the point at infinity is a no-op
+        if z1 == 0:
+            return p2
+        if z2 == 0:
+            return p1
+        # Adding an Affine to a Jacobian is more efficient since we save field multiplications and squarings when z = 1
+        if z1 == 1:
+            return self.add_mixed(p2, p1)
+        if z2 == 1:
+            return self.add_mixed(p1, p2)
+        z1_2 = (z1**2) % self.p
+        z1_3 = (z1_2 * z1) % self.p
+        z2_2 = (z2**2) % self.p
+        z2_3 = (z2_2 * z2) % self.p
+        u1 = (x1 * z2_2) % self.p
+        u2 = (x2 * z1_2) % self.p
+        s1 = (y1 * z2_3) % self.p
+        s2 = (y2 * z1_3) % self.p
+        if u1 == u2:
+            if (s1 != s2):
+                # p1 and p2 are inverses. Return the point at infinity.
+                return (0, 1, 0)
+            # p1 == p2. The formulas below fail when the two points are equal.
+            return self.double(p1)
+        h = u2 - u1
+        r = s2 - s1
+        h_2 = (h**2) % self.p
+        h_3 = (h_2 * h) % self.p
+        u1_h_2 = (u1 * h_2) % self.p
+        x3 = (r**2 - h_3 - 2*u1_h_2) % self.p
+        y3 = (r*(u1_h_2 - x3) - s1*h_3) % self.p
+        z3 = (h*z1*z2) % self.p
+        return (x3, y3, z3)
 
+    def mul(self, ps):
+        """Compute a (multi) point multiplication
+
+        ps is a list of (Jacobian tuple, scalar) pairs.
+        """
+        r = (0, 1, 0)
+        for i in range(255, -1, -1):
+            r = self.double(r)
+            for (p, n) in ps:
+                if ((n >> i) & 1):
+                    r = self.add(r, p)
+        return r
+
+SECP256K1 = EllipticCurve(2**256 - 2**32 - 977, 0, 7)
+SECP256K1_G = (0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798, 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8, 1)
 SECP256K1_ORDER = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
 SECP256K1_ORDER_HALF = SECP256K1_ORDER // 2
 
-# Thx to Sam Devlin for the ctypes magic 64-bit fix.
+class ECPubKey():
-def _check_result(val, func, args):
+    """A secp256k1 public key"""
-    if val == 0:
-        raise ValueError
-    else:
-        return ctypes.c_void_p (val)
-
-ssl.EC_KEY_new_by_curve_name.restype = ctypes.c_void_p
-ssl.EC_KEY_new_by_curve_name.errcheck = _check_result
-
-class CECKey():
-    """Wrapper around OpenSSL's EC_KEY"""
 
     def __init__(self):
-        self.k = ssl.EC_KEY_new_by_curve_name(NID_secp256k1)
+        """Construct an uninitialized public key"""
+        self.valid = False
 
-    def __del__(self):
+    def set(self, data):
-        if ssl:
+        """Construct a public key from a serialization in compressed or uncompressed format"""
-            ssl.EC_KEY_free(self.k)
+        if (len(data) == 65 and data[0] == 0x04):
-        self.k = None
+            p = (int.from_bytes(data[1:33], 'big'), int.from_bytes(data[33:65], 'big'), 1)
-
+            self.valid = SECP256K1.on_curve(p)
-    def set_secretbytes(self, secret):
+            if self.valid:
-        priv_key = ssl.BN_bin2bn(secret, 32, ssl.BN_new())
+                self.p = p
-        group = ssl.EC_KEY_get0_group(self.k)
+                self.compressed = False
-        pub_key = ssl.EC_POINT_new(group)
+        elif (len(data) == 33 and (data[0] == 0x02 or data[0] == 0x03)):
-        ctx = ssl.BN_CTX_new()
+            x = int.from_bytes(data[1:33], 'big')
-        if not ssl.EC_POINT_mul(group, pub_key, priv_key, None, None, ctx):
+            if SECP256K1.is_x_coord(x):
-            raise ValueError("Could not derive public key from the supplied secret.")
+                p = SECP256K1.lift_x(x)
-        ssl.EC_POINT_mul(group, pub_key, priv_key, None, None, ctx)
+                # if the oddness of the y co-ord isn't correct, find the other
-        ssl.EC_KEY_set_private_key(self.k, priv_key)
+                # valid y
-        ssl.EC_KEY_set_public_key(self.k, pub_key)
+                if (p[1] & 1) != (data[0] & 1):
-        ssl.EC_POINT_free(pub_key)
+                    p = SECP256K1.negate(p)
-        ssl.BN_CTX_free(ctx)
+                self.p = p
-        return self.k
+                self.valid = True
-
+                self.compressed = True
-    def set_privkey(self, key):
+            else:
-        self.mb = ctypes.create_string_buffer(key)
+                self.valid = False
-        return ssl.d2i_ECPrivateKey(ctypes.byref(self.k), ctypes.byref(ctypes.pointer(self.mb)), len(key))
-
-    def set_pubkey(self, key):
-        self.mb = ctypes.create_string_buffer(key)
-        return ssl.o2i_ECPublicKey(ctypes.byref(self.k), ctypes.byref(ctypes.pointer(self.mb)), len(key))
-
-    def get_privkey(self):
-        size = ssl.i2d_ECPrivateKey(self.k, 0)
-        mb_pri = ctypes.create_string_buffer(size)
-        ssl.i2d_ECPrivateKey(self.k, ctypes.byref(ctypes.pointer(mb_pri)))
-        return mb_pri.raw
-
-    def get_pubkey(self):
-        size = ssl.i2o_ECPublicKey(self.k, 0)
-        mb = ctypes.create_string_buffer(size)
-        ssl.i2o_ECPublicKey(self.k, ctypes.byref(ctypes.pointer(mb)))
-        return mb.raw
-
-    def get_raw_ecdh_key(self, other_pubkey):
-        ecdh_keybuffer = ctypes.create_string_buffer(32)
-        r = ssl.ECDH_compute_key(ctypes.pointer(ecdh_keybuffer), 32,
-                                 ssl.EC_KEY_get0_public_key(other_pubkey.k),
-                                 self.k, 0)
-        if r != 32:
-            raise Exception('CKey.get_ecdh_key(): ECDH_compute_key() failed')
-        return ecdh_keybuffer.raw
-
-    def get_ecdh_key(self, other_pubkey, kdf=lambda k: hashlib.sha256(k).digest()):
-        # FIXME: be warned it's not clear what the kdf should be as a default
-        r = self.get_raw_ecdh_key(other_pubkey)
-        return kdf(r)
-
-    def sign(self, hash, low_s = True):
-        # FIXME: need unit tests for below cases
-        if not isinstance(hash, bytes):
-            raise TypeError('Hash must be bytes instance; got %r' % hash.__class__)
-        if len(hash) != 32:
-            raise ValueError('Hash must be exactly 32 bytes long')
-
-        sig_size0 = ctypes.c_uint32()
-        sig_size0.value = ssl.ECDSA_size(self.k)
-        mb_sig = ctypes.create_string_buffer(sig_size0.value)
-        result = ssl.ECDSA_sign(0, hash, len(hash), mb_sig, ctypes.byref(sig_size0), self.k)
-        assert 1 == result
-        assert mb_sig.raw[0] == 0x30
-        assert mb_sig.raw[1] == sig_size0.value - 2
-        total_size = mb_sig.raw[1]
-        assert mb_sig.raw[2] == 2
-        r_size = mb_sig.raw[3]
-        assert mb_sig.raw[4 + r_size] == 2
-        s_size = mb_sig.raw[5 + r_size]
-        s_value = int.from_bytes(mb_sig.raw[6+r_size:6+r_size+s_size], byteorder='big')
-        if (not low_s) or s_value <= SECP256K1_ORDER_HALF:
-            return mb_sig.raw[:sig_size0.value]
         else:
-            low_s_value = SECP256K1_ORDER - s_value
+            self.valid = False
-            low_s_bytes = (low_s_value).to_bytes(33, byteorder='big')
-            while len(low_s_bytes) > 1 and low_s_bytes[0] == 0 and low_s_bytes[1] < 0x80:
-                low_s_bytes = low_s_bytes[1:]
-            new_s_size = len(low_s_bytes)
-            new_total_size_byte = (total_size + new_s_size - s_size).to_bytes(1,byteorder='big')
-            new_s_size_byte = (new_s_size).to_bytes(1,byteorder='big')
-            return b'\x30' + new_total_size_byte + mb_sig.raw[2:5+r_size] + new_s_size_byte + low_s_bytes
-
-    def verify(self, hash, sig):
-        """Verify a DER signature"""
-        return ssl.ECDSA_verify(0, hash, len(hash), sig, len(sig), self.k) == 1
-
-
-class CPubKey(bytes):
-    """An encapsulated public key
-
-    Attributes:
-
-    is_valid      - Corresponds to CPubKey.IsValid()
-    is_fullyvalid - Corresponds to CPubKey.IsFullyValid()
-    is_compressed - Corresponds to CPubKey.IsCompressed()
-    """
-
-    def __new__(cls, buf, _cec_key=None):
-        self = super(CPubKey, cls).__new__(cls, buf)
-        if _cec_key is None:
-            _cec_key = CECKey()
-        self._cec_key = _cec_key
-        self.is_fullyvalid = _cec_key.set_pubkey(self) != 0
-        return self
-
-    @property
-    def is_valid(self):
-        return len(self) > 0
 
     @property
     def is_compressed(self):
-        return len(self) == 33
+        return self.compressed
 
-    def verify(self, hash, sig):
+    @property
-        return self._cec_key.verify(hash, sig)
+    def is_valid(self):
+        return self.valid
 
-    def __str__(self):
+    def get_bytes(self):
-        return repr(self)
+        assert(self.valid)
+        p = SECP256K1.affine(self.p)
+        if p is None:
+            return None
+        if self.compressed:
+            return bytes([0x02 + (p[1] & 1)]) + p[0].to_bytes(32, 'big')
+        else:
+            return bytes([0x04]) + p[0].to_bytes(32, 'big') + p[1].to_bytes(32, 'big')
 
-    def __repr__(self):
+    def verify_ecdsa(self, sig, msg, low_s=True):
-        return '%s(%s)' % (self.__class__.__name__, super(CPubKey, self).__repr__())
+        """Verify a strictly DER-encoded ECDSA signature against this pubkey.
 
+        See https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm for the
+        ECDSA verifier algorithm"""
+        assert(self.valid)
+
+        # Extract r and s from the DER formatted signature. Return false for
+        # any DER encoding errors.
+        if (sig[1] + 2 != len(sig)):
+            return False
+        if (len(sig) < 4):
+            return False
+        if (sig[0] != 0x30):
+            return False
+        if (sig[2] != 0x02):
+            return False
+        rlen = sig[3]
+        if (len(sig) < 6 + rlen):
+            return False
+        if rlen < 1 or rlen > 33:
+            return False
+        if sig[4] >= 0x80:
+            return False
+        if (rlen > 1 and (sig[4] == 0) and not (sig[5] & 0x80)):
+            return False
+        r = int.from_bytes(sig[4:4+rlen], 'big')
+        if (sig[4+rlen] != 0x02):
+            return False
+        slen = sig[5+rlen]
+        if slen < 1 or slen > 33:
+            return False
+        if (len(sig) != 6 + rlen + slen):
+            return False
+        if sig[6+rlen] >= 0x80:
+            return False
+        if (slen > 1 and (sig[6+rlen] == 0) and not (sig[7+rlen] & 0x80)):
+            return False
+        s = int.from_bytes(sig[6+rlen:6+rlen+slen], 'big')
+
+        # Verify that r and s are within the group order
+        if r < 1 or s < 1 or r >= SECP256K1_ORDER or s >= SECP256K1_ORDER:
+            return False
+        if low_s and s >= SECP256K1_ORDER_HALF:
+            return False
+        z = int.from_bytes(msg, 'big')
+
+        # Run verifier algorithm on r, s
+        w = modinv(s, SECP256K1_ORDER)
+        u1 = z*w % SECP256K1_ORDER
+        u2 = r*w % SECP256K1_ORDER
+        R = SECP256K1.affine(SECP256K1.mul([(SECP256K1_G, u1), (self.p, u2)]))
+        if R is None or R[0] != r:
+            return False
+        return True
+
+class ECKey():
+    """A secp256k1 private key"""
+
+    def __init__(self):
+        self.valid = False
+
+    def set(self, secret, compressed):
+        """Construct a private key object with given 32-byte secret and compressed flag."""
+        assert(len(secret) == 32)
+        secret = int.from_bytes(secret, 'big')
+        self.valid = (secret > 0 and secret < SECP256K1_ORDER)
+        if self.valid:
+            self.secret = secret
+            self.compressed = compressed
+
+    def generate(self, compressed=True):
+        """Generate a random private key (compressed or uncompressed)."""
+        self.set(random.randrange(1, SECP256K1_ORDER).to_bytes(32, 'big'), compressed)
+
+    def get_bytes(self):
+        """Retrieve the 32-byte representation of this key."""
+        assert(self.valid)
+        return self.secret.to_bytes(32, 'big')
+
+    @property
+    def is_valid(self):
+        return self.valid
+
+    @property
+    def is_compressed(self):
+        return self.compressed
+
+    def get_pubkey(self):
+        """Compute an ECPubKey object for this secret key."""
+        assert(self.valid)
+        ret = ECPubKey()
+        p = SECP256K1.mul([(SECP256K1_G, self.secret)])
+        ret.p = p
+        ret.valid = True
+        ret.compressed = self.compressed
+        return ret
+
+    def sign_ecdsa(self, msg, low_s=True):
+        """Construct a DER-encoded ECDSA signature with this key.
+
+        See https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm for the
+        ECDSA signer algorithm."""
+        assert(self.valid)
+        z = int.from_bytes(msg, 'big')
+        # Note: no RFC6979, but a simple random nonce (some tests rely on distinct transactions for the same operation)
+        k = random.randrange(1, SECP256K1_ORDER)
+        R = SECP256K1.affine(SECP256K1.mul([(SECP256K1_G, k)]))
+        r = R[0] % SECP256K1_ORDER
+        s = (modinv(k, SECP256K1_ORDER) * (z + self.secret * r)) % SECP256K1_ORDER
+        if low_s and s > SECP256K1_ORDER_HALF:
+            s = SECP256K1_ORDER - s
+        # Represent in DER format. The byte representations of r and s have
+        # length rounded up (255 bits becomes 32 bytes and 256 bits becomes 33
+        # bytes).
+        rb = r.to_bytes((r.bit_length() + 8) // 8, 'big')
+        sb = s.to_bytes((s.bit_length() + 8) // 8, 'big')
+        return b'\x30' + bytes([4 + len(rb) + len(sb), 2, len(rb)]) + rb + bytes([2, len(sb)]) + sb