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ecc.py

from io import BytesIO
from random import randint
from unittest import TestCase

import hashlib
import hmac

from helper import encode_base58_checksum, hash160
 

class FieldElement:

    def __init__(self, num, prime):
        if num >= prime or num < 0:
            error = 'Num {} not in field range 0 to {}'.format(
                num, prime - 1)
            raise ValueError(error)
        self.num = num
        self.prime = prime

    def __repr__(self):
        return 'FieldElement_{}({})'.format(self.prime, self.num)

    def __eq__(self, other):
        if other is None:
            return False
        return self.num == other.num and self.prime == other.prime

    def __ne__(self, other):
        # this should be the inverse of the == operator
        return not (self == other)

    def __add__(self, other):
        if self.prime != other.prime:
            raise TypeError('Cannot add two numbers in different Fields')
        # self.num and other.num are the actual values
        # self.prime is what we need to mod against
        num = (self.num + other.num) % self.prime
        # We return an element of the same class
        return self.__class__(num, self.prime)

    def __sub__(self, other):
        if self.prime != other.prime:
            raise TypeError('Cannot subtract two numbers in different Fields')
        # self.num and other.num are the actual values
        # self.prime is what we need to mod against
        num = (self.num - other.num) % self.prime
        # We return an element of the same class
        return self.__class__(num, self.prime)

    def __mul__(self, other):
        if self.prime != other.prime:
            raise TypeError('Cannot multiply two numbers in different Fields')
        # self.num and other.num are the actual values
        # self.prime is what we need to mod against
        num = (self.num * other.num) % self.prime
        # We return an element of the same class
        return self.__class__(num, self.prime)

    def __pow__(self, exponent):
        n = exponent % (self.prime - 1)
        num = pow(self.num, n, self.prime)
        return self.__class__(num, self.prime)

    def __truediv__(self, other):
        if self.prime != other.prime:
            raise TypeError('Cannot divide two numbers in different Fields')
        # self.num and other.num are the actual values
        # self.prime is what we need to mod against
        # use fermat's little theorem:
        # self.num**(p-1) % p == 1
        # this means:
        # 1/n == pow(n, p-2, p)
        num = (self.num * pow(other.num, self.prime - 2, self.prime)) % self.prime
        # We return an element of the same class
        return self.__class__(num, self.prime)

    def __rmul__(self, coefficient):
        num = (self.num * coefficient) % self.prime
        return self.__class__(num=num, prime=self.prime)


class FieldElementTest(TestCase):

    def test_ne(self):
        a = FieldElement(2, 31)
        b = FieldElement(2, 31)
        c = FieldElement(15, 31)
        self.assertEqual(a, b)
        self.assertTrue(a != c)
        self.assertFalse(a != b)

    def test_add(self):
        a = FieldElement(2, 31)
        b = FieldElement(15, 31)
        self.assertEqual(a + b, FieldElement(17, 31))
        a = FieldElement(17, 31)
        b = FieldElement(21, 31)
        self.assertEqual(a + b, FieldElement(7, 31))

    def test_sub(self):
        a = FieldElement(29, 31)
        b = FieldElement(4, 31)
        self.assertEqual(a - b, FieldElement(25, 31))
        a = FieldElement(15, 31)
        b = FieldElement(30, 31)
        self.assertEqual(a - b, FieldElement(16, 31))

    def test_mul(self):
        a = FieldElement(24, 31)
        b = FieldElement(19, 31)
        self.assertEqual(a * b, FieldElement(22, 31))

    def test_rmul(self):
        a = FieldElement(24, 31)
        b = 2
        self.assertEqual(b * a, a + a)

    def test_pow(self):
        a = FieldElement(17, 31)
        self.assertEqual(a**3, FieldElement(15, 31))
        a = FieldElement(5, 31)
        b = FieldElement(18, 31)
        self.assertEqual(a**5 * b, FieldElement(16, 31))

    def test_div(self):
        a = FieldElement(3, 31)
        b = FieldElement(24, 31)
        self.assertEqual(a / b, FieldElement(4, 31))
        a = FieldElement(17, 31)
        self.assertEqual(a**-3, FieldElement(29, 31))
        a = FieldElement(4, 31)
        b = FieldElement(11, 31)
        self.assertEqual(a**-4 * b, FieldElement(13, 31))


class Point:

    def __init__(self, x, y, a, b):
        self.a = a
        self.b = b
        self.x = x
        self.y = y
        # x being None and y being None represents the point at infinity
        # Check for that here since the equation below won't make sense
        # with None values for both.
        if self.x is None and self.y is None:
            return
        # make sure that the elliptic curve equation is satisfied
        # y**2 == x**3 + a*x + b
        if self.y**2 != self.x**3 + a * x + b:
            # if not, throw a ValueError
            raise ValueError('({}, {}) is not on the curve'.format(x, y))

    def __eq__(self, other):
        return self.x == other.x and self.y == other.y \
            and self.a == other.a and self.b == other.b

    def __ne__(self, other):
        # this should be the inverse of the == operator
        return not (self == other)

    def __repr__(self):
        if self.x is None:
            return 'Point(infinity)'
        elif isinstance(self.x, FieldElement):
            return 'Point({},{})_{}_{} FieldElement({})'.format(
                self.x.num, self.y.num, self.a.num, self.b.num, self.x.prime)
        else:
            return 'Point({},{})_{}_{}'.format(self.x, self.y, self.a, self.b)

    def __add__(self, other):
        if self.a != other.a or self.b != other.b:
            raise TypeError('Points {}, {} are not on the same curve'.format(self, other))
        # Case 0.0: self is the point at infinity, return other
        if self.x is None:
            return other
        # Case 0.1: other is the point at infinity, return self
        if other.x is None:
            return self

        # Case 1: self.x == other.x, self.y != other.y
        # Result is point at infinity
        if self.x == other.x and self.y != other.y:
            return self.__class__(None, None, self.a, self.b)

        # Case 2: self.x ≠ other.x
        # Formula (x3,y3)==(x1,y1)+(x2,y2)
        # s=(y2-y1)/(x2-x1)
        # x3=s**2-x1-x2
        # y3=s*(x1-x3)-y1
        if self.x != other.x:
            s = (other.y - self.y) / (other.x - self.x)
            x = s**2 - self.x - other.x
            y = s * (self.x - x) - self.y
            return self.__class__(x, y, self.a, self.b)

        # Case 4: if we are tangent to the vertical line,
        # we return the point at infinity
        # note instead of figuring out what 0 is for each type
        # we just use 0 * self.x
        if self == other and self.y == 0 * self.x:
            return self.__class__(None, None, self.a, self.b)

        # Case 3: self == other
        # Formula (x3,y3)=(x1,y1)+(x1,y1)
        # s=(3*x1**2+a)/(2*y1)
        # x3=s**2-2*x1
        # y3=s*(x1-x3)-y1
        if self == other:
            s = (3 * self.x**2 + self.a) / (2 * self.y)
            x = s**2 - 2 * self.x
            y = s * (self.x - x) - self.y
            return self.__class__(x, y, self.a, self.b)

    def __rmul__(self, coefficient):
        coef = coefficient
        current = self
        result = self.__class__(None, None, self.a, self.b)
        while coef:
            if coef & 1:
                result += current
            current += current
            coef >>= 1
        return result


class PointTest(TestCase):

    def test_ne(self):
        a = Point(x=3, y=-7, a=5, b=7)
        b = Point(x=18, y=77, a=5, b=7)
        self.assertTrue(a != b)
        self.assertFalse(a != a)

    def test_on_curve(self):
        with self.assertRaises(ValueError):
            Point(x=-2, y=4, a=5, b=7)
        # these should not raise an error
        Point(x=3, y=-7, a=5, b=7)
        Point(x=18, y=77, a=5, b=7)

    def test_add0(self):
        a = Point(x=None, y=None, a=5, b=7)
        b = Point(x=2, y=5, a=5, b=7)
        c = Point(x=2, y=-5, a=5, b=7)
        self.assertEqual(a + b, b)
        self.assertEqual(b + a, b)
        self.assertEqual(b + c, a)

    def test_add1(self):
        a = Point(x=3, y=7, a=5, b=7)
        b = Point(x=-1, y=-1, a=5, b=7)
        self.assertEqual(a + b, Point(x=2, y=-5, a=5, b=7))

    def test_add2(self):
        a = Point(x=-1, y=1, a=5, b=7)
        self.assertEqual(a + a, Point(x=18, y=-77, a=5, b=7))


class ECCTest(TestCase):

    def test_on_curve(self):
        # tests the following points whether they are on the curve or not
        # on curve y^2=x^3-7 over F_223:
        # (192,105) (17,56) (200,119) (1,193) (42,99)
        # the ones that aren't should raise a ValueError
        prime = 223
        a = FieldElement(0, prime)
        b = FieldElement(7, prime)

        valid_points = ((192, 105), (17, 56), (1, 193))
        invalid_points = ((200, 119), (42, 99))

        # iterate over valid points
        for x_raw, y_raw in valid_points:
            x = FieldElement(x_raw, prime)
            y = FieldElement(y_raw, prime)
            # Creating the point should not result in an error
            Point(x, y, a, b)

        # iterate over invalid points
        for x_raw, y_raw in invalid_points:
            x = FieldElement(x_raw, prime)
            y = FieldElement(y_raw, prime)
            with self.assertRaises(ValueError):
                Point(x, y, a, b)

    def test_add(self):
        # tests the following additions on curve y^2=x^3-7 over F_223:
        # (192,105) + (17,56)
        # (47,71) + (117,141)
        # (143,98) + (76,66)
        prime = 223
        a = FieldElement(0, prime)
        b = FieldElement(7, prime)

        additions = (
            # (x1, y1, x2, y2, x3, y3)
            (192, 105, 17, 56, 170, 142),
            (47, 71, 117, 141, 60, 139),
            (143, 98, 76, 66, 47, 71),
        )
        # iterate over the additions
        for x1_raw, y1_raw, x2_raw, y2_raw, x3_raw, y3_raw in additions:
            x1 = FieldElement(x1_raw, prime)
            y1 = FieldElement(y1_raw, prime)
            p1 = Point(x1, y1, a, b)
            x2 = FieldElement(x2_raw, prime)
            y2 = FieldElement(y2_raw, prime)
            p2 = Point(x2, y2, a, b)
            x3 = FieldElement(x3_raw, prime)
            y3 = FieldElement(y3_raw, prime)
            p3 = Point(x3, y3, a, b)
            # check that p1 + p2 == p3
            self.assertEqual(p1 + p2, p3)

    def test_rmul(self):
        # tests the following scalar multiplications
        # 2*(192,105)
        # 2*(143,98)
        # 2*(47,71)
        # 4*(47,71)
        # 8*(47,71)
        # 21*(47,71)
        prime = 223
        a = FieldElement(0, prime)
        b = FieldElement(7, prime)

        multiplications = (
            # (coefficient, x1, y1, x2, y2)
            (2, 192, 105, 49, 71),
            (2, 143, 98, 64, 168),
            (2, 47, 71, 36, 111),
            (4, 47, 71, 194, 51),
            (8, 47, 71, 116, 55),
            (21, 47, 71, None, None),
        )

        # iterate over the multiplications
        for s, x1_raw, y1_raw, x2_raw, y2_raw in multiplications:
            x1 = FieldElement(x1_raw, prime)
            y1 = FieldElement(y1_raw, prime)
            p1 = Point(x1, y1, a, b)
            # initialize the second point based on whether it's the point at infinity
            if x2_raw is None:
                p2 = Point(None, None, a, b)
            else:
                x2 = FieldElement(x2_raw, prime)
                y2 = FieldElement(y2_raw, prime)
                p2 = Point(x2, y2, a, b)

            # check that the product is equal to the expected point
            self.assertEqual(s * p1, p2)


A = 0
B = 7
P = 2**256 - 2**32 - 977
N = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141


class S256Field(FieldElement):

    def __init__(self, num, prime=None):
        super().__init__(num=num, prime=P)

    def __repr__(self):
        return '{:x}'.format(self.num).zfill(64)

    # tag::source2[]
    def sqrt(self):
        return self**((P + 1) // 4)
    # end::source2[]


class S256Point(Point):

    def __init__(self, x, y, a=None, b=None):
        a, b = S256Field(A), S256Field(B)
        if type(x) == int:
            super().__init__(x=S256Field(x), y=S256Field(y), a=a, b=b)
        else:
            super().__init__(x=x, y=y, a=a, b=b)

    def __repr__(self):
        if self.x is None:
            return 'S256Point(infinity)'
        else:
            return 'S256Point({}, {})'.format(self.x, self.y)

    def __rmul__(self, coefficient):
        coef = coefficient % N
        return super().__rmul__(coef)

    def verify(self, z, sig):
        # By Fermat's Little Theorem, 1/s = pow(s, N-2, N)
        s_inv = pow(sig.s, N - 2, N)
        # u = z / s
        u = z * s_inv % N
        # v = r / s
        v = sig.r * s_inv % N
        # u*G + v*P should have as the x coordinate, r
        total = u * G + v * self
        return total.x.num == sig.r

    # tag::source1[]
    def sec(self, compressed=True):
        '''returns the binary version of the SEC format'''
        if compressed:
            if self.y.num % 2 == 0:
                return b'\x02' + self.x.num.to_bytes(32, 'big')
            else:
                return b'\x03' + self.x.num.to_bytes(32, 'big')
        else:
            return b'\x04' + self.x.num.to_bytes(32, 'big') + \
                self.y.num.to_bytes(32, 'big')
    # end::source1[]

    # tag::source5[]
    def hash160(self, compressed=True):
        return hash160(self.sec(compressed))

    def address(self, compressed=True, testnet=False):
        '''Returns the address string'''
        h160 = self.hash160(compressed)
        if testnet:
            prefix = b'\x6f'
        else:
            prefix = b'\x00'
        return encode_base58_checksum(prefix + h160)
    # end::source5[]

    # tag::source3[]
    @classmethod
    def parse(self, sec_bin):
        '''returns a Point object from a SEC binary (not hex)'''
        if sec_bin[0] == 4:  # <1>
            x = int.from_bytes(sec_bin[1:33], 'big')
            y = int.from_bytes(sec_bin[33:65], 'big')
            return S256Point(x=x, y=y)
        is_even = sec_bin[0] == 2  # <2>
        x = S256Field(int.from_bytes(sec_bin[1:], 'big'))
        # right side of the equation y^2 = x^3 + 7
        alpha = x**3 + S256Field(B)
        # solve for left side
        beta = alpha.sqrt()  # <3>
        if beta.num % 2 == 0:  # <4>
            even_beta = beta
            odd_beta = S256Field(P - beta.num)
        else:
            even_beta = S256Field(P - beta.num)
            odd_beta = beta
        if is_even:
            return S256Point(x, even_beta)
        else:
            return S256Point(x, odd_beta)
    # end::source3[]


G = S256Point(
    0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798,
    0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8)


class S256Test(TestCase):

    def test_order(self):
        point = N * G
        self.assertIsNone(point.x)

    def test_pubpoint(self):
        # write a test that tests the public point for the following
        points = (
            # secret, x, y
            (7, 0x5cbdf0646e5db4eaa398f365f2ea7a0e3d419b7e0330e39ce92bddedcac4f9bc, 0x6aebca40ba255960a3178d6d861a54dba813d0b813fde7b5a5082628087264da),
            (1485, 0xc982196a7466fbbbb0e27a940b6af926c1a74d5ad07128c82824a11b5398afda, 0x7a91f9eae64438afb9ce6448a1c133db2d8fb9254e4546b6f001637d50901f55),
            (2**128, 0x8f68b9d2f63b5f339239c1ad981f162ee88c5678723ea3351b7b444c9ec4c0da, 0x662a9f2dba063986de1d90c2b6be215dbbea2cfe95510bfdf23cbf79501fff82),
            (2**240 + 2**31, 0x9577ff57c8234558f293df502ca4f09cbc65a6572c842b39b366f21717945116, 0x10b49c67fa9365ad7b90dab070be339a1daf9052373ec30ffae4f72d5e66d053),
        )

        # iterate over points
        for secret, x, y in points:
            # initialize the secp256k1 point (S256Point)
            point = S256Point(x, y)
            # check that the secret*G is the same as the point
            self.assertEqual(secret * G, point)

    def test_verify(self):
        point = S256Point(
            0x887387e452b8eacc4acfde10d9aaf7f6d9a0f975aabb10d006e4da568744d06c,
            0x61de6d95231cd89026e286df3b6ae4a894a3378e393e93a0f45b666329a0ae34)
        z = 0xec208baa0fc1c19f708a9ca96fdeff3ac3f230bb4a7ba4aede4942ad003c0f60
        r = 0xac8d1c87e51d0d441be8b3dd5b05c8795b48875dffe00b7ffcfac23010d3a395
        s = 0x68342ceff8935ededd102dd876ffd6ba72d6a427a3edb13d26eb0781cb423c4
        self.assertTrue(point.verify(z, Signature(r, s)))
        z = 0x7c076ff316692a3d7eb3c3bb0f8b1488cf72e1afcd929e29307032997a838a3d
        r = 0xeff69ef2b1bd93a66ed5219add4fb51e11a840f404876325a1e8ffe0529a2c
        s = 0xc7207fee197d27c618aea621406f6bf5ef6fca38681d82b2f06fddbdce6feab6
        self.assertTrue(point.verify(z, Signature(r, s)))

    def test_sec(self):
        coefficient = 999**3
        uncompressed = '049d5ca49670cbe4c3bfa84c96a8c87df086c6ea6a24ba6b809c9de234496808d56fa15cc7f3d38cda98dee2419f415b7513dde1301f8643cd9245aea7f3f911f9'
        compressed = '039d5ca49670cbe4c3bfa84c96a8c87df086c6ea6a24ba6b809c9de234496808d5'
        point = coefficient * G
        self.assertEqual(point.sec(compressed=False), bytes.fromhex(uncompressed))
        self.assertEqual(point.sec(compressed=True), bytes.fromhex(compressed))
        coefficient = 123
        uncompressed = '04a598a8030da6d86c6bc7f2f5144ea549d28211ea58faa70ebf4c1e665c1fe9b5204b5d6f84822c307e4b4a7140737aec23fc63b65b35f86a10026dbd2d864e6b'
        compressed = '03a598a8030da6d86c6bc7f2f5144ea549d28211ea58faa70ebf4c1e665c1fe9b5'
        point = coefficient * G
        self.assertEqual(point.sec(compressed=False), bytes.fromhex(uncompressed))
        self.assertEqual(point.sec(compressed=True), bytes.fromhex(compressed))
        coefficient = 42424242
        uncompressed = '04aee2e7d843f7430097859e2bc603abcc3274ff8169c1a469fee0f20614066f8e21ec53f40efac47ac1c5211b2123527e0e9b57ede790c4da1e72c91fb7da54a3'
        compressed = '03aee2e7d843f7430097859e2bc603abcc3274ff8169c1a469fee0f20614066f8e'
        point = coefficient * G
        self.assertEqual(point.sec(compressed=False), bytes.fromhex(uncompressed))
        self.assertEqual(point.sec(compressed=True), bytes.fromhex(compressed))

    def test_address(self):
        secret = 888**3
        mainnet_address = '148dY81A9BmdpMhvYEVznrM45kWN32vSCN'
        testnet_address = 'mieaqB68xDCtbUBYFoUNcmZNwk74xcBfTP'
        point = secret * G
        self.assertEqual(
            point.address(compressed=True, testnet=False), mainnet_address)
        self.assertEqual(
            point.address(compressed=True, testnet=True), testnet_address)
        secret = 321
        mainnet_address = '1S6g2xBJSED7Qr9CYZib5f4PYVhHZiVfj'
        testnet_address = 'mfx3y63A7TfTtXKkv7Y6QzsPFY6QCBCXiP'
        point = secret * G
        self.assertEqual(
            point.address(compressed=False, testnet=False), mainnet_address)
        self.assertEqual(
            point.address(compressed=False, testnet=True), testnet_address)
        secret = 4242424242
        mainnet_address = '1226JSptcStqn4Yq9aAmNXdwdc2ixuH9nb'
        testnet_address = 'mgY3bVusRUL6ZB2Ss999CSrGVbdRwVpM8s'
        point = secret * G
        self.assertEqual(
            point.address(compressed=False, testnet=False), mainnet_address)
        self.assertEqual(
            point.address(compressed=False, testnet=True), testnet_address)


class Signature:

    def __init__(self, r, s):
        self.r = r
        self.s = s

    def __repr__(self):
        return 'Signature({:x},{:x})'.format(self.r, self.s)

    # tag::source4[]
    def der(self):
        rbin = self.r.to_bytes(32, byteorder='big')
        # remove all null bytes at the beginning
        rbin = rbin.lstrip(b'\x00')
        # if rbin has a high bit, add a \x00
        if rbin[0] & 0x80:
            rbin = b'\x00' + rbin
        result = bytes([2, len(rbin)]) + rbin  # <1>
        sbin = self.s.to_bytes(32, byteorder='big')
        # remove all null bytes at the beginning
        sbin = sbin.lstrip(b'\x00')
        # if sbin has a high bit, add a \x00
        if sbin[0] & 0x80:
            sbin = b'\x00' + sbin
        result += bytes([2, len(sbin)]) + sbin
        return bytes([0x30, len(result)]) + result
    # end::source4[]

    @classmethod
    def parse(cls, signature_bin):
        s = BytesIO(signature_bin)
        compound = s.read(1)[0]
        if compound != 0x30:
            raise SyntaxError("Bad Signature")
        length = s.read(1)[0]
        if length + 2 != len(signature_bin):
            raise SyntaxError("Bad Signature Length")
        marker = s.read(1)[0]
        if marker != 0x02:
            raise SyntaxError("Bad Signature")
        rlength = s.read(1)[0]
        r = int.from_bytes(s.read(rlength), 'big')
        marker = s.read(1)[0]
        if marker != 0x02:
            raise SyntaxError("Bad Signature")
        slength = s.read(1)[0]
        s = int.from_bytes(s.read(slength), 'big')
        if len(signature_bin) != 6 + rlength + slength:
            raise SyntaxError("Signature too long")
        return cls(r, s)


class SignatureTest(TestCase):

    def test_der(self):
        testcases = (
            (1, 2),
            (randint(0, 2**256), randint(0, 2**255)),
            (randint(0, 2**256), randint(0, 2**255)),
        )
        for r, s in testcases:
            sig = Signature(r, s)
            der = sig.der()
            sig2 = Signature.parse(der)
            self.assertEqual(sig2.r, r)
            self.assertEqual(sig2.s, s)


class PrivateKey:

    def __init__(self, secret):
        self.secret = secret
        self.point = secret * G

    def hex(self):
        return '{:x}'.format(self.secret).zfill(64)

    def sign(self, z):
        k = self.deterministic_k(z)
        # r is the x coordinate of the resulting point k*G
        r = (k * G).x.num
        # remember 1/k = pow(k, N-2, N)
        k_inv = pow(k, N - 2, N)
        # s = (z+r*secret) / k
        s = (z + r * self.secret) * k_inv % N
        if s > N / 2:
            s = N - s
        # return an instance of Signature:
        # Signature(r, s)
        return Signature(r, s)

    def deterministic_k(self, z):
        k = b'\x00' * 32
        v = b'\x01' * 32
        if z > N:
            z -= N
        z_bytes = z.to_bytes(32, 'big')
        secret_bytes = self.secret.to_bytes(32, 'big')
        s256 = hashlib.sha256
        k = hmac.new(k, v + b'\x00' + secret_bytes + z_bytes, s256).digest()
        v = hmac.new(k, v, s256).digest()
        k = hmac.new(k, v + b'\x01' + secret_bytes + z_bytes, s256).digest()
        v = hmac.new(k, v, s256).digest()
        while True:
            v = hmac.new(k, v, s256).digest()
            candidate = int.from_bytes(v, 'big')
            if candidate >= 1 and candidate < N:
                return candidate
            k = hmac.new(k, v + b'\x00', s256).digest()
            v = hmac.new(k, v, s256).digest()

    # tag::source6[]
    def wif(self, compressed=True, testnet=False):
        secret_bytes = self.secret.to_bytes(32, 'big')
        if testnet:
            prefix = b'\xef'
        else:
            prefix = b'\x80'
        if compressed:
            suffix = b'\x01'
        else:
            suffix = b''
        return encode_base58_checksum(prefix + secret_bytes + suffix)
    # end::source6[]


class PrivateKeyTest(TestCase):

    def test_sign(self):
        pk = PrivateKey(randint(0, N))
        z = randint(0, 2**256)
        sig = pk.sign(z)
        self.assertTrue(pk.point.verify(z, sig))

    def test_wif(self):
        pk = PrivateKey(2**256 - 2**199)
        expected = 'L5oLkpV3aqBJ4BgssVAsax1iRa77G5CVYnv9adQ6Z87te7TyUdSC'
        self.assertEqual(pk.wif(compressed=True, testnet=False), expected)
        pk = PrivateKey(2**256 - 2**201)
        expected = '93XfLeifX7Jx7n7ELGMAf1SUR6f9kgQs8Xke8WStMwUtrDucMzn'
        self.assertEqual(pk.wif(compressed=False, testnet=True), expected)
        pk = PrivateKey(0x0dba685b4511dbd3d368e5c4358a1277de9486447af7b3604a69b8d9d8b7889d)
        expected = '5HvLFPDVgFZRK9cd4C5jcWki5Skz6fmKqi1GQJf5ZoMofid2Dty'
        self.assertEqual(pk.wif(compressed=False, testnet=False), expected)
        pk = PrivateKey(0x1cca23de92fd1862fb5b76e5f4f50eb082165e5191e116c18ed1a6b24be6a53f)
        expected = 'cNYfWuhDpbNM1JWc3c6JTrtrFVxU4AGhUKgw5f93NP2QaBqmxKkg'
        self.assertEqual(pk.wif(compressed=True, testnet=True), expected)

图像的颜色一直是研究和关注的热点，也是特征工程不可或缺的feature，今天就简单介绍一种非常基础的颜色的rgb特征。

1. 目的：

日常工作中，分析师会接到一些专项分析的需求，首先会搜索脑中的分析体系，根据业务需求构建相应的分析模型（不只是机器学习模型），根据模型填充相应维度表，这些维度特征表能够被使用的前提是假设已经清洗干净了。

上一课我们涉及到用来装东西用的“容器”，可以说，当我们使用魔法咒语创建出各种东西都建议用一个容器来装起来，容器的使用非常简单：起个名，并用“=”连接要装的东西。例如我要用容器number来装一个数字1：