Question

我目前正在尝试从压缩的公钥中派生出比特币未压缩的ECDSA公钥。

根据这个link on the Bitcoin wiki，有可能这样做......但是如何？

为您提供更多详细信息：截至目前，我在比特币网络上收集了压缩密钥（33字节长）。

它们具有以下格式：＆lt; 1字节长前缀＆gt;＆lt; 32字节长X＆gt;。从那里，我想获得一个未压缩的密钥（65字节长），其格式为：＆lt; 1字节长前缀＆gt;＆lt; 32字节长X＆gt;＆lt; 32字节长Y＆gt;

根据这个other link on the Bitcoin wiki，它应该像解决方程一样简单：

Y ^ 2 = X ^ 3 + 7

然而，我似乎无法到达那里。我对Y的价值远远不够。这是我的代码（公钥的值来自Bitcoin wiki example）：

import binascii
from decimal import *

expected_uncompressed_key_hex = '0450863AD64A87AE8A2FE83C1AF1A8403CB53F53E486D8511DAD8A04887E5B23522CD470243453A299FA9E77237716103ABC11A1DF38855ED6F2EE187E9C582BA6'
expected_y_hex = expected_uncompressed_key_hex[-64:]
expected_y_dec = int(expected_y_hex, 16)
x_hex = expected_uncompressed_key_hex[2:66]
if expected_y_dec % 2 == 0:
    prefix = "02"
else:
    prefix = "03"

artificial_compressed_key = prefix + x_hex

getcontext().prec = 500
test_dec = Decimal(int(x_hex, 16))
y_square_dec = test_dec**3 + 7
if prefix == "02":
    y_dec = - Decimal(y_square_dec).sqrt()
else:
    y_dec = Decimal(y_square_dec).sqrt()

computed_y_hex = hex(int(y_dec))
computed_uncompressed_key = "04" + x + computed_y_hex

有关信息，我的输出是：

computed_y_hex = '0X2D29684BD207BF6D809F7D0EB78E4FD61C3C6700E88AB100D1075EFA8F8FD893080F35E6C7AC2E2214F8F4D088342951'
expected_y_hex = '2CD470243453A299FA9E77237716103ABC11A1DF38855ED6F2EE187E9C582BA6'

感谢您的帮助！

Answer 1

您需要在字段 $\mathbb{Z}_p$ 中进行计算，这主要意味着您必须在每次计算后用p除以后将数字减少到余数。计算它被称为取模，并在python中写为% p。

在这个领域中的指数可以比仅仅倍增和减少多次的天真方式更有效地完成。这称为模幂运算。 Python内置的指数函数pow（n，e，p）可以解决这个问题。

剩下的问题是找到平方根。幸运的是，secp256k1以特殊的方式选择（ $p\equiv3\pmod 4$ ），因此取平方根很容易：x的平方根是 $x^{\frac{p+1}{4}} \pmod p$ 。

因此，代码的简化版本将变为：

import binascii

p_hex = 'FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F'
p = int(p_hex, 16)
compressed_key_hex = '0250863AD64A87AE8A2FE83C1AF1A8403CB53F53E486D8511DAD8A04887E5B2352'
x_hex = compressed_key_hex[2:66]
x = int(x_hex, 16)
prefix = compressed_key_hex[0:2]

y_square = (pow(x, 3, p)  + 7) % p
y_square_square_root = pow(y_square, (p+1)/4, p)
if (prefix == "02" and y_square_square_root & 1) or (prefix == "03" and not y_square_square_root & 1):
    y = (-y_square_square_root) % p
else:
    y = y_square_square_root

computed_y_hex = format(y, '064x')
computed_uncompressed_key = "04" + x_hex + computed_y_hex

print computed_uncompressed_key

Answer 2

椭圆曲线的场不在实数域上。它超过了有限域模数的一些素数。

对于Secp256k1，素数p = 2 ^ 256 - 2 ^ 32 - 2 ^ 9 - 2 ^ 8 - 2 ^ 7 - 2 ^ 6 - 2 ^ 4 - 1.

因此：y ^ 2 =（x ^ 3）+ 7（mod p）

没有直接解决方程的方法，你需要使用Cipolla的算法：https://en.wikipedia.org/wiki/Cipolla%27s_algorithm

Answer 3

以下示例代码没有任何第三方python库：

IoT core

请参阅比特币谈话：https://bitcointalk.org/index.php?topic=644919.0

Answer 4

我知道这个问题已经回答了，我实际上从这个回答中受益，所以谢谢。问题是我在C＃中寻找相同的解决方案时发现了3次这些答案，而我实际上并没有在python中编写代码:)。因此，对于任何尝试解决此问题的人，这里都是C＃解决方案，请尽情享受！：）（它使用BouncyCastle库）。

using System;
using System.Collections.Generic;
using System.Linq;
using MoreLinq;
using NBitcoin;
using Org.BouncyCastle.Asn1.X9;
using Org.BouncyCastle.Crypto;
using Org.BouncyCastle.Crypto.Parameters;
using Org.BouncyCastle.Math;
using Org.BouncyCastle.Math.EC;

namespace BitcoinPublicKeyDecompression
{
    public class Program
    {
        public static void Main()
        {
            const string cPubKey = "0250863ad64a87ae8a2fe83c1af1a8403cb53f53e486d8511dad8a04887e5b2352";
            var uPubKey = cPubKey.ToHexByteArray().BitcoinDecompressPublicKey().ToHexString();
            var expectedUPubKey = new PubKey(cPubKey).Decompress().ToString();

            Console.WriteLine($"Public Key:\n\n{cPubKey}\n\nhas been {(uPubKey == expectedUPubKey ? "correctly" : "incorrectly")} decompressed to:\n\n{uPubKey}");

            Console.WriteLine("\nPress any key to quit...");
            Console.ReadKey();
        }
    }

    public static class Extensions
    {
        public static readonly byte[] EmptyByteArray = new byte[0];

        public static byte[] BitcoinDecompressPublicKey(this byte[] bPubC)
        {
            var ecPubKey = bPubC.BitcoinCompressedPublicKeyToECPublicKey();
            return ecPubKey.ToBitcoinUncompressedPublicKey();
        }

        public static ECPublicKeyParameters BitcoinCompressedPublicKeyToECPublicKey(this byte[] bPubC)
        {
            var pubKey = bPubC.Skip(1).ToArray();

            var curve = ECNamedCurveTable.GetByName("secp256k1");
            var domainParams = new ECDomainParameters(curve.Curve, curve.G, curve.N, curve.H, curve.GetSeed());

            var yParity = new BigInteger(bPubC.Take(1).ToArray()).Subtract(BigInteger.Two);
            var x = new BigInteger(1, pubKey);
            var p = ((FpCurve)curve.Curve).Q;
            var a = x.ModPow(new BigInteger("3"), p).Add(new BigInteger("7")).Mod(p);
            var y = a.ModPow(p.Add(BigInteger.One).FloorDivide(new BigInteger("4")), p);

            if (!y.Mod(BigInteger.Two).Equals(yParity))
                y = y.Negate().Mod(p);

            var q = curve.Curve.CreatePoint(x, y);
            return new ECPublicKeyParameters(q, domainParams);
        }

        public static byte[] ToBitcoinUncompressedPublicKey(this AsymmetricKeyParameter ecPublicKey)
        {
            var publicKey = ((ECPublicKeyParameters)ecPublicKey).Q;
            var xs = publicKey.AffineXCoord.ToBigInteger().ToByteArrayUnsigned().PadStart(32);
            var ys = publicKey.AffineYCoord.ToBigInteger().ToByteArrayUnsigned().PadStart(32);
            return new byte[] { 0x04 }.ConcatMany(xs, ys).ToArray();
        }

        public static BigInteger FloorDivide(this BigInteger a, BigInteger b)
        {
            if (a.CompareTo(BigInteger.Zero) > 0 ^ b.CompareTo(BigInteger.Zero) < 0 && !a.Mod(b).Equals(BigInteger.Zero))
                return a.Divide(b).Subtract(BigInteger.One);

            return a.Divide(b);
        }

        public static byte[] ToHexByteArray(this string str)
        {
            byte[] bytes;
            if (string.IsNullOrEmpty(str))
                bytes = EmptyByteArray;
            else
            {
                var string_length = str.Length;
                var character_index = str.StartsWith("0x", StringComparison.Ordinal) ? 2 : 0;
                var number_of_characters = string_length - character_index;
                var add_leading_zero = false;

                if (0 != number_of_characters % 2)
                {
                    add_leading_zero = true;
                    number_of_characters += 1;
                }

                bytes = new byte[number_of_characters / 2];

                var write_index = 0;
                if (add_leading_zero)
                {
                    bytes[write_index++] = CharacterToByte(str[character_index], character_index);
                    character_index += 1;
                }

                for (var read_index = character_index; read_index < str.Length; read_index += 2)
                {
                    var upper = CharacterToByte(str[read_index], read_index, 4);
                    var lower = CharacterToByte(str[read_index + 1], read_index + 1);

                    bytes[write_index++] = (byte)(upper | lower);
                }
            }

            return bytes;
        }

        public static byte CharacterToByte(char character, int index, int shift = 0)
        {
            var value = (byte)character;
            if (0x40 < value && 0x47 > value || 0x60 < value && 0x67 > value)
            {
                if (0x40 != (0x40 & value))
                    return value;
                if (0x20 == (0x20 & value))
                    value = (byte)((value + 0xA - 0x61) << shift);
                else
                    value = (byte)((value + 0xA - 0x41) << shift);
            }
            else if (0x29 < value && 0x40 > value)
                value = (byte)((value - 0x30) << shift);
            else
                throw new InvalidOperationException($"Character '{character}' at index '{index}' is not valid alphanumeric character.");

            return value;
        }

        public static string ToHexString(this byte[] value, bool prefix = false)
        {
            var strPrex = prefix ? "0x" : "";
            return strPrex + string.Concat(value.Select(b => b.ToString("x2")).ToArray());
        }

        public static IEnumerable<T> ConcatMany<T>(this IEnumerable<T> enumerable, params IEnumerable<T>[] enums)
        {
            return enumerable.Concat(enums.SelectMany(x => x));
        }
    } 
}

结果：

从压缩的公钥中派生ECDSA未压缩的公钥

4 个答案: