Question

我正在尝试实现NIST P256点压缩和解压缩，当我求解y = sqrt（x ^ 3 + ax + b）时，我总是得到错误的y坐标。

我认为对减压的一个很好的测试是采用由NIST（https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf，D.1.2.3节）定义的基点G，并使用其x坐标来验证y坐标是否为计算是正确的。

这就是我在做什么：

 Public Shared Sub TestBasePoint()
    'Parameters for the curve as defined by NIST: https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf D.1.2.3
    Dim p As BigInteger = BigInteger.Pow(New BigInteger(2), 256) - BigInteger.Pow(New BigInteger(2), 224) + BigInteger.Pow(New BigInteger(2), 192) + BigInteger.Pow(New BigInteger(2), 96) - 1
    Dim a As BigInteger = BigInteger.Parse("3")
    Dim b As BigInteger = BigInteger.Parse("41058363725152142129326129780047268409114441015993725554835256314039467401291")

    'The base point 
    Dim Gx As BigInteger = BigInteger.Parse("48439561293906451759052585252797914202762949526041747995844080717082404635286") 'Gx = 6b17d1f2 e12c4247 f8bce6e5 63a440f2 77037d81 2deb33a0 f4a13945 d898c296
    Dim Gy As BigInteger = BigInteger.Parse("36134250956749795798585127919587881956611106672985015071877198253568414405109") 'Gy = 4fe342e2 fe1a7f9b 8ee7eb4a 7c0f9e16 2bce3357 6b315ece cbb64068 37bf51f5

    'Compute Sqrt(x^3+ax+b) to get possible values
    Dim x3 As BigInteger = BigInteger.Pow(Gx, 3) Mod p
    Dim ax As BigInteger = (a * Gx) Mod p
    Dim x3ax As BigInteger = (x3 - ax) Mod p
    Dim alpha As BigInteger = (x3ax + b) Mod p
    Dim y_solution1 As BigInteger = Sqrt(alpha) Mod p '110978579505207128328462658488647051134659543018045598806500827547749176924776
    Dim y_solution2 As BigInteger = (p - y_solution1) '4813509705149120434234788460760522395426600397244715389032803761117920929175

End Sub

我得到了可能的要点

110978579505207128328462658488647051134659543018045598806500827547749176924776
4813509705149120434234788460760522395426600397244715389032803761117920929175

都不是预期的y基点

36134250956749795798585127919587881956611106672985015071877198253568414405109

有什么我想念的吗？

Answer 1

您不能仅在此处取普通平方根，还需要取 modular 平方根。换句话说，给定整数 n 和素数 p ，您需要找到整数 r 使得 r ²≡ n mod p 。

我不知道VB是否提供模块化的平方根函数，它看起来并不像它。某些语言确实在其标准库中提供了它，例如Go。

假设它没有一个，您将需要找到一个或创建自己的一个。首先是研究Tonelli-Shanks。

NIST p256点解压缩：寻找基点的Y坐标

1 个答案: