我如何计算在密码学对数？ [英] How do I compute logarithms in cryptography?

问题描述

我想对字节进行非线性函数来实现SAFER +。该算法需要在字节计算基础-45对数，我不知道该怎么做。

I am trying to perform non-linear functions on bytes to implement SAFER+. The algorithm requires computing base-45 logarithm on bytes, and I don't understand how to do it.

登录<子> 45 （201）= 1.39316393

log₄₅(201) = 1.39316393

当我分配给字节，值被截断为1，且无法恢复的确切的结果。

When I assign this to a byte, the value is truncated to 1, and I can't recover the exact result.

我应该怎么来处理这个

推荐答案

加密经常使用的黄金领域，在这种情况下，GF（257）。的创建一个看起来像一个幂表这样的：

Cryptography often uses prime fields, in this case, GF(257). Create an exponentiation table that looks like this:

exp | log
----+----
  0 |   1
  1 |  45
  2 | 226
  3 | 147
... | ...
128 |   0
... | ...
255 |  40
---------

日志值45 ^EXP ％257.你需要用 modPow 功能的高精度计算库（养数量的电力，模一定的价值）来构建这个表。你可以看到，对于值EXP128是一种特殊的情况下，由于通常的零的对数是不确定的。

The "log" values are 45^exp % 257. You'll need an arbitrary precision arithmetic library with a modPow function (raise a number to a power, modulo some value) to build this table. You can see that the value for "exp" 128 is a special case, since normally the logarithm of zero is undefined.

通过找到它计算一个数的对数在日志栏目;在该行的EXP列中的值的对数。

Compute the logarithm of a number by finding the it in the "log" column; the value in the "exp" column of that row is the logarithm.

下面是初始化的草图：

BigInteger V45 = new BigInteger(45);
BigInteger V257 = new BigInteger(257);
byte[] exp = new byte[256];
for (int idx = 0; idx < 256; ++idx)
  exp[idx] = BigInteger.ModPow(V45, new BigInteger(idx), V257) % 256;
byte[] log = new byte[256];
for (int idx = 0; idx < 256; ++idx)
  log[exp[idx]] = idx;

通过这种设置，例如，日志 ₄₅ （131）= 日志[131] = 63，和45 ³⁸ = EXP [38] = 59。

With this setup, for example, log₄₅(131) = log[131] = 63, and 45³⁸ = exp[38] = 59.

（我从来没有写C＃;我只是从的 的BigInteger 文档;也有可能是与数据类型的错误）

(I've never written C#; I'm just guessing from the BigInteger documentation; there are likely to be errors with the data types.)

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