Mathematica溢出[]错误:为什么以及如何绕过？

Question

我从未在Mathematica中出现溢出错误,发生了以下情况.

我演示了RSA加密的原理如下:

 n = 11*13
 m = EulerPhi[n]
 e = 7
 GCD[e, m]
 d = PowerMod[e, -1, m]
 cipher2[m_String] := Map[Mod[#^e, n] &, ToCharacterCode[m]]
 decipher2[x_Integer] := FromCharacterCode[Map[Mod[#^d, n] &, x]]

 In[207]:= cipher2["StackOverflow"]
 decipher2[cipher2["StackOverflow"]]
 Out[207]= {8,129,59,44,68,40,79,62,49,119,4,45,37}
 Out[208]= StackOverflow

没有问题.

然后我将质数更改为更现实,但仍然非常适中.

 n = 252097800611*252097800629

 In[236]:= cipher2["StackOverflow"]
 decipher2[cipher2["StackOverflow"]]

 Out[236]= {27136050989627, 282621973446656, 80798284478113, \
 93206534790699, 160578147647843, 19203908986159, 318547390056832, \
 107213535210701, 250226879128704, 114868566764928, 171382426877952, \
 207616015289871, 337931541778439}

 During evaluation of In[236]:= General::ovfl: Overflow occurred in computation. >>

 During evaluation of In[236]:= General::ovfl: Overflow occurred in computation. >>

 Out[237]= FromCharacterCode[{Overflow[], Overflow[], Overflow[], 
   Overflow[], Overflow[], Overflow[], Overflow[], Overflow[], 
   Overflow[], Overflow[], Overflow[], Overflow[], Overflow[]}]

问题:我是否已经完成了Mathematica的限制？我使用了不正确的方法吗？什么是旁路,如果有的话？

Answer 1

尝试PowerMod在decyphering操作中使用:

n = 252097800611*252097800629;
m = EulerPhi[n];
e = 7;
Print[GCD[e, m]];
d = PowerMod[e, -1, m];
Print[{"n" -> n, "m" -> m, "e" -> e, "d" -> d}];
Grid[
 Join[{
  {"Input", "Encrypted", "Decrypt with Mod", "Decrypt with PowerMod"}}, 
  Table[{i, (j = Mod[i^e, n]), Mod[j^d, n], PowerMod[j, d, n]}, {i, 40}]], 
 Frame -> All]

Answer 2

是的,你已经完成了Mathematica的限制.可以在特定版本的Mathematica中的系统上表示的最大数量显示为$MaxNumber.在你的第二个例子中,d=18158086021982021938023因此27136050989627^d比大于$MaxNumber.

您可以PowerMod在第二步中使用,就像您所做的那样d,它将a^b mod n比计算更有效Mod.有了decipher2[x_List] := FromCharacterCode[Map[PowerMod[#, d, n] &, x]],你得到:

cipher2["StackOverflow"]
decipher2[cipher2["StackOverflow"]]

Out[1]= {27136050989627, 282621973446656, 80798284478113, \
93206534790699, 160578147647843, 19203908986159, 318547390056832, \
107213535210701, 250226879128704, 114868566764928, 171382426877952, \
207616015289871, 337931541778439}

Out[2]= "StackOverflow"