为什么在NTL库中使用CRT算法时显示"InvMod: inverse undefined"？

Question

我尝试使用 NTL 库来实现我的密码算法。但是，它向我展示了一些关于 CRT 算法的问题。 CRT是Incremental Chinese Remaindering的缩写，定义如下：

long CRT(ZZ& a, ZZ& p, const ZZ& A, const ZZ& P);
long CRT(ZZ& a, ZZ& p, long A, long P);

// 0 <= A < P, (p, P) = 1; computes a' such that a' = a mod p, 
// a' = A mod P, and -p*P/2 < a' <= p*P/2; sets a := a', p := p*P, and
// returns 1 if a's value has changed, otherwise 0

我尝试打印一些信息：

gp=66390828312651972973888361332364813613478546962956830307987136462065800330260786554389230936903970584093871405496526397214600259238471440684133633914352198492704004953513651583287557225250450851122687829965259605518687677556714602317993643325112503010372335796589795208660430772233273662752270218833954206912
p=98506238639786519141405322641812326504550334169475033833322673238957789026078361022953456930946799879629297087483487500221235712451981623924607546413344684294639373306430972887026133598448706655301440705253664704851471711197893571186626487501140015523931128287493592419972025673134065648304689552713586835457
gq=6667000274529267578982370009668139148348778725432024700386402060811738943238338877475151419711063751789939702415071471603295111504118310469114424237809527
q=11378762793182988817702088080680425474862067132459357853502156713056912474438574734825983540828752440930678737903819042152986539729875178831578851503505409
InvMod: inverse undefined

我调用 CRT 如下：

// ...
std::cout << "gp=" << gp << std::endl;
std::cout << "p=" << p << std::endl;
std::cout << "gq=" << gq << std::endl;
std::cout << "q=" << q << std::endl;
NTL::CRT(gp, p, gq, q);

Answer 1

来自ntl/ZZ.cpp:1307，针对CRT函数给出了以下评论：

// Chinese Remaindering.
[...]
// This function takes as input g, a, G, p,
// such that a > 0, 0 <= G < p, and gcd(a, p) = 1.
[...]
long CRT(ZZ& gg, ZZ& a, long G, long p)

在您的情况下，我们需要 gcd(p, q) = 1，但使用您的值 gcd(p, q) = q，即 p = r * q，其中

r = 8657025410425249234452045797441790045805303296607662\
201360892817104202207969817487169683905119008751440550350704\
462002980862924347497458107362187486953473

结果是使用Big Number Calculator online获得的。

换句话说，p 和 q 必须互质 - 但它们不是。

相对素数测试在 InvMod:

内执行

long InvMod(long a, long n)
{
   long d, s, t;

   XGCD(d, s, t, a, n);
   if (d != 1) {
      InvModError("InvMod: inverse undefined");
   }
   if (s < 0)
      return s + n;
   else
      return s;
}

而InvMod是invoked from CRT as follows:

long CRT(ZZ& gg, ZZ& a, long G, long p)
{
   if (p >= NTL_SP_BOUND) {
      ZZ GG, pp;
      conv(GG, G);
      conv(pp, p);
      return CRT(gg, a, GG, pp);
   }

   [...]

   long a_inv;
   a_inv = rem(a, p);
   a_inv = InvMod(a_inv, p);

   [...]
}

由于 p 和 q 不互质，rem(p, q) = 0，InvMod(0, q) 调用失败。

为什么在NTL库中使用CRT算法时显示"InvMod: inverse undefined"？

Why did it show "InvMod: inverse undefined" when used CRT algorithm in NTL library?

c++

ntl