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Is there any chance to write a faster GCD than the built-in one in Mathematica?

@Mr.Wizard has written one in this question (although it's not for this purpose) which is 6 times slower on a 100k digits:

In[1]:= whileGCD=Module[{a=#,b=#2},While[b!=0,{a,b}={b,Mod[a,b]}];a]&;
In[2]:= Primo[n_]:=Product[i,{i,Prime[Range[n]]}]

In[10]:= prod=Primo[500000];
In[17]:= m=22Primo[20500] +1;

In[16]:= Timing[GCD[m,prod]]
Out[16]= {0.25,1}

In[12]:= Timing[whileGCD[m,prod]]
Out[12]= {1.609,1}
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    $\begingroup$ The short answer is no. The longer answer is maybe, if you are a GMP developer with access to some fairly low level NTT (number theory transform) code. Even then it will be difficult. $\endgroup$ Commented Jan 6, 2013 at 20:49
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    $\begingroup$ I don't have it installed at the moment so I cannot compare them but you may wish to try PARI/GP. I have found that to be much faster than Mathematica's built-in functions in at least one instance. $\endgroup$
    – Mr.Wizard
    Commented Jan 12, 2013 at 10:52
  • $\begingroup$ @Mr.Wizard I've tried it for Timing[GCD[100000! + 1, 1000000!]] and it is 9 times slower than Mathematica $\endgroup$ Commented Jan 12, 2013 at 13:17
  • $\begingroup$ I'm actually happy to hear that, though of course that isn't any help to you. $\endgroup$
    – Mr.Wizard
    Commented Jan 12, 2013 at 13:26

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Daniel Lichtblau answered this question in the comments

The short answer is no. The longer answer is maybe, if you are a GMP developer with access to some fairly low level NTT (number theory transform) code. Even then it will be difficult.

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