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Old 2019-01-23, 21:44   #1
paulunderwood
 
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Sep 2002
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Default Possible obfuscation for Mersennes

Carried out on an idle Pi Zero W:

Code:
? gettime();p=21701;n=2^p-1;e=n;r=Mod(3,n)^((e))-3;print([length(e),lift(r)]);gettime()
[679, 0]
54996
? gettime();p=21701;n=2^p-1;e=lift(Mod(2,n-1)^(p-1));r=Mod(3,n)^((e))+3;print([length(e),lift(r)]);gettime()
[679, 0]
54455
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Old 2019-01-23, 22:10   #2
paulunderwood
 
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Quote:
Originally Posted by paulunderwood View Post
Carried out on an idle Pi Zero W:

Code:
? gettime();p=21701;n=2^p-1;e=n;r=Mod(3,n)^((e))-3;print([length(e),lift(r)]);gettime()
[679, 0]
54996
? gettime();p=21701;n=2^p-1;e=lift(Mod(2,n-1)^(p-1));r=Mod(3,n)^((e))+3;print([length(e),lift(r)]);gettime()
[679, 0]
54455
Oh this is poor. I should have used e=n+1. Other than this the test was 3^((n+1)/2)+3 == 0 (mod n)
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Old 2019-01-23, 22:10   #3
R. Gerbicz
 
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"Robert Gerbicz"
Oct 2005
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Quote:
Originally Posted by paulunderwood View Post
Carried out on an idle Pi Zero W:

Code:
? gettime();p=21701;n=2^p-1;e=n;r=Mod(3,n)^((e))-3;print([length(e),lift(r)]);gettime()
[679, 0]
54996
? gettime();p=21701;n=2^p-1;e=lift(Mod(2,n-1)^(p-1));r=Mod(3,n)^((e))+3;print([length(e),lift(r)]);gettime()
[679, 0]
54455
Not surprising, in the 1st case e=2^p-1, while in the 2nd case e=2^(p-1) and that enables a simple repeated squaring at powmod. Actually we're doing the 2nd variant with error checking, probably it is better to do 3^(2^p) mod mp to allow a fast space efficient cofactor test for mp/d.
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Old 2019-01-24, 03:14   #4
CRGreathouse
 
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Aug 2006

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Incidentally, rather than
Code:
? gettime(); foo; print(bar); gettime()
I would write
Code:
? #
? foo; bar
where # turns the timer on (or off). In your case this would have resulted in
Code:
? #
   timer = 1 (on)
? p=21701;n=2^p-1;e=n;r=Mod(3,n)^((e))-3;[length(e),lift(r)]
time = 54,996 ms.
%1 = [679, 0]
? p=21701;n=2^p-1;e=lift(Mod(2,n-1)^(p-1));r=Mod(3,n)^((e))+3;[length(e),lift(r)]
time = 54,455 ms.
%2 = [679, 0]
(Apologies if this was known/obvious.)
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