Possible obfuscation for Mersennes

[paulunderwood]

Carried out on an idle Pi Zero W:

? 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

[paulunderwood]

Originally Posted by paulunderwood

Carried out on an idle Pi Zero W:

? 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)

[R. Gerbicz]

Originally Posted by paulunderwood

Carried out on an idle Pi Zero W:

? 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.

[CRGreathouse]

Incidentally, rather than

? gettime(); foo; print(bar); gettime()

I would write

? #
? foo; bar

where # turns the timer on (or off). In your case this would have resulted in

? #
   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.)