20180307, 11:36  #1 
Feb 2018
2^{5}×3 Posts 
Whats is a fast algorithm for compute znorder(Mod(2,p)).
whats is a fast algorithm for compute znorder(Mod(2,p)).
for p prime but big. Last fiddled with by JM Montolio A on 20180307 at 11:45 
20180307, 11:40  #2 
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 

20180307, 11:46  #3 
Feb 2018
2^{5}·3 Posts 
You never sleeps ?

20180307, 15:10  #4  
Aug 2006
1011101010100_{2} Posts 
Quote:
For example, let's try to find the order of 2 mod 101. m = 100 = 2^2 * 5^2, so we know that 2^100 = 1 mod 101. 2^(100/2) mod 101 is 100 which is not 1, so skip 2. 2^(100/5) = 95 which is not 1, so skip 5, and we're done: the order is 100. Let's try 103. m = 102 = 2 * 3 * 17. 2^(102/2) = 1 mod 103, so m becomes 102/2 = 51. We don't have any more 2s though so we'll go on to 3. 2^(51/3) = 56 mod 103 which is not 1, and 2^(51/17) = 8 mod 103 which is also not 1, so we're done: the order of 2 mod 103 is 51. 

20190612, 01:44  #6 
Apr 2019
5×41 Posts 
Um, if p is prime, isn't the answer always p1?
edit: nevermind, i'm obviously confused Last fiddled with by hansl on 20190612 at 01:50 
20190612, 16:09  #7 
Aug 2006
2^{2}·1,493 Posts 

20190612, 17:46  #8  
"Robert Gerbicz"
Oct 2005
Hungary
2·7·103 Posts 
Quote:
Code:
res_i=2^((p1)/q^i) mod p for i=e,e1,..1,0 Multiply these primepowers to get the order. Note that you really need some intelligence here also, because there could be more than polynomial number of divisors of p1. 

20190612, 19:02  #9 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
3×3,109 Posts 

20190613, 11:46  #10  
Feb 2017
Nowhere
2·17·127 Posts 
Quote:
As is well known (supplement to the quadratic reciprocity law), znorder(Mod(2,p)) divides (p1)/2 if p == 1 or 7 (mod 8), but not otherwise. Since 101 == 5 (mod 8) we know that znorder(Mod(2,101)) does not divide 100/2 = 50. Since 103 = 7 (mod 8) we know that znorder(Mod(2,103)) does divide 102/2 = 51. Alas, the classical criterion of when znorderMod(2,p)) divides (p1)/4 when 8 divides p1 [p = x^2 + 64*y^2] likely takes longer to decide than whether Mod(2,p)^((p1)/4) == Mod(1,p). Likewise, the classical criterion for whether znorder(Mod(2,p)) divides (p1)/3 when 3 divides p1 [p = x^2 + 27*y^2] is likely also slower than computing Mod(2,p)^(p1)/3 when 3 divides p1. 

20190614, 03:13  #11 
Aug 2006
2^{2}·1,493 Posts 
We’re not disagreeing here of course: the factorization takes up asymptotically all the time (in the worst/average case), but for me the heart of the problem is what I described in my post, and the factorization is a (difficult) subproblem that must be resolved before moving on to that core part.

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