A big number modulo a prime

GatorPower

Im looking through old exams for a course in Cryptography and have beaten my head against the wall for a long time on one of the questions:

p = 683 is a prime, p-1 = 2*11*31. What is x = 4^11112 mod p?

When i did chinese remainder theorem on primes 2,11,31 i got that x = 16 mod 682, but so far i have not found a way to use this in determining x mod 683..

Also, when computing discrete logs I have found that one goes from mod p to mod (p-1) alot, why is that?

mfb

Can you use a computer to calculate it?
The common general approach to calculate a^b mod c will give you the answer in milliseconds.

GatorPower

It is supposedly solvable by hand, but I'm wondering if it was meant to calculate mod p-1 instead of p. Atleast thats what I'll conclude if no one finds another answer.

Also, it seems that the answer is 16 mod 683 as well. Perhaps there is some once-in-a-lifetime connection between 682 and 683 for this task?

mfb

4^11 = 2^22 = 1 mod 683

I would expect that there is some way to get this from p-1=2*11*31.
The rest is easy, as 11112 = 11*1010 + 2.

4^6 = -2 mod 683 is quite interesting, too.
It is possible to calculate that by hand, but then you don't need the hint.

GatorPower

4 is quadratic residue, so 4^(p-1)/2 = 1 mod p, but then i get 4^11*31 = 1 mod p, and i need to get rid of 31.. Otherwise i get stuck with 4^200 mod p. I'll try to figure something out, thanks for the help!

ramsey2879

The solution is to use Fermat's little theorm. To use that you need 11112 mod 682 not x mod 682.

GatorPower

Fermat gives 4^11112 = 1*4^200 mod p, so that gets me a little further, but not quite there.

ramsey2879

Knowing that 4^200 =(((4^5)^2)^2)^5 should help

GatorPower

That does help, its just that i want to find a better way of doing it! Usually these things turn out to be some number squared or something, so I feel like im doing something wrong if i compute too much :p

dodo

p-1=2*11*31 is also phi(p), and the order of any number (for example, 2) mod p is going to be a divisor of phi(p). So there are not many combinations to try in order to find a divisor d of phi(p) such that 2^d =1 mod p.

mfb

I think this is the trick. You still have to calculate 4^2 mod p (trivial) and 4^6 mod p (you can stop calculating 4^11 here if you see 4^6 = -2 mod p and therefore 4^11 = 1 mod p), but you know that you don't have to calculate anything beyond 4^11.

GatorPower

Indeed, that does help. Thank you everyone for helping out!

johnqwertyful

The way I did it was
4^682=1 mod 683
(4^219)(4^463)=1mod 683
Then 4^11=1 mod 683
You can surely simplify 4^219 by using that

The rest should be clear.