We've looked at multiplication and division in modular arithmetic, but what does it mean to take the square root modulo an integer?
For the following discussion, let's work modulo p = 29
. We can take the integer a = 11
and calculate a^{2} = 5 mod 29
.
As a = 11, a^{2} = 5
, we say the square root of 5
is 11
.
This feels good, but now let's think about the square root of 18
. From the above, we know we need to find some integer a
such that a^{2} = 18
Your first idea might be to start with a = 1
and loop to a = p-1
. In this discussion p
isn't too large and we can quickly look.
Have a go, try coding this and see what you find. If you've coded it right, you'll find that for all a ∈ F_{p}^{*}
you never find an a
such that a^{2} = 18
.
What we are seeing, is that for the elements of F^{*}_{p}
, not every element has a square root. In fact, what we find is that for roughly one half of the elements of F_{p}^{*}
, there is no square root.
We say that an integer x
is a Quadratic Residue if there exists an a
such that a^{2} = x mod p
. If there is no such solution, then the integer is a Quadratic Non-Residue.
In other words, x
is a quadratic residue when it is possible to take the square root of x
modulo an integer p
.
In the below list there are two non-quadratic residues and one quadratic residue.
Find the quadratic residue and then calculate its square root. Of the two possible roots, submit the smaller one as the flag.
If a^{2} = x
then (-a)^{2} = x. So if x
is a quadratic residue in some finite field, then there are always two solutions for a
.
p = 29
ints = [14, 6, 11]
You must be logged in to submit your flag.
You are now level Current level