#Greatest Common Divisor ==> https://en.wikipedia.org/wiki/Euclidean_algorithm
print("Greatest Common Divisor")
a=66528
b=52920
def gcd(a, b):
    if a:
        return gcd(b%a, a)
    else:
        return b
    # while True:
    #     if b == 0:
    #         break
    #     c = max(a, b) % min(a, b)
    #     a = b
    #     b = c
    # return a
print(f"GCD({a}, {b}) = {gcd(a, b)}")
print()

#Extended GCD
print("Extended GCD")
p = 26513
q = 32321
def egcd(a, b):
    r = [a, b]
    s = [1, 0]
    t = [0, 1]
    while True:
        if r[1] == 0:
            break
        q = r[0]//r[1]
        r.append(r[0] % r[1])
        s.append(s[0]-q*s[1])
        t.append(t[0]-q*t[1])
        r.pop(0)
        s.pop(0)
        t.pop(0)
    return r[0], s[0], t[0]
r, s, t = egcd(p, q)
print(f"EGCD({p}, {q}) : GCD = {r} = ({s}*{p}) + ({t}*{q})")
print(f"flag = {min(s, t)}")
print()

#Modular Arithmetic 1
print("Modular Arithmetic 1")
a = 11
m = 6
x = a%m
a = 8146798528947
m = 17
y = a%m
print("x = ", x)
print("y = ", y)
print(f"flag = {min(x, y)}")
print()

#Modular Arithmetic 2
print("Modular Arithmetic 2")
def fermat(x, p):
    print(f"({x}^{p-1}) % {p} = {(x**(p-1))%p}")
fermat(7, 17)
fermat(273246787654, 65537)
print()

#Modular Inverting
print("Modular Inverting")
'''
Looking again at Fermat's little theorem...
if p is prime, for every integer a:
        pow(a, p) = a mod p
and, if p is prime and a is an integer coprime with p:
        pow(a, p-1) = 1 mod p
We can do some magic like this:
Note: i'll use math notation, so a^b means pow(a,b)
        a^(p-1) = 1 (mod p)
        a^(p-1) * a^-1 = a^-1 (mod p)
        a^(p-2) * a * a^-1 = a^-1 (mod p)
        a^(p-2) * 1 = a^-1 (mod p)
So finally we have:
        a^(p-2) = a^-1 (mod p)
So, doing a^(p-2) and then (mod p) we can achieve
our result
'''
a = 3
p = 13
print(f"flag = {(a**(p-2))%p}")
print()

#Quadratic Residues
p = 29
ints = [14, 6, 11]
for i in range(p):
    x = (i**2)%p
    print(f"{i}^2 % {p} = {x} ({x} is a Quadratic Residue {'***' if x in ints else ''})")

#Legendre Symbol
# Quadratic Residue * Quadratic Residue = Quadratic Residue
# Quadratic Residue * Quadratic Non-residue = Quadratic Non-residue
# Quadratic Non-residue * Quadratic Non-residue = Quadratic Residue