Modular Arithmetic

course_modular_arithmetic.py

@@ -0,0 +1,87 @@
+#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()