p = 7
q = 13
n = p*q
phi = (p-1)*(q-1)
print(phi)
e = 5
assert gcd(e,phi) == 1

d = 1/e % phi

n,e 	# public
p,q,d	# private

m = 50
c = m^e % n
c

m_ = c^d % n
m_
assert m_ == m

all((m^e)^d % n == m for m in range(n))

# alternativ geht die Arithemtik auch so:
Zn = Integers(n)
Zphi = Integers(phi)

e = Zphi(5)
assert gcd(e,phi) == 1

d = 1/e

n,e 	# public
p,q,d	# private

m = Zn(50)
c = m^e
c

m_ = c^d
m_
assert m_ == m

all((m^e)^d == m for m in Zn)