def ord(n, b): # Multiplicative order
"""Return the smallest positive integer e such that b**e=1(mod n)."""
if gcd(n, b) != 1 or n == 1:
raise Exception("Multiplicative order is not defined");
e = 1;
power = b%n;
while power != 1:
power = (power*b)%n;
e += 1;
return e;
def linear(n, a, b):
"""Give general solution to linear congruence ax=b (mod n)"""
if a == 0:
raise Exception("Not a linear congruence.");
g = gcd(a, n);
if b % g:
return [];
a, b, s = a//g, b//g, n//g;
x = (multiplicative_inverse(s, a)*b) % n;
solutions = [];
for i in range(g):
solutions.append(x);
x = (x+s) % n;
solutions.sort();
return solutions;
def generate_keys(cls, digit):
"""Generate public and private keys.
The length of each number is roughly twice the value of 'digit'"""
p = generate_large_prime(digit);
q = generate_large_prime(digit);
n = p * q;
period = (p-1) * (q-1);
while(True):
e = randint(1, n-1);
if gcd(period, e) == 1:
break;
# e has to be invertible
d = multiplicative_inverse(period, e);
return (RSA_public_key(n, e), RSA_private_key(n, d));
def fermat_test(n, trials=DEFAULT_TRIALS, show_witness=False):
"""Using Fermet test, to check (with certain confidence) whether a
positive integer 'n'(>=3) is prime.
Carmichael numbers, which are composites satisfying the korselt criterion,
can fool this algorithm."""
for i in range(trials):
while True:
witness = randint(2, n - 1)
if gcd(n, witness) == 1:
break
if (mod(n, witness, n - 1) != 1):
if show_witness:
print("Witness:", witness)
return 0
return 1
def euclid():
"""Compute the gcd and lcm of two numbers"""
# render the template "euclid"
if request.method == "GET":
return render_template("euclid.html")
if request.method == "POST":
# if the first number is not provided, return apology
if not request.form.get("first"):
return apology("must provide a number")
# if the second number not provided, return apology
if not request.form.get("second"):
return apology("must provide a number")
# extract the numbers from the request form
first = int(request.form.get("first"))
second = int(request.form.get("second"))
# if numbers less than 0, return apology
if first <= 0:
return apology("must provide a positive integer")
if second <= 0:
return apology("must provide a positive integer")
# calculate gcd and lcm
divisor = gcd(first, second)
multiplicity = lcm(first, second)
# render a template that will show the results
return render_template("gcd_lcm.html",
x=first,
y=second,
gcd=divisor,
lcm=multiplicity)