Welcome to MathCrypto
MathCrypto is a library of useful funtions used in cryptography. Do not use this library for improving the security of your application, it is not safe or powerful enough to provide that.
Current version is 0.2.4.
Includes
- Multiplicative group operations
Generating a group from modulus
Get inverse element any element of the group
Get element order of any element in group
- Math functions
Classic number primality check
Fermat’s primality test
Euler’s Totient function (Phi)
Euclidean algorithm (GCD)
Simple number factorization
Chinese Remainder Theorem
Extended Euclidean Algorithm
- Cryptography algorithms
- Diffie-Hellman Key exchange generation and cracking
Multithreaded Brute-force cracking
Baby-step Giant-step algorithm cracking
Installation
MathCrypto is avalaible through Python Package Index (PyPI) using pip:
$ python3 -m pip install mathcrypto
To uninstall using pip:
$ python3 -m pip uninstall mathcrypto
Getting Started
from mathcrypto import MathFunctions, MultiplicativeGroup
def main():
print("Playing with math functions")
print()
print("Is 137 prime?")
print(MathFunctions.is_prime(137))
print("What are the factors of 134?")
print(MathFunctions.factorize(134))
print("What is the greatest common divisor of 135 and 186?")
print(MathFunctions.euclid_gcd(135, 186))
print("What is the Euler's Totient function of 65?")
print(MathFunctions.phi(65))
print()
print("Playing with multiplicative groups")
print()
group = MultiplicativeGroup(13)
print("Group modulus:")
print(group.mod)
print("Group order:")
print(group.order)
print("Group elements:")
print(group.elements)
print("Group generators:") # If there are any
print(group.generators)
main()
This outputs:
Playing with math functions
Is 137 prime?
True
What are the factors of 134?
[2, 67]
What is the greatest common divisor of 135 and 186?
3
What is the Euler's Totient function of 65?
48
Playing with multiplicative groups
Group modulus:
13
Group order:
12
Group elements:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
Group generators:
[2, 6, 7, 11]
Development
Source code repository is available on GitHub. Feel free to contribute. Bug reports and suggestions are welcome.
License
MathCrypto is licensed under the MIT License.
Cryptography module
Primes
- class mathcrypto.cryptography.primes.Primes[source]
Bases:
object- classmethod factorize(num: int) list[source]
Classic number factorization
Tests divisibility by 2 and then every odd number up to sqrt(num) while appending the factors. If the number contains multiple instances of a factor, this function returns a list with duplicates. Very fast unless the number is a compound of more than one large prime (more than 7 digits, 8 is still acceptable).
- Parameters
num (int) – Number to factorize
- Returns
List of factors including duplicates
- Return type
list
- classmethod get_prime(bit_length: int) int[source]
Get a n-bit prime
- Parameters
bit_length (int) – Bit size of the desired prime number
- Returns
desired prime number
- Return type
int
- classmethod is_prime(num: int) bool[source]
Classic number modulus check
Tests divisibility by 2 and then every odd number up to sqrt(num). Takes longer to compute than Fermat’s primality test but has 100% certainty. Cannot handle numbers larger than the system maxint.
- Parameters
num (int) – Number to test
- Raises
OverflowError – If the number is too large
- Returns
True if
numis prime- Return type
bool
- classmethod is_probable_prime_fermat(num: int, rounds: int = 5) bool[source]
Automatic Fermat’s primality test
This test can not provide 100% certainty that the number is indeed prime, so more than 1 round should be required. If determined that the number is not prime, that is on the other hand 100% certain. Tests for the number of rounds specified, 2-3 rounds are generally enough. If any round returns a result that is not 1, the number is not prime.
- Parameters
num (int) – Number to be tested
rounds (int) – How many rounds of testing to perform
verbose (bool, optional) – Whether to return optional list of [<number it was tested against>: int,<result>: int]. Defaults to False.
- Returns
True if probably prime
- Return type
bool
- If verbose, returns:
(tuple): tuple containing:
bool: True if probably prime
list: List of [<number it was tested against>: int,<result>: int]
Diffie Hellmann
- class mathcrypto.cryptography.diffie_hellman.DHCracker[source]
Bases:
object- classmethod baby_step(crack_me) int[source]
Discrete logarithm problem solution using the Baby-step Giant-step algorithm. RAM intensive.
- Parameters
crack_me (DHCryptosystem object) – Object containing the publicly know values of the cryptosystem.
- Returns
int if a key was found, else None
- Return type
int or None
- classmethod brute_force(crack_me, num_cpus: int) int[source]
Calculates the DHCryptosystem key by utilizing multiprocessing enhanced brute force. CPU and time intensive.
- Parameters
crack_me (DHCryptosystem object) – Needs to be containing the publicly known values of the cryptosystem.
num_cpus (int) – Number of CPU cores to utilize. Do not exceed the number of logical cores your CPU has, this will result in slower execution.
- Returns
int if a key was found, else None
- Return type
int or None
- class mathcrypto.cryptography.diffie_hellman.DHCryptosystem(prime: Optional[int] = None, generator: Optional[int] = None, alice_secret: Optional[int] = None, bob_secret: Optional[int] = None, alice_sends: Optional[int] = None, bob_sends: Optional[int] = None, key: Optional[int] = None)[source]
Bases:
objectObject containing all values of the cryptosystem
- generate_from(bit_length: Optional[int] = None, prime: Optional[int] = None)[source]
Generates the DHCryptosystem values (If not passed == if they are None) or assigns them
- Parameters
bit_size (int, optional) – Bit size of the prime. Not necessary if prime also passed.
prime (int, optional) – Prime number base of the cryptosystem.
- Raises
ValueError – If neither
bit_sizeorprimeis passed. At least one of these is required.
Elliptic Curves
This module wouldn’t be possible to be created this fast if I didn’t leverage the code that was written by xnomas and Baka-Git in their repository at:
https://github.com/Baka-Git/Crypto_Math
I want to thank them for allowing me to use their code.
- class mathcrypto.cryptography.elliptic_curves.EllipticCurve(a0: int, a1: int, a2: int, a3: int, a4: int, a5: int, a6: int, field: Optional[int] = None, point_px: Optional[int] = None, point_py: Optional[int] = None)[source]
Bases:
objectElliptic curve objects
- Parameters
a0-a6 (int) – Curve attributes (using the equation a0*y^2 + a1*y + a2*y*x = a3*x^3 + a4*x^2 + a5*x+a6)
field (int, optional) – The curves field
point_px (int, optional) – X coordinate of point P
point_py (int, optional) – Y coordinate of point P
- add_point(point_qx: int, point_qy: int)[source]
Adds point P and point Q (of given coordinates) on the curve.
- Parameters
point_qx (int) – X coordinate of point Q
point_qy (int) – Y coordinate of point Q
- Raises
ValueError – If curve field is not set.
ValueError – If the curve is not elliptic.
ValueError – If point P is not set.
ValueError – If one or both poits are not on the curve.
- Returns
Resulting point coordinates
- Return type
list
- get_all_point_order()[source]
Gets orders of all points on the curve
- Raises
ValueError – If field is not set
- Returns
list of lists[order, point]
- Return type
list of lists
- get_curve_order(get_points: bool = False)[source]
Gets the order of the curve.
- Parameters
get_points (bool, optional) – Whether or not to return the curve points as well. Defaults to False.
- Raises
ValueError – If curve field is not set or the curve is not elliptic.
- Returns
False if this EC is not supported. int: Elliptic curve order if get_points is not set to True (tuple):If get_points is set to true, returns a tuple containing:
int: Elliptic curve order
list: List of points on curve
- Return type
bool
- get_point_order(point_x: Optional[int] = None, point_y: Optional[int] = None)[source]
Gets the order of point of given coordinates or point P if set. Given coordinates take precedence.
- Parameters
point_x (int, optional) – X coordinate of the point
point_y (int, optional) – Y coordinate of the point
- Raises
ValueError – Neither valid parameters were passed and point P was not set
- Returns
Order of the point
- Return type
int
- classmethod get_possible_orders(order: int, new_order: Optional[int] = None)[source]
Gets the possible orders of points on a curve of certain order or if a curves order was changed to a given value.
- Parameters
order (int) – Order of the curve
new_order (int, optional) – New order of the curve. Defaults to None.
- Returns
List of possible orders
- Return type
list
- is_elliptic_curve()[source]
Checks if the curve is elliptic
- Returns
True if curve with these attributes is elliptic
- Return type
bool
- is_point_on_elliptic_curve(x: int, y: int)[source]
Checks if point of given coordinates is on the curve.
- Parameters
x (int) – X coordinate of the point
y (int) – Y coordinate of the point
- Raises
ValueError – If curve field is not set or the curve is not elliptic.
- Returns
True if point is on the curve
- Return type
bool
Math module
Functions
- class mathcrypto.math.funcs.MathFunctions[source]
Bases:
objectA collection of useful mathematical functions
- classmethod crt(lis) int[source]
Chinese remainder theorem
- Solves x for problems like:
x ≡ 8 mod 9
x ≡ 3 mod 5
- Parameters
lis – list of [int, int]:
- Example input:
[[8, 9],[3, 5]] for the example problem
- Returns
Solution for x
- Return type
int
- classmethod eea(modulus: int, number: int, verbose: bool = False) int[source]
Extended Euclidean Algorithm
Get multiplicative inverse of a number in modulus
- Parameters
modulus (int) – modulus of a multiplicative group
number (int) – number to get inverse to
verbose (bool, optional) – Whether to return a graphical solution. Defaults to False.
- Raises
ValueError – If number is not an element of the group defined by
modulus- Returns
resulting inverse
- Return type
int
- If verbose, returns:
str: Graphical solution of the problem.
Groups
- class mathcrypto.math.groups.MultiplicativeGroup(mod)[source]
Bases:
objectMultiplicative group objects
- mod
Modulus of the group
- Type
int
- elements
List of elements in the group
- Type
list
- order
Order of the group
- Type
int
- generators
List of generators of the group
- Type
list
- get_element_order(element) int[source]
Gets the order of an element in the group
- Parameters
element (int) – Element of the group
- Raises
ValueError – When the
elementdoes not belong to the group- Returns
Returns the order of
elementin the group- Return type
int