Cryptography module

Primes

class mathcrypto.cryptography.primes.Primes

Bases: object

classmethod factorize(num: int) → list

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

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

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 num is prime

Return type

bool

classmethod is_probable_prime_fermat(num: int, rounds: int = 5) → bool

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

Bases: object

classmethod baby_step(crack_me) → int

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

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

classmethod mov_attack(secret: int, g: int, order: int) → int

The MOV attack on Elliptic curve DH.

Parameters

• secret (int) – Secret to be cracked

• g (int) –

• order (int) –

Returns

int if secret was cracked, 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)

Bases: object

Object containing all values of the cryptosystem

generate_from(bit_length: Optional[int] = None, prime: Optional[int] = None)

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_size or prime is passed. At least one of these is required.

generate_rest()

Generates the missing attributes of the DHCryptosystem attributes if possible.

Raises

ValueError – If the attributes that were already in the DHCryptosystem object are calculated from values that were autogenerated. Then you wouldn’t have a valid DHCryptosystem.

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)

Bases: object

Elliptic 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)

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()

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)

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)

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)

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()

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)

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

Bases: object

A collection of useful mathematical functions

classmethod crt(lis) → int

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

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.

classmethod euclid_gcd(num_a: int, num_b: int) → int

Euclidean algorithm

Calculates the Greatest Common Divisor of two numbers.

Parameters

• num_a (int) – First number

• num_b (int) – Second number

Returns

Greatest Common Divisor of the two numbers

Return type

int

classmethod phi(num: int) → int

Euler’s Totient function Phi. If the number is not prime, the execution time depends on the speed of factorization.

Parameters

num (int) – Any positive whole number

Returns

How many elements belong to a multiplicative group set by this number.

Return type

int

Groups

class mathcrypto.math.groups.MultiplicativeGroup(mod)

Bases: object

Multiplicative 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

Gets the order of an element in the group

Parameters

element (int) – Element of the group

Raises

ValueError – When the element does not belong to the group

Returns

Returns the order of element in the group

Return type

int

get_element_subgroup(element) → int

Gets the subgroup of any element in the group

Parameters

element (int) – Element of the group

Raises

ValueError – When the element does not belong to the group

Returns

Returns the order of element in the group

Return type

list

get_inverse_element(element: int) → int

Gets the inverse to an element in the group

Parameters

element (int) – Element of the group

Raises

ValueError – When the element does not belong to the group

Returns

Inverse element to element

Return type

int