Solving modular linear congruences for large numbers

Suppose that for some reason the linear congruence equations you'll be 'attacking' come up 'empty' (no solutions) often enough to be a design criteria for your algorithm.

It turns out that you can use just (with any real overhead) the residue operations to answer that binary question -

There exist solutions XOR There are no solutions

This might have utility in cryptography; see also the abstract,

Introduction of the Residue Number Arithmetic Logic Unit
With Brief Computational Complexity Analysis

Once you determine that a solution exist, you can use back substitution
and the ALU to determine a solution.

Also, you'll have calculated the gcd(a,m) and can construct the coefficients of Bézout's identity
(if you need them).

Following is python program that incorporates the above ideas; it calculates the minimal solution when it exists and prints out Bézout's identity.

test_data = [ 
(32,12,82), 
(9,3,23), 
(17,41,73), 
(227,1,2011), 
(25,15,29), 
(2,22,71), 
(7,10,21), 
(124,58,900), 
(46, 12, 240), 
]

for lc in test_data:
    LC = lc
    back_sub_List = []
    while True:
        back_sub_List.append(LC)
        n_mod_a = LC[2] % LC[0]
        if n_mod_a == 0:
            break
        LC = (n_mod_a, -LC[1] % LC[0], LC[0])
    gcd_of_a0_n0 = LC[0]
    if LC[1] % LC[0] != 0:
        print(f"No solution          for {back_sub_List[0][0]}x = {back_sub_List[0][1]} (mod {back_sub_List[0][2]})")
    else:
        k = 0
        for LC in back_sub_List[::-1]: # solve with back substitution
            a,b,m = LC
            k = (b + k*m) // a         # optimize calculation since the remainder is zero?
        print(f"The minimal solution for {back_sub_List[0][0]}x = {back_sub_List[0][1]} (mod {back_sub_List[0][2]}) is equal to {k}")
    # get bezout
    S = [1,0]
    T = [0,1]
    for LC in back_sub_List:    
        a,b,n = LC
        q = n // a
        s = S[0] - q * S[1]
        S = [S[1], s]
        t = T[0] - q * T[1]
        T = [T[1], t]
    print(f"  Bézout's identity:     ({S[0]})({lc[2]}) + ({T[0]})({lc[0]}) = {gcd_of_a0_n0}")

PROGRAM OUTPUT

The minimal solution for 32x = 12 (mod 82) is equal to 26
  Bézout's identity:     (-7)(82) + (18)(32) = 2
The minimal solution for 9x = 3 (mod 23) is equal to 8
  Bézout's identity:     (2)(23) + (-5)(9) = 1
The minimal solution for 17x = 41 (mod 73) is equal to 11
  Bézout's identity:     (7)(73) + (-30)(17) = 1
The minimal solution for 227x = 1 (mod 2011) is equal to 1320
  Bézout's identity:     (78)(2011) + (-691)(227) = 1
The minimal solution for 25x = 15 (mod 29) is equal to 18
  Bézout's identity:     (-6)(29) + (7)(25) = 1
The minimal solution for 2x = 22 (mod 71) is equal to 11
  Bézout's identity:     (1)(71) + (-35)(2) = 1
No solution          for 7x = 10 (mod 21)
  Bézout's identity:     (0)(21) + (1)(7) = 7
No solution          for 124x = 58 (mod 900)
  Bézout's identity:     (4)(900) + (-29)(124) = 4
The minimal solution for 46x = 12 (mod 240) is equal to 42
  Bézout's identity:     (-9)(240) + (47)(46) = 2

If you have Python 3.8 or later, you can do everything you need to with a very small number of lines of code.

First some mathematics: I'm assuming that you want to solve ax = b (mod m) for an integer x, given integers a, b and m. I'm also assuming that m is positive.

The first thing you need to compute is the greatest common divisor g of a and m. There are two cases:

• if b is not a multiple of g, then the congruence has no solutions (if ax + my = b for some integers x and y, then any common divisor of a and m must also be a divisor of b)

• if b is a multiple of g, then the congruence is exactly equivalent to (a/g)x = (b/g) (mod (m/g)). Now a/g and m/g are relatively prime, so we can compute an inverse to a/g modulo m/g. Multiplying that inverse by b/g gives a solution, and the general solution can be obtained by adding an arbitrary multiple of m/g to that solution.

Python's math module has had a gcd function since Python 3.5, and the built-in pow function can be used to compute modular inverses since Python 3.8.

Putting it all together, here's some code. First a function that finds the general solution, or raises an exception if no solution exists. If it succeeds, it returns two integers. The first gives a particular solution; the second gives the modulus that provides the general solution.

def solve_linear_congruence(a, b, m):
    """ Describe all solutions to ax = b  (mod m), or raise ValueError. """
    g = math.gcd(a, m)
    if b % g:
        raise ValueError("No solutions")
    a, b, m = a//g, b//g, m//g
    return pow(a, -1, m) * b % m, m

And then some driver code, to demonstrate how to use the above.

def print_solutions(a, b, m):
    print(f"Solving the congruence: {a}x = {b}  (mod {m})")
    try:
        x, mx = solve_linear_congruence(a, b, m)
    except ValueError:
        print("No solutions")
    else:
        print(f"Particular solution: x = {x}")
        print(f"General solution: x = {x}  (mod {mx})")

Example use:

>>> print_solutions(272, 256, 1009)
Solving the congruence: 272x = 256  (mod 1009)
Particular solution: x = 179
General solution: x = 179  (mod 1009)
>>> print_solutions(98, 105, 1001)
Solving the congruence: 98x = 105  (mod 1001)
Particular solution: x = 93
General solution: x = 93  (mod 143)
>>> print_solutions(98, 107, 1001)
Solving the congruence: 98x = 107  (mod 1001)
No solutions