Implement Extended Euclidean Algorithm with igcdex function and add c…

…orresponding tests

- Introduced a new function `igcdex` in `math_utils.py` to compute the extended greatest common divisor, returning coefficients and the gcd.
- Added unit tests for `igcdex` in `test_math_utils.py` to ensure correctness of the implementation.
- Removed the outdated `sympy` dependency from `requirements.txt` to clean up the requirements.

jesse/math_utils.py

@@ -0,0 +1,39 @@
+def igcdex(a, b):
+    """Returns x, y, g such that g = x*a + y*b = gcd(a, b).
+    >>> igcdex(2, 3)
+    (-1, 1, 1)
+    >>> igcdex(10, 12)
+    (-1, 1, 2)
+
+    >>> x, y, g = igcdex(100, 2004)
+    >>> x, y, g
+    (-20, 1, 4)
+    >>> x*100 + y*2004
+    4
+
+    """
+    if (not a) and (not b):
+        return (0, 1, 0)
+
+    if not a:
+        return (0, b // abs(b), abs(b))
+    if not b:
+        return (a // abs(a), 0, abs(a))
+
+    if a < 0:
+        a, x_sign = -a, -1
+    else:
+        x_sign = 1
+
+    if b < 0:
+        b, y_sign = -b, -1
+    else:
+        y_sign = 1
+
+    x, y, r, s = 1, 0, 0, 1
+
+    while b:
+        (c, q) = (a % b, a // b)
+        (a, b, r, s, x, y) = (b, c, x - q * r, y - q * s, r, s)
+
+    return (x * x_sign, y * y_sign, a)

requirements.txt

@@ -28,5 +28,4 @@ aiofiles~=0.7.0
 numba~=0.61.0rc2
 PyJWT~=2.8.0
 cryptography~=42.0.5
-sympy~=1.6
 ecdsa>=0.16.0

tests/test_math_utils.py

@@ -0,0 +1,6 @@
+import jesse.math_utils as mu
+
+
+def test_igcdex():
+    assert mu.igcdex(2, 3) == (-1, 1, 1)
+    assert mu.igcdex(10, 12) == (-1, 1, 2)