Editing Abstract Algebra (section)

== <span style="color: #FFFFFF;">Applying</span> ==
'''Abstract algebra in cryptography and coding:'''
<syntaxhighlight lang="python">
# Abstract algebra provides the foundations of modern cryptography.
# We implement key group/field operations.

# === Group Theory: Modular Arithmetic Group ===
class ModularGroup:
    """The additive group ℤ/nℤ."""
    def __init__(self, n: int):
        self.n = n
        self.elements = list(range(n))
        self.identity = 0
    
    def operation(self, a: int, b: int) -> int:
        return (a + b) % self.n
    
    def inverse(self, a: int) -> int:
        return (self.n - a) % self.n
    
    def order_of_element(self, a: int) -> int:
        """Order of a: smallest k > 0 such that a*k ≡ 0 (mod n)."""
        current = a
        for k in range(1, self.n + 1):
            if current % self.n == 0:
                return k
            current += a
        return -1  # Should never reach here
    
    def subgroups(self) -> list:
        """All subgroups of ℤ/nℤ are generated by divisors of n (by Lagrange)."""
        from math import gcd
        return [d for d in range(1, self.n + 1) if self.n % d == 0]

g = ModularGroup(12)
print(f"ℤ/12ℤ subgroups (orders): {g.subgroups()}")  # [1, 2, 3, 4, 6, 12]
print(f"Order of element 3 in ℤ/12ℤ: {g.order_of_element(3)}")  # 4

# === Field Theory: Galois Fields ===
class GF2:
    """Galois Field GF(2) = {0, 1} with XOR as addition, AND as multiplication."""
    @staticmethod
    def add(a: int, b: int) -> int: return a ^ b  # XOR
    @staticmethod
    def mul(a: int, b: int) -> int: return a & b  # AND
    @staticmethod
    def inverse_add(a: int) -> int: return a      # a + a = 0 in GF(2)

# GF(2^8) for AES — polynomial arithmetic mod irreducible polynomial
# x^8 + x^4 + x^3 + x + 1 (AES uses 0x11b)
AES_IRREDUCIBLE = 0x11b

def gf256_mul(a: int, b: int) -> int:
    """Multiplication in GF(2^8) — the core of AES MixColumns."""
    p = 0
    while b > 0:
        if b & 1:
            p ^= a
        hi_bit = a & 0x80
        a = (a << 1) & 0xFF
        if hi_bit:
            a ^= 0x1b  # x^8 mod (irreducible poly) reduction
        b >>= 1
    return p

# Demonstrate: in GF(2^8), multiplication is done mod the irreducible polynomial
print(f"GF(2^8): 0x57 × 0x83 = 0x{gf256_mul(0x57, 0x83):02x}")  # Expected: 0xc1

# === Elliptic Curve Group (over finite field) ===
class EllipticCurve:
    """
    Elliptic curve: y² = x³ + ax + b (mod p)
    Points form an abelian group — the basis of ECC cryptography.
    """
    def __init__(self, a: int, b: int, p: int):
        self.a, self.b, self.p = a, b, p
        assert (4*a_'3 + 27*b_'2) % p != 0, "Singular curve"
    
    def point_add(self, P, Q):
        """Add two points on the elliptic curve (group operation)."""
        if P is None: return Q  # Point at infinity is identity
        if Q is None: return P
        x1, y1 = P; x2, y2 = Q
        if x1 == x2 and y1 != y2: return None  # P + (-P) = point at infinity
        if P == Q:
            if y1 == 0: return None
            lam = (3*x1**2 + self.a) * pow(2*y1, -1, self.p) % self.p
        else:
            lam = (y2 - y1) * pow(x2 - x1, -1, self.p) % self.p
        x3 = (lam**2 - x1 - x2) % self.p
        y3 = (lam*(x1 - x3) - y1) % self.p
        return (x3, y3)
    
    def scalar_mul(self, k: int, P) -> tuple:
        """Double-and-add: compute kP efficiently — core of ECC."""
        R = None
        Q = P
        while k > 0:
            if k & 1: R = self.point_add(R, Q)
            Q = self.point_add(Q, Q)
            k >>= 1
        return R
</syntaxhighlight>

; Key texts and theorists
: '''Group theory''' → Cauchy, Galois, Sylow, Burnside (''Theory of Groups''), Hall
: '''Rings and ideals''' → Dedekind, Noether (''Theory of Ideals''), van der Waerden (''Modern Algebra'')
: '''Galois theory''' → Évariste Galois, Emil Artin (''Galois Theory'')
: '''Finite fields''' → Galois; Moore; applications: Reed-Solomon, AES, elliptic curves
: '''Modern texts''' → Dummit & Foote (''Abstract Algebra''), Herstein (''Topics in Algebra''), Lang (''Algebra'')
</div>

<div style="background-color: #8B4500; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">