#!/usr/bin/env python3 # ============================================================================== # COPYRIGHT NO ONE EVERYWHERE LLC (WYOMING HOLDING COMPANY) # PROJECT: SOVEREIGN STACK # This artifact is entirely proprietary and cryptographically proven. # Open-Source usage requires explicit permission from Brandon Scott Schneider. # ============================================================================== """ SHA256 Mathematical Derivation and Security Analysis Educational derivation of SHA256 structure, security properties, and theoretical attack complexity WARNING: This is for EDUCATIONAL PURPOSES ONLY - SHA256 is cryptographically secure - No practical attack exists with current technology - Brute force requires 2^256 operations (physically impossible) - This code demonstrates structure, NOT exploitation """ import sys import os sys.path.insert(0, os.path.abspath(os.path.join(os.path.dirname(__file__), ".."))) from math_harness_compat import xp, AnyArray import hashlib import json import time from dataclasses import dataclass, field from typing import List, Dict, Optional, Tuple from pathlib import Path import sys # ============================================================================ # SHA256 CONSTANTS (First 32 bits of fractional parts of cube roots of primes) # ============================================================================ SHA256_K = [ 0x428a2f98, 0x71374491, 0xb5c0fbcf, 0xe9b5dba5, 0x3956c25b, 0x59f111f1, 0x923f82a4, 0xab1c5ed5, 0xd807aa98, 0x12835b01, 0x243185be, 0x550c7dc3, 0x72be5d74, 0x80deb1fe, 0x9bdc06a7, 0xc19bf174, 0xe49b69c1, 0xefbe4786, 0x0fc19dc6, 0x240ca1cc, 0x2de92c6f, 0x4a7484aa, 0x5cb0a9dc, 0x76f988da, 0x983e5152, 0xa831c66d, 0xb00327c8, 0xbf597fc7, 0xc6e00bf3, 0xd5a79147, 0x06ca6351, 0x14292967, 0x27b70a85, 0x2e1b2138, 0x4d2c6dfc, 0x53380d13, 0x650a7354, 0x766a0abb, 0x81c2c92e, 0x92722c85, 0xa2bfe8a1, 0xa81a664b, 0xc24b8b70, 0xc76c51a3, 0xd192e819, 0xd6990624, 0xf40e3585, 0x106aa070, 0x19a4c116, 0x1e376c08, 0x2748774c, 0x34b0bcb5, 0x391c0cb3, 0x4ed8aa4a, 0x5b9cca4f, 0x682e6ff3, 0x748f82ee, 0x78a5636f, 0x84c87814, 0x8cc70208, 0x90befffa, 0xa4506ceb, 0xbef9a3f7, 0xc67178f2 ] # Initial hash values (First 32 bits of fractional parts of square roots of first 8 primes) SHA256_H0 = [ 0x6a09e667, 0xbb67ae85, 0x3c6ef372, 0xa54ff53a, 0x510e527f, 0x9b05688c, 0x1f83d9ab, 0x5be0cd19 ] # ============================================================================ # BITWISE OPERATIONS (SHA256 PRIMITIVES) # ============================================================================ def rotr(x: int, n: int) -> int: """Right rotation for 32-bit integers""" return ((x >> n) | (x << (32 - n))) & 0xFFFFFFFF def shr(x: int, n: int) -> int: """Right shift for 32-bit integers""" return x >> n def ch(x: int, y: int, z: int) -> int: """Choice function: if x then y else z (bitwise)""" return (x & y) ^ (~x & z) & 0xFFFFFFFF def maj(x: int, y: int, z: int) -> int: """Majority function: majority of bits in x, y, z""" return (x & y) ^ (x & z) ^ (y & z) & 0xFFFFFFFF def sigma0(x: int) -> int: """Σ0 function for compression""" return rotr(x, 2) ^ rotr(x, 13) ^ rotr(x, 22) def sigma1(x: int) -> int: """Σ1 function for compression""" return rotr(x, 6) ^ rotr(x, 11) ^ rotr(x, 25) def gamma0(x: int) -> int: """σ0 function for message schedule""" return rotr(x, 7) ^ rotr(x, 18) ^ shr(x, 3) def gamma1(x: int) -> int: """σ1 function for message schedule""" return rotr(x, 17) ^ rotr(x, 19) ^ shr(x, 10) # ============================================================================ # SHA256 COMPRESSION FUNCTION (DERIVATION) # ============================================================================ @dataclass class SHA256State: """Represents the internal state of SHA256 during compression""" # Working variables (a-h) a: int = SHA256_H0[0] b: int = SHA256_H0[1] c: int = SHA256_H0[2] d: int = SHA256_H0[3] e: int = SHA256_H0[4] f: int = SHA256_H0[5] g: int = SHA256_H0[6] h: int = SHA256_H0[7] # Round number (0-63) round: int = 0 # Message schedule word for current round w_t: int = 0 # Intermediate values t1: int = 0 t2: int = 0 def to_list(self) -> List[int]: return [self.a, self.b, self.c, self.d, self.e, self.f, self.g, self.h] def copy(self) -> 'SHA256State': return SHA256State( a=self.a, b=self.b, c=self.c, d=self.d, e=self.e, f=self.f, g=self.g, h=self.h, round=self.round ) class SHA256Derivation: """ Step-by-step SHA256 derivation with security analysis """ def __init__(self): self.state = SHA256State() self.message_schedule: List[int] = [] self.round_trace: List[SHA256State] = [] def pad_message(self, message: bytes) -> bytes: """ SHA256 padding: 1. Append bit '1' to message 2. Append k zeros where k is smallest non-negative solution to: (message_length + 1 + k) ≡ 448 (mod 512) 3. Append original length in bits as 64-bit big-endian integer """ msg_len = len(message) msg_bit_len = msg_len * 8 # Append '1' bit (0x80 = 10000000) message += b'\x80' # Append zeros until length ≡ 448 (mod 512) bits = 56 (mod 64) bytes while (len(message) % 64) != 56: message += b'\x00' # Append original length as 64-bit big-endian message += msg_bit_len.to_bytes(8, 'big') return message def parse_message_blocks(self, padded_message: bytes) -> List[bytes]: """Parse padded message into 512-bit (64-byte) blocks""" return [padded_message[i:i+64] for i in range(0, len(padded_message), 64)] def compute_message_schedule(self, block: bytes) -> List[int]: """ Compute message schedule W[0..63] For t = 0..15: W[t] = M[t] (direct from message block) For t = 16..63: W[t] = σ1(W[t-2]) + W[t-7] + σ0(W[t-15]) + W[t-16] This is where diffusion begins - each word depends on 4 previous words """ W = [] # First 16 words from message block (big-endian) for i in range(16): W.append(int.from_bytes(block[i*4:(i+1)*4], 'big')) # Extend to 64 words using message schedule expansion for t in range(16, 64): s0 = gamma0(W[t-15]) s1 = gamma1(W[t-2]) W.append((W[t-16] + s0 + W[t-7] + s1) & 0xFFFFFFFF) self.message_schedule = W return W def compression_round(self, t: int) -> SHA256State: """ Perform one round of SHA256 compression T1 = h + Σ1(e) + Ch(e,f,g) + K[t] + W[t] T2 = Σ0(a) + Maj(a,b,c) h = g g = f f = e e = d + T1 d = c c = b b = a a = T1 + T2 All operations mod 2^32 """ state = self.state.copy() state.round = t # Get message schedule word W_t = self.message_schedule[t] state.w_t = W_t # Compute T1 S1 = sigma1(state.e) ch_result = ch(state.e, state.f, state.g) state.t1 = (state.h + S1 + ch_result + SHA256_K[t] + W_t) & 0xFFFFFFFF # Compute T2 S0 = sigma0(state.a) maj_result = maj(state.a, state.b, state.c) state.t2 = (S0 + maj_result) & 0xFFFFFFFF # Update working variables state.h = state.g state.g = state.f state.f = state.e state.e = (state.d + state.t1) & 0xFFFFFFFF state.d = state.c state.c = state.b state.b = state.a state.a = (state.t1 + state.t2) & 0xFFFFFFFF self.state = state self.round_trace.append(state.copy()) return state def compress(self, block: bytes) -> SHA256State: """ Full compression function for one 512-bit block This is the core one-way function - easy to compute forward, computationally infeasible to invert """ # Reset state for new block self.state = SHA256State() self.round_trace = [] # Compute message schedule self.compute_message_schedule(block) # Perform 64 rounds for t in range(64): self.compression_round(t) return self.state def final_hash(self, state: SHA256State) -> bytes: """ Compute final hash by adding compressed state to initial values """ h0 = (SHA256_H0[0] + state.a) & 0xFFFFFFFF h1 = (SHA256_H0[1] + state.b) & 0xFFFFFFFF h2 = (SHA256_H0[2] + state.c) & 0xFFFFFFFF h3 = (SHA256_H0[3] + state.d) & 0xFFFFFFFF h4 = (SHA256_H0[4] + state.e) & 0xFFFFFFFF h5 = (SHA256_H0[5] + state.f) & 0xFFFFFFFF h6 = (SHA256_H0[6] + state.g) & 0xFFFFFFFF h7 = (SHA256_H0[7] + state.h) & 0xFFFFFFFF return b''.join(h.to_bytes(4, 'big') for h in [h0, h1, h2, h3, h4, h5, h6, h7]) def compute_full_sha256(self, message: bytes) -> Tuple[bytes, Dict]: """ Compute complete SHA256 hash with derivation metadata """ # Pad message padded = self.pad_message(message) # Parse into blocks blocks = self.parse_message_blocks(padded) # Process each block current_state = SHA256State() for i, block in enumerate(blocks): # Compute message schedule W = self.compute_message_schedule(block) # Set initial state (for first block) or use previous hash if i == 0: self.state = SHA256State() else: # Add previous hash to current state self.state.a = (self.state.a + current_state.a) & 0xFFFFFFFF self.state.b = (self.state.b + current_state.b) & 0xFFFFFFFF self.state.c = (self.state.c + current_state.c) & 0xFFFFFFFF self.state.d = (self.state.d + current_state.d) & 0xFFFFFFFF self.state.e = (self.state.e + current_state.e) & 0xFFFFFFFF self.state.f = (self.state.f + current_state.f) & 0xFFFFFFFF self.state.g = (self.state.g + current_state.g) & 0xFFFFFFFF self.state.h = (self.state.h + current_state.h) & 0xFFFFFFFF # Compress for t in range(64): self.compression_round(t) current_state = self.state.copy() # Final hash final_hash = self.final_hash(current_state) metadata = { "original_length": len(message), "padded_length": len(padded), "num_blocks": len(blocks), "total_rounds": len(blocks) * 64, "hash_hex": final_hash.hex() } return final_hash, metadata # ============================================================================ # SECURITY ANALYSIS AND ATTACK COMPLEXITY # ============================================================================ @dataclass class SecurityAnalysis: """Analysis of SHA256 security properties and attack complexity""" # Hash output size output_bits: int = 256 # Preimage resistance: Given h, find m such that H(m) = h preimage_complexity: int = 2**256 # Second preimage resistance: Given m1, find m2 such that H(m1) = H(m2) second_preimage_complexity: int = 2**256 # Collision resistance: Find any m1, m2 such that H(m1) = H(m2) # Birthday paradox: sqrt of search space collision_complexity: int = 2**128 # Avalanche effect: Changing 1 bit changes ~50% of output avalanche_factor: float = 0.5 def compute_attack_feasibility(self, attack_type: str) -> Dict: """ Compute feasibility of various attack types """ # Physical limits bremermann_limit = 1.36e50 # bits/sec/kg (maximum computational rate) earth_mass = 5.97e24 # kg age_of_universe_seconds = 4.35e17 # seconds (13.8 billion years) if attack_type == "brute_force_preimage": operations_needed = self.preimage_complexity time_on_earth_computer = operations_needed / (bremermann_limit * earth_mass) time_as_universe_age = time_on_earth_computer / age_of_universe_seconds return { "attack_type": "Brute Force Preimage Attack", "operations_needed": f"2^256 ≈ {operations_needed:.2e}", "time_on_planetary_computer": f"{time_on_earth_computer:.2e} seconds", "time_as_universe_ages": f"{time_as_universe_age:.2e} universe ages", "feasibility": "PHYSICALLY IMPOSSIBLE" } elif attack_type == "birthday_collision": operations_needed = self.collision_complexity time_on_earth_computer = operations_needed / (bremermann_limit * earth_mass) time_as_universe_age = time_on_earth_computer / age_of_universe_seconds return { "attack_type": "Birthday Collision Attack", "operations_needed": f"2^128 ≈ {operations_needed:.2e}", "time_on_planetary_computer": f"{time_on_earth_computer:.2e} seconds", "time_as_universe_ages": f"{time_as_universe_age:.2e} universe ages", "feasibility": "PHYSICALLY IMPOSSIBLE" } elif attack_type == "quantum_grover": # Grover's algorithm: sqrt of classical search operations_needed = 2**128 # Still 2^128 for 256-bit hash time_on_earth_computer = operations_needed / (bremermann_limit * earth_mass) return { "attack_type": "Quantum Attack (Grover's Algorithm)", "operations_needed": f"2^128 ≈ {operations_needed:.2e}", "note": "Grover provides quadratic speedup, but 2^128 is still infeasible", "feasibility": "PHYSICALLY IMPOSSIBLE with foreseeable technology" } elif attack_type == "differential_cryptanalysis": return { "attack_type": "Differential Cryptanalysis", "status": "No practical differential path found for full 64-round SHA256", "best_known_attack": "Reduced-round variants only (up to ~46 rounds)", "full_sha256_security": "256 bits (no known weakness)", "feasibility": "NO KNOWN PRACTICAL ATTACK" } return {"error": f"Unknown attack type: {attack_type}"} def analyze_avalanche_effect(self, message: bytes, deriv: SHA256Derivation) -> Dict: """ Analyze avalanche effect - changing 1 bit should change ~50% of output """ # Compute original hash original_hash, _ = deriv.compute_full_sha256(message) # Flip one bit message_bytes = bytearray(message) message_bytes[0] ^= 0x01 # Flip least significant bit of first byte modified_message = bytes(message_bytes) # Compute modified hash modified_hash, _ = deriv.compute_full_sha256(modified_message) # Count differing bits differing_bits = 0 for i in range(len(original_hash)): xor = original_hash[i] ^ modified_hash[i] differing_bits += bin(xor).count('1') total_bits = len(original_hash) * 8 avalanche_percentage = differing_bits / total_bits return { "original_hash": original_hash.hex(), "modified_hash": modified_hash.hex(), "bits_changed": differing_bits, "total_bits": total_bits, "avalanche_percentage": f"{avalanche_percentage*100:.1f}%", "ideal": "50%", "assessment": "EXCELLENT" if 0.45 <= avalanche_percentage <= 0.55 else "DEVIATION" } # ============================================================================ # MAIN EXECUTION # ============================================================================ def main(): """Demonstrate SHA256 derivation and security analysis""" print("=" * 70) print(" SHA256 MATHEMATICAL DERIVATION AND SECURITY ANALYSIS") print(" Educational Derivation - NOT FOR EXPLOITATION") print("=" * 70) print() # Initialize derivation engine deriv = SHA256Derivation() security = SecurityAnalysis() # Test message test_message = b"Bitcoin block header data for SHA256 derivation analysis" print("[SECTION 1] SHA256 ALGORITHM DERIVATION") print() print(" Initial Hash Values (H0):") print(" (First 32 bits of fractional parts of square roots of first 8 primes)") for i, h in enumerate(SHA256_H0): prime = [2, 3, 5, 7, 11, 13, 17, 19][i] print(f" H{chr(97+i)} = 0x{h:08x} ← √{prime}") print() print(" Round Constants (K):") print(" (First 32 bits of fractional parts of cube roots of first 64 primes)") print(f" K[0..7] = {[f'0x{k:08x}' for k in SHA256_K[:8]]}") print(f" ... (64 total constants)") print() print(" Compression Function:") print(" T1 = h + Σ1(e) + Ch(e,f,g) + K[t] + W[t]") print(" T2 = Σ0(a) + Maj(a,b,c)") print(" Where:") print(" Σ1(e) = ROTR⁶(e) ⊕ ROTR¹¹(e) ⊕ ROTR²⁵(e)") print(" Ch(e,f,g) = (e ∧ f) ⊕ (¬e ∧ g)") print(" Σ0(a) = ROTR²(a) ⊕ ROTR¹³(a) ⊕ ROTR²²(a)") print(" Maj(a,b,c) = (a ∧ b) ⊕ (a ∧ c) ⊕ (b ∧ c)") print() print("[SECTION 2] MESSAGE SCHEDULE EXPANSION") print() # Pad and parse message padded = deriv.pad_message(test_message) blocks = deriv.parse_message_blocks(padded) print(f" Original message length: {len(test_message)} bytes") print(f" Padded message length: {len(padded)} bytes") print(f" Number of 512-bit blocks: {len(blocks)}") print() # Compute message schedule for first block W = deriv.compute_message_schedule(blocks[0]) print(" Message Schedule W[0..15] (direct from message):") for i in range(16): print(f" W[{i:2d}] = 0x{W[i]:08x}") print() print(" Message Schedule W[16..63] (expanded using σ0, σ1):") print(" W[t] = σ1(W[t-2]) + W[t-7] + σ0(W[t-15]) + W[t-16]") for i in range(16, 24): print(f" W[{i:2d}] = 0x{W[i]:08x}") print(" ... (64 total words)") print() print("[SECTION 3] COMPRESSION ROUNDS (64 iterations)") print() # Run compression deriv.compress(blocks[0]) print(" Round trace (first 8 rounds):") for i in range(min(8, len(deriv.round_trace))): trace = deriv.round_trace[i] print(f" Round {i:2d}: a=0x{trace.a:08x} e=0x{trace.e:08x} T1=0x{trace.t1:08x}") print(" ... (64 rounds total)") print() print("[SECTION 4] FINAL HASH COMPUTATION") print() final_hash, metadata = deriv.compute_full_sha256(test_message) print(f" Final SHA256 Hash: {final_hash.hex()}") print(f" Metadata:") print(f" Original length: {metadata['original_length']} bytes") print(f" Padded length: {metadata['padded_length']} bytes") print(f" Total rounds: {metadata['total_rounds']}") print() # Verify against standard library standard_hash = hashlib.sha256(test_message).hexdigest() match = "✓ MATCH" if final_hash.hex() == standard_hash else "✗ MISMATCH" print(f" Verification against hashlib: {match}") print() print("[SECTION 5] AVALANCHE EFFECT ANALYSIS") print() avalanche = security.analyze_avalanche_effect(test_message, deriv) print(f" Original: {avalanche['original_hash']}") print(f" Modified: {avalanche['modified_hash']} (1 bit flipped)") print(f" Bits changed: {avalanche['bits_changed']}/{avalanche['total_bits']}") print(f" Avalanche: {avalanche['avalanche_percentage']} (ideal: 50%)") print(f" Assessment: {avalanche['assessment']}") print() print("[SECTION 6] SECURITY ANALYSIS AND ATTACK COMPLEXITY") print() attacks = [ "brute_force_preimage", "birthday_collision", "quantum_grover", "differential_cryptanalysis" ] for attack_type in attacks: result = security.compute_attack_feasibility(attack_type) print(f" {result['attack_type']}:") if "operations_needed" in result: print(f" Operations: {result['operations_needed']}") if "time_on_planetary_computer" in result: print(f" Time: {result['time_on_planetary_computer']}") if "time_as_universe_ages" in result: print(f" Universe ages: {result['time_as_universe_ages']}") if "status" in result: print(f" Status: {result['status']}") if "feasibility" in result: print(f" Feasibility: {result['feasibility']}") print() print("=" * 70) print(" CONCLUSION: SHA256 IS COMPUTATIONALLY SECURE") print(" No known practical attack exists for the full 64-round algorithm") print(" Brute force requires more energy than exists in the observable universe") print("=" * 70) # Save results results = { "derivation": { "initial_values": SHA256_H0, "round_constants_count": len(SHA256_K), "message_schedule_words": 64, "compression_rounds": 64 }, "test_hash": { "message": test_message.decode(), "hash": final_hash.hex(), "verification": "match" if final_hash.hex() == standard_hash else "mismatch" }, "avalanche_effect": avalanche, "security_analysis": { "preimage_resistance": "2^256 operations", "collision_resistance": "2^128 operations (birthday bound)", "quantum_resistance": "2^128 operations (Grover's algorithm)", "best_known_attack": "None for full 64-round SHA256" }, "timestamp": time.time() } output_path = Path(__file__).resolve().parent.parent / "out" / "sha256_derivation_results.json" output_path.parent.mkdir(parents=True, exist_ok=True) with open(output_path, "w") as f: json.dump(results, f, indent=2) print(f"\n[+] Results saved to: {output_path}") return 0 if __name__ == "__main__": sys.exit(main())