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