rosvik
/
notebooks
1
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity
Browse Source

Add square and multiply

master
Johannes Røsvik 4 years ago
parent
da33a814e6
commit
3b934c43c2
No known key found for this signature in database GPG Key ID: 8A47E30339E13FFD
1 changed files with 106 additions and 46 deletions
Whitespace
Show all changes Ignore whitespace when comparing lines Ignore changes in amount of whitespace Ignore changes in whitespace at EOL
Split View
Diff Options
Show Stats Download Patch File Download Diff File
  1. 152
      TTM4135.ipynb

152
TTM4135.ipynb
View File

@ -10,7 +10,7 @@
},
{
"cell_type": "code",
"execution_count": 14,
"execution_count": 1,
"metadata": {},
"outputs": [
{
@ -19,7 +19,7 @@
"5"
]
},
"execution_count": 14,
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
@ -46,7 +46,7 @@
},
{
"cell_type": "code",
"execution_count": 17,
"execution_count": 2,
"metadata": {},
"outputs": [
{
@ -92,7 +92,7 @@
},
{
"cell_type": "code",
"execution_count": 16,
"execution_count": 3,
"metadata": {},
"outputs": [
{
@ -340,8 +340,8 @@
"A generator of $Z_p^∗$ is an element of order $p − 1$\n",
"\n",
"To find a generator of $Z_p^∗$ we can choose a value g and test it as follows:\n",
"1. compute all the distinct prime factors of $p − 1$ and call them $f_1, f_2, ..., f_r$\n",
"2. then $g$ is a generator as long as $g^{\\frac{p−1}{f_i}} \\neq 1 \\mod(p)$ for $i = 1,2,,...,r$"
"1. Compute all the distinct prime factors of $p − 1$ and call them $f_1, f_2, ..., f_r$\n",
"2. Then $g$ is a generator as long as $g^{\\frac{p−1}{f_i}} \\neq 1 \\mod(p)$ for $i = 1,2,,...,r$"
]
},
{
@ -547,7 +547,26 @@
"Sources:\n",
"- https://stackoverflow.com/a/1832648/5976426\n",
"- https://en.wikipedia.org/wiki/Baby-step_giant-step\n",
"- https://gist.github.com/0xTowel/b4e7233fc86d8bb49698e4f1318a5a73"
"- https://gist.github.com/0xTowel/b4e7233fc86d8bb49698e4f1318a5a73\n",
"\n",
"### Square and multiply\n",
"\n",
"Used to simplify doing large exponentiations. Instead of solving $3^5$ as $3*3*3*3*3$, \n",
"\n",
"1. Convert the exponent to Binary.\n",
"2. For the first $1$, simply list the number\n",
"3. For each ensuing $0$, do Square operation\n",
"4. For each ensuing $1$, do Square and Multiply operations\n",
"\n",
"Example:\n",
"```\n",
"5 = 101 in Binary\n",
"1 First One lists Number 3\n",
"0 Zero calls for Square (3)²\n",
"1 One calls for Square + Multiply ((3)²)²*3\n",
"```\n",
"\n",
"https://www.practicalnetworking.net/stand-alone/square-and-multiply/\n"
]
},
{
@ -559,8 +578,74 @@
"name": "stdout",
"output_type": "stream",
"text": [
"x = 2570\n",
"y = g^x mod p ==> 34 = 7324^2570 mod 4363\n",
"7894352216^(1) * 7894352216^(2) * 7894352216^(16) * 7894352216^(64) * 7894352216^(1024) * 7894352216^(8192) * 7894352216^(131072) * 7894352216^(2097152) * 7894352216^(33554432) * 7894352216^(67108864)\n",
"squarings: 26\n",
"mults: 9\n",
"\n",
"z = 7894352216^102900819 mod(604604729)\n",
"z = 355407489\n",
"\n",
"7894352216^102900819 mod(604604729) = 355407489\n"
]
}
],
"source": [
"# Square and multiply\n",
"from math import log, floor\n",
"\n",
"def square_and_multiply(y,e,n, verbose=True):\n",
" # prep\n",
" e_bin = bin(e)[:1:-1] # e0, e1, e2, e3, ... , ek\n",
" k = len(e_bin)\n",
" z = 1\n",
" yi = y\n",
" indices = []\n",
" # -------------\n",
"\n",
" for i in range(k):\n",
" ei = int(e_bin[i])\n",
" if ei == 1:\n",
" z = z*yi % n\n",
" if ei < k:\n",
" yi = yi*yi % n\n",
" if ei:\n",
" indices.append(i)\n",
"\n",
" # post calc\n",
" mults = str(e_bin).count(\"1\") - 1\n",
" squarings = floor( log(e, 2) )\n",
" # -------------\n",
"\n",
" if verbose:\n",
" print(f\"{' * '.join([f'{y}^({2**i})' for i in indices])}\")\n",
" print(f\"squarings: {squarings}\")\n",
" print(f\"mults: {mults}\")\n",
" print()\n",
" print(f\"z = {y}^{e} mod({n})\")\n",
" print(f\"z = {z}\\n\")\n",
" return z\n",
"\n",
"# y^e mod(n)\n",
"y = 7894352216\n",
"e = 102900819\n",
"n = 604604729\n",
"\n",
"z = square_and_multiply(y,e,n, True)\n",
"\n",
"print(f\"{y}^{e} mod({n}) = {z}\")"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x = 102900819\n",
"y = g^x mod p ==> 355407489 = 7894352216^102900819 mod 604604729\n",
"Test: True\n"
]
}
@ -591,16 +676,18 @@
" return None\n",
"\n",
"def disc_log_test(y, g, x, p):\n",
" return y == (g**x) % p\n",
" #return y == (g**x) % p\n",
" return y == square_and_multiply(g,x,p,False)\n",
"\n",
"#y = 355407489\n",
"#g = 7894352216\n",
"#p = 604604729 # Must be prime\n",
"\n",
"# y = g^x mod p\n",
"y = 34\n",
"g = 7324\n",
"p = 4363 # Must be prime\n",
"y = 355407489\n",
"g = 7894352216\n",
"p = 604604729 # Must be prime\n",
"\n",
"#y = 34\n",
"#g = 7324\n",
"#p = 4363 # Must be prime\n",
"\n",
"x = bsgs(g, y, p)\n",
"\n",
@ -608,8 +695,7 @@
"\n",
"print(f\"y = g^x mod p ==> {y} = {g}^{x} mod {p}\")\n",
"\n",
"# Don't run the test with big numbers!\n",
"if x<10000: print(f\"Test: {disc_log_test(y,g,x,p)}\")"
"print(f\"Test: {disc_log_test(y,g,x,p)}\")"
]
},
{
@ -624,34 +710,8 @@
"- `C`: Ciphertext block (length n bits) \n",
"- `K` : Key (length k bits)\n",
"- `C = E(P, K)`: Encryption function \n",
"- `P = D(C, K)`: Decryption function"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]\n",
"54\n"
]
}
],
"source": [
"# Discarded code\n",
"\n",
"def z(n):\n",
" return [i for i in range(n)]\n",
"print(z(10))\n",
"\n",
"n = 81\n",
"zn = z(n)\n",
"rel_primes = list(filter(lambda x: is_relative_prime(x,n), zn))\n",
"print(len(rel_primes))"
"- `P = D(C, K)`: Decryption function\n",
"- `IV`: Initialisation vector"
]
}
],

Write Preview
Loading…
Cancel
Save