Added public key compress and uncompress

Author: Vizeet Srivastava

BitcoinEllipticCurveCryptography.ipynb

@@ -30,7 +30,7 @@
     {
      "data": {
       "application/vnd.jupyter.widget-view+json": {
-       "model_id": "9ece5999755e498b90475d64ba6270f2"
+       "model_id": "92eefc2b63a1450e927aeec68465492e"
       }
      },
      "metadata": {},
@@ -209,9 +209,14 @@
     "λ = (yq-yp)/(xq-xp) % P<br>\n",
     "To obtain λ we need to find multiplicative-inverse of (xq-xp) in finite field<br>\n",
     "=> (yq-yp) \\* Multiplicate-Inverse(xq-xp) % P<br>\n",
-    "In Python Multiplicative inverse of x in finite field by modulo prime P is:<br>\n",
-    "pow(x, P-2, P)<br>\n",
-    "=> (yq-yp) \\* pow(xq-xp, P - 2, P) % P<br>\n",
+    "########### <br>\n",
+    "Multiplicative inverse i of a number N in finite field is calculated by :<br>\n",
+    "N * i % P = 1 <br>\n",
+    "since N^(P-1) % P = 1 <br>\n",
+    "=> i = N^(P-2) <br>\n",
+    "=> (1/N) % P = N^(P-2) % P <br>\n",
+    "########## <br>\n",
+    "=> ((yq-yp) \\* ((xq-xp)^(P - 2) % P)) % P<br>\n",
     "From (7)<br>\n",
     "xr = (λ^2 - xp - xq) % P <br>\n",
     "From (6)<br>\n",
@@ -442,7 +447,155 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<h2> Compressing Point on Elliptic Curve </h2>"
+    "<h2> Compressing/Uncompressing Point on Elliptic Curve secp256k1 </h2>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<h3> Compressing Point </h3>\n",
+    "<p>\n",
+    "For a given x0 there are two values of y i.e. y0, -y0 <br>\n",
+    "In finite field Ƒp: <br>\n",
+    "The two values of y are y0 and P - y0 <br>\n",
+    "This gives us equation: y1 = P - y0 <br>\n",
+    "Since P is odd in secp256k1 for any given x: y is either odd or even. <br>\n",
+    "So y is removed by adding prefix denoting whether y is odd or even. <br>\n",
+    "0x02 prefix denote y is even and 0x03 prefix denote y is odd. <br>\n",
+    "0x04 prefix denote point is uncompressed.\n",
+    "</p>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "6e81e321819442e68344276458ce1050"
+      }
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import ipywidgets as widgets\n",
+    "from IPython.display import display \n",
+    "import binascii\n",
+    "\n",
+    "def compressPubkey(pubkey: bytes):\n",
+    "    if pubkey[0:1] == b'\\x04':\n",
+    "        x_b = pubkey[1:33]\n",
+    "        y_b = pubkey[33:65]\n",
+    "        if (y_b[31] & 0x01) == 0: # even\n",
+    "                compressed_pubkey = b'\\x02' + x_b\n",
+    "        else:\n",
+    "                compressed_pubkey = b'\\x03' + x_b\n",
+    "    elif pubkey[0:1] == b'\\x02' or pubkey[0:1] == b'\\x03':\n",
+    "        compressed_pubkey = pubkey\n",
+    "    else:\n",
+    "        raise ValueError('Invalid Elliptic Curve Point prefix %s' % pubkey[0])\n",
+    "    return compressed_pubkey\n",
+    "\n",
+    "pubkey_s = widgets.Text(description = 'Public Key')\n",
+    "pubkey_s"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0250863ad64a87ae8a2fe83c1af1a8403cb53f53e486d8511dad8a04887e5b2352\n"
+     ]
+    }
+   ],
+   "source": [
+    "compressed_pubkey = compressPubkey(binascii.unhexlify(pubkey_s.value))\n",
+    "compressed_pubkey_s = bytes.decode(binascii.hexlify(compressed_pubkey))\n",
+    "print(compressed_pubkey_s)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<h3> Uncompressing Point </h3>\n",
+    "<p>\n",
+    "Curve E: y^2 = x^3 + 7 over Ƒp <br>\n",
+    "and P % 4 = 3 <br>\n",
+    "=> y % p = y^(2*(p + 1)/4) % p <br>\n",
+    "</p>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "e5e6ade26d2e4d989461d337f05fe47d"
+      }
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import ipywidgets as widgets\n",
+    "from IPython.display import display \n",
+    "import binascii\n",
+    "\n",
+    "# field prime\n",
+    "P = 2 ** 256 - 2 ** 32 - 2 ** 9 - 2 ** 8 - 2 ** 7 - 2 ** 6 - 2 ** 4 - 1\n",
+    "\n",
+    "def uncompressPubkey(x_b: bytes):\n",
+    "        prefix = x_b[0:1]\n",
+    "        x_b = x_b[1:33]\n",
+    "        x = int.from_bytes(x_b, byteorder='big')\n",
+    "\n",
+    "        y_square = (pow(x, 3, P)  + 7) % P\n",
+    "        y_square_square_root = pow(y_square, ((P+1) >> 2), P)\n",
+    "        if (prefix == b\"\\x02\" and y_square_square_root & 1) or (prefix == b\"\\x03\" and not y_square_square_root & 1):\n",
+    "            y = (-y_square_square_root) % P\n",
+    "        else:\n",
+    "            y = y_square_square_root\n",
+    "\n",
+    "        y_b = y.to_bytes(32, 'big')\n",
+    "        full_pubkey_b = b''.join([b'\\x04', x_b, y_b])\n",
+    "        return full_pubkey_b\n",
+    "    \n",
+    "pubkey_s = widgets.Text(description = 'Public Key')\n",
+    "pubkey_s"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0450863ad64a87ae8a2fe83c1af1a8403cb53f53e486d8511dad8a04887e5b23522cd470243453a299fa9e77237716103abc11a1df38855ed6f2ee187e9c582ba6\n"
+     ]
+    }
+   ],
+   "source": [
+    "uncompressed_pubkey = uncompressPubkey(binascii.unhexlify(pubkey_s.value))\n",
+    "uncompressed_pubkey_s = bytes.decode(binascii.hexlify(uncompressed_pubkey))\n",
+    "print(uncompressed_pubkey_s)"
    ]
   },
   {