Skinny.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "9d9ef19f",
   "metadata": {},
   "outputs": [],
   "source": [
    "import spnutils\n",
    "import embedding\n",
    "import crv"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4b8af2dd",
   "metadata": {},
   "source": [
    "### Find suitable field for polynomial interpolation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "44578b75",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Best: 14 non-zeros\n",
      "modulus: x^4 + x^3 + 1\n"
     ]
    }
   ],
   "source": [
    "SKINNY_S4 = [0xc, 0x6, 0x9, 0x0, 0x1, 0xa, 0x2, 0xb, 0x3, 0x8, 0x5, 0xd, 0x4, 0xe, 0x7, 0xf]\n",
    "fields, non_zero = spnutils.find_field_for_sparse_interpolation(4, list(range(16)), SKINNY_S4)\n",
    "print(f'Best: {non_zero} non-zeros')\n",
    "for f in fields:\n",
    "    print(f'modulus: {f.modulus()}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "10925ff2",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Best: 13 non-zeros\n",
      "modulus: x^4 + x + 1\n",
      "modulus: x^4 + x^3 + 1\n"
     ]
    }
   ],
   "source": [
    "SKINNY_S4_inv = [0x3, 0x4, 0x6, 0x8, 0xc, 0xa, 0x1, 0xe, 0x9, 0x2, 0x5, 0x7, 0x0, 0xb, 0xd, 0xf]\n",
    "fields, non_zero = spnutils.find_field_for_sparse_interpolation(4, list(range(16)), SKINNY_S4_inv)\n",
    "print(f'Best: {non_zero} non-zeros')\n",
    "for f in fields:\n",
    "    print(f'modulus: {f.modulus()}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a270831e",
   "metadata": {},
   "source": [
    "### Find suitable embedding"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "17a7b930",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = polygen(GF(2), 'x')\n",
    "Y = polygen(GF(2), 'Y')\n",
    "F4 = GF(2**4, name='x', modulus=x**4 + x**3 + 1)\n",
    "F40 = GF(2**40, name='Y', modulus=Y^40 + Y^20 + Y^15 + Y^10 + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "413d9904",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Embedding from Finite Field in x of size 2^4 to Finite Field in Y of size 2^40 via Y^35 + Y^20 + Y^5 + 1\n"
     ]
    }
   ],
   "source": [
    "emb, _ = embedding.find_min_cost_embedding(F4, F40)\n",
    "print(emb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "8c6ad876",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Forward embedding\n",
      "[1 1 1 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 1 1 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 1]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 1 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 1 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 1]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 0 0 0]\n",
      "[0 1 0 0]\n",
      "Inverse embedding\n",
      "([1 1 0 0]\n",
      "[0 1 0 1]\n",
      "[0 0 0 1]\n",
      "[0 0 1 0], [0, 5, 10, 15])\n"
     ]
    }
   ],
   "source": [
    "print('Forward embedding')\n",
    "print(emb.bit_embedding())\n",
    "\n",
    "print('Inverse embedding')\n",
    "print(emb.inverse_bit_embedding())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "e8279d5a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 0 1 1]\n",
      "[0 0 0 1]\n",
      "[0 1 0 1]\n",
      "[0 0 1 1]\n"
     ]
    }
   ],
   "source": [
    "print(spnutils.gf_squaring(F4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "23794e71",
   "metadata": {},
   "outputs": [],
   "source": [
    "SKINNY_S4_F = [spnutils.natural_encoding(F4, x) for x in SKINNY_S4]\n",
    "SKINNY_S4_inv_F = [spnutils.natural_encoding(F4, x) for x in SKINNY_S4_inv]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d3da4cd9",
   "metadata": {},
   "source": [
    "### MUL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "a587de35",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0xc z^0 + 0x8 z^1 + 0x3 z^2 + 0xd z^3 + 0xf z^4 + 0x4 z^5 + 0x8 z^6 + 0x6 z^7 + 0x1 z^8 + 0x9 z^9 + 0x8 z^10 + 0x0 z^11 + 0xe z^12 + 0xc z^13 + 0xb z^14 + 0x0 z^15\n"
     ]
    }
   ],
   "source": [
    "c = spnutils.interpolate(F4, [spnutils.natural_encoding(F4, i) for i in range(16)], SKINNY_S4_F)\n",
    "s = \" + \".join(f'0x{spnutils.natural_encoding_to_int(ci):x} z^{i}' for i,ci in enumerate(c) if c != 0)\n",
    "print(s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "af9479f1",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0x3 z^0 + 0xa z^1 + 0x6 z^2 + 0x7 z^3 + 0x5 z^4 + 0xa z^5 + 0x0 z^6 + 0x7 z^7 + 0x9 z^8 + 0x6 z^9 + 0x9 z^10 + 0x0 z^11 + 0x4 z^12 + 0xd z^13 + 0xb z^14 + 0x0 z^15\n"
     ]
    }
   ],
   "source": [
    "c = spnutils.interpolate(F4, [spnutils.natural_encoding(F4, i) for i in range(16)], SKINNY_S4_inv_F)\n",
    "s = \" + \".join(f'0x{spnutils.natural_encoding_to_int(ci):x} z^{i}' for i,ci in enumerate(c) if c != 0)\n",
    "print(s)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e62face7",
   "metadata": {},
   "source": [
    "### CRV decomposition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "b4b981b0",
   "metadata": {},
   "outputs": [],
   "source": [
    "#instance = crv.CRV(F4, [0,1,3])\n",
    "#instance.find_q_polynomials(t=2)\n",
    "\n",
    "# set q polynomial from paper\n",
    "instance = crv.CRV(F4, [0,1,3])\n",
    "q0 = [spnutils.natural_encoding(F4, c) for c in [0x0,0x6,0x4,0xd,0x3,0x4,0x8,0xb,0x8]]\n",
    "instance.set_q_polynomials([q0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "60b90e9e",
   "metadata": {},
   "outputs": [],
   "source": [
    "res = instance.polynomial_decomposition(SKINNY_S4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "62dd8ca8",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "alphas = [0, 1, 3]\n",
      "L = [0, 1, 2, 3, 4, 6, 8, 9, 12]\n",
      "t = 2\n"
     ]
    }
   ],
   "source": [
    "print(f'alphas = {res.alphas}')\n",
    "print(f'L = {res.L}')\n",
    "print(f't = {res.t}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "34993581",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0x6 z^1,0x4 z^2,0xd z^3,0x3 z^4,0x4 z^6,0x8 z^8,0xb z^9,0x8 z^12\n"
     ]
    }
   ],
   "source": [
    "for q in res.q:\n",
    "    print(','.join(f'0x{spnutils.natural_encoding_to_int(c):x} z^{i}' for i,c in enumerate(q.coefficients(sparse=False)) if c != 0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "fde437c3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p_0: 0x1 z^0 + 0xc z^1 + 0xf z^2 + 0x1 z^3 + 0x5 z^4 + 0x2 z^5 + 0xb z^6 + 0xb z^7\n",
      "p_1: 0xc z^0 + 0x3 z^1 + 0x8 z^3 + 0xe z^4 + 0xa z^5 + 0x9 z^7\n"
     ]
    }
   ],
   "source": [
    "for j,p in enumerate(res.p):\n",
    "    pi = [F4(list(c)[0]) for c in p]\n",
    "    print(f'p_{j}: ' + ' + '.join(f'0x{spnutils.natural_encoding_to_int(c):x} z^{i}' for i,c in enumerate(pi) if c != 0))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "db4d5e7f",
   "metadata": {},
   "source": [
    "### CRV decomposition for inverse S-box"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "9a16468c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p0: 0x7 + 0x7 + 0x7 + 0x9 + 0x3 + 0xa + 0xc + 0xf + 0x7\n",
      "p1: 0x3 + 0xd + 0x4 + 0xf + 0x5 + 0x7 + 0x0 + 0x6 + 0x0\n"
     ]
    }
   ],
   "source": [
    "res_inv = instance.polynomial_decomposition(SKINNY_S4_inv)\n",
    "for i,p in enumerate(res_inv.p):\n",
    "    print(f'p{i}: ' + ' + '.join(f'0x{spnutils.natural_encoding_to_int(pi[0]):x}' for pi in p))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "aa50dd78",
   "metadata": {},
   "outputs": [],
   "source": [
    "# test correctness of the decomposition\n",
    "\n",
    "Z = polygen(GF(2), 'Z')\n",
    "p0 = sum(spnutils.natural_encoding(F4, c) * Z**i for i,c in enumerate([0x1,0xc,0xf,0x1,0x5,0x2,0xb,0xb,0x0]))\n",
    "p1 = sum(spnutils.natural_encoding(F4, c) * Z**i for i,c in enumerate([0xc,0x3,0x0,0x8,0xe,0xa,0x0,0x9,0x0]))\n",
    "q0 = sum(spnutils.natural_encoding(F4, c) * Z**i for i,c in enumerate([0x0,0x6,0x4,0xd,0x3,0x4,0x8,0xb,0x8]))\n",
    "\n",
    "def eval_crv(x):\n",
    "    return p0(x) * q0(x) + p1(x)\n",
    "\n",
    "for x,y in enumerate(SKINNY_S4):\n",
    "    x_f = spnutils.natural_encoding(F4, x)\n",
    "    e = res.compute(x_f)[0]#eval_crv(x_f)\n",
    "    y_f = spnutils.natural_encoding(F4, y)\n",
    "    #e = spnutils.natural_encoding_to_int(e)\n",
    "    assert e == y_f, f'x=0x{x:x}, e=0x{e}, y=0x{y_f}'"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6cd47b01",
   "metadata": {},
   "source": [
    "# SKINNY's 8-bit S-box"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "6a1f3035",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Best: 244 non-zeros\n",
      "modulus: x^8 + x^6 + x^5 + x^4 + x^3 + x + 1\n",
      "modulus: x^8 + x^7 + x^6 + x^5 + x^4 + x^2 + 1\n"
     ]
    }
   ],
   "source": [
    "SKINNY_S8 = [\n",
    "    0x65, 0x4c, 0x6a, 0x42, 0x4b, 0x63, 0x43, 0x6b, 0x55, 0x75, 0x5a, 0x7a, 0x53, 0x73, 0x5b,\n",
    "    0x7b, 0x35, 0x8c, 0x3a, 0x81, 0x89, 0x33, 0x80, 0x3b, 0x95, 0x25, 0x98, 0x2a, 0x90, 0x23,\n",
    "    0x99, 0x2b, 0xe5, 0xcc, 0xe8, 0xc1, 0xc9, 0xe0, 0xc0, 0xe9, 0xd5, 0xf5, 0xd8, 0xf8, 0xd0,\n",
    "    0xf0, 0xd9, 0xf9, 0xa5, 0x1c, 0xa8, 0x12, 0x1b, 0xa0, 0x13, 0xa9, 0x05, 0xb5, 0x0a, 0xb8,\n",
    "    0x03, 0xb0, 0x0b, 0xb9, 0x32, 0x88, 0x3c, 0x85, 0x8d, 0x34, 0x84, 0x3d, 0x91, 0x22, 0x9c,\n",
    "    0x2c, 0x94, 0x24, 0x9d, 0x2d, 0x62, 0x4a, 0x6c, 0x45, 0x4d, 0x64, 0x44, 0x6d, 0x52, 0x72,\n",
    "    0x5c, 0x7c, 0x54, 0x74, 0x5d, 0x7d, 0xa1, 0x1a, 0xac, 0x15, 0x1d, 0xa4, 0x14, 0xad, 0x02,\n",
    "    0xb1, 0x0c, 0xbc, 0x04, 0xb4, 0x0d, 0xbd, 0xe1, 0xc8, 0xec, 0xc5, 0xcd, 0xe4, 0xc4, 0xed,\n",
    "    0xd1, 0xf1, 0xdc, 0xfc, 0xd4, 0xf4, 0xdd, 0xfd, 0x36, 0x8e, 0x38, 0x82, 0x8b, 0x30, 0x83,\n",
    "    0x39, 0x96, 0x26, 0x9a, 0x28, 0x93, 0x20, 0x9b, 0x29, 0x66, 0x4e, 0x68, 0x41, 0x49, 0x60,\n",
    "    0x40, 0x69, 0x56, 0x76, 0x58, 0x78, 0x50, 0x70, 0x59, 0x79, 0xa6, 0x1e, 0xaa, 0x11, 0x19,\n",
    "    0xa3, 0x10, 0xab, 0x06, 0xb6, 0x08, 0xba, 0x00, 0xb3, 0x09, 0xbb, 0xe6, 0xce, 0xea, 0xc2,\n",
    "    0xcb, 0xe3, 0xc3, 0xeb, 0xd6, 0xf6, 0xda, 0xfa, 0xd3, 0xf3, 0xdb, 0xfb, 0x31, 0x8a, 0x3e,\n",
    "    0x86, 0x8f, 0x37, 0x87, 0x3f, 0x92, 0x21, 0x9e, 0x2e, 0x97, 0x27, 0x9f, 0x2f, 0x61, 0x48,\n",
    "    0x6e, 0x46, 0x4f, 0x67, 0x47, 0x6f, 0x51, 0x71, 0x5e, 0x7e, 0x57, 0x77, 0x5f, 0x7f, 0xa2,\n",
    "    0x18, 0xae, 0x16, 0x1f, 0xa7, 0x17, 0xaf, 0x01, 0xb2, 0x0e, 0xbe, 0x07, 0xb7, 0x0f, 0xbf,\n",
    "    0xe2, 0xca, 0xee, 0xc6, 0xcf, 0xe7, 0xc7, 0xef, 0xd2, 0xf2, 0xde, 0xfe, 0xd7, 0xf7, 0xdf,\n",
    "    0xff]\n",
    "fields, non_zero = spnutils.find_field_for_sparse_interpolation(8, list(range(256)), SKINNY_S8)\n",
    "print(f'Best: {non_zero} non-zeros')\n",
    "for f in fields:\n",
    "    print(f'modulus: {f.modulus()}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "034cc764",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Best: 241 non-zeros\n",
      "modulus: x^8 + x^7 + x^6 + x^5 + x^4 + x^2 + 1\n"
     ]
    }
   ],
   "source": [
    "SKINNY_S8_inv = [SKINNY_S8.index(i) for i in range(256)]\n",
    "fields, non_zero = spnutils.find_field_for_sparse_interpolation(8, list(range(256)), SKINNY_S8_inv)\n",
    "print(f'Best: {non_zero} non-zeros')\n",
    "for f in fields:\n",
    "    print(f'modulus: {f.modulus()}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "9892341a",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = polygen(GF(2), 'x')\n",
    "Y = polygen(GF(2), 'Y')\n",
    "F8 = GF(2**8, name='x', modulus=x^8 + x^7 + x^6 + x^5 + x^4 + x^2 + 1)\n",
    "F40 = GF(2**40, name='Y', modulus=Y^40 + Y^20 + Y^15 + Y^10 + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "5f0a37aa",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "38\n",
      "Embedding from Finite Field in x of size 2^8 to Finite Field in Y of size 2^40 via Y^35 + Y^30 + Y^25 + Y^20 + Y^10 + Y^5\n"
     ]
    }
   ],
   "source": [
    "emb, cost = embedding.find_min_cost_embedding(F8, F40)\n",
    "print(cost)\n",
    "print(emb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "451e6bc3",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Forward embedding\n",
      "[1 0 1 0 0 1 1 1]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 1 1 1 1 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 1 0 0 0 0 1 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 1 0 0 1 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 1 0 1 0 1 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 1 1 1 1 0 1 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 1 0 0 1 0 0 1]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 0 0 0 0 0 0 0]\n",
      "[0 1 1 0 0 0 0 0]\n",
      "Inverse embedding\n",
      "([1 1 1 0 1 0 1 0]\n",
      "[0 1 1 0 0 1 0 0]\n",
      "[0 1 1 0 0 1 0 1]\n",
      "[0 1 0 1 0 1 0 0]\n",
      "[0 0 0 1 0 1 0 1]\n",
      "[0 0 1 1 1 0 0 0]\n",
      "[0 1 0 0 0 1 0 0]\n",
      "[0 1 1 1 0 0 1 1], [0, 5, 10, 15, 20, 25, 30, 35])\n"
     ]
    }
   ],
   "source": [
    "print('Forward embedding')\n",
    "print(emb.bit_embedding())\n",
    "\n",
    "print('Inverse embedding')\n",
    "print(emb.inverse_bit_embedding())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "67e6c97d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y0 = x0 + x2 + x5 + x6 + x7\n",
      "y5 = x1 + x2 + x3 + x4\n",
      "y10 = x1 + x6\n",
      "y15 = x3 + x6\n",
      "y20 = x1 + x3 + x5\n",
      "y25 = x1 + x2 + x3 + x4 + x6\n",
      "y30 = x1 + x4 + x7\n",
      "y35 = x1 + x2\n"
     ]
    }
   ],
   "source": [
    "emb.print_impl_forward()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "e78b351a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x0 = y0 + y5 + y10 + y20 + y30\n",
      "x1 = y5 + y10 + y25\n",
      "x2 = y5 + y10 + y25 + y35\n",
      "x3 = y5 + y15 + y25\n",
      "x4 = y15 + y25 + y35\n",
      "x5 = y10 + y15 + y20\n",
      "x6 = y5 + y25\n",
      "x7 = y5 + y10 + y15 + y30 + y35\n"
     ]
    }
   ],
   "source": [
    "emb.print_impl_backward()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "8a983a49",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1 0 0 0 1 0 0 0]\n",
       "[0 0 0 0 0 1 0 1]\n",
       "[0 1 0 0 1 1 0 0]\n",
       "[0 0 0 0 0 1 1 1]\n",
       "[0 0 1 0 1 1 1 0]\n",
       "[0 0 0 0 1 1 1 0]\n",
       "[0 0 0 1 1 0 1 0]\n",
       "[0 0 0 0 1 0 1 0]"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spnutils.gf_squaring(F8)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6be15cbe-8003-4ca2-9019-33073471e1a5",
   "metadata": {},
   "source": [
    "# CRV decomposition paramters for the 8-bit S-box"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "36c75ebe",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "round 1: [0, 1, 2, 4, 8, 16, 32, 64, 128]\n"
     ]
    }
   ],
   "source": [
    "def next_cyclotomic_classes(L, start):\n",
    "    \"\"\" \n",
    "    Prints the number of elements that will be added to the set L, i.e., the computable powers in the decomposition, for each base value\n",
    "    ranging from start until 254\n",
    "    \"\"\"\n",
    "    L = set(L)\n",
    "    for i in range(start, 2**8-1):\n",
    "        class_i = set(crv._generate_cyclotomic_class(8,i))\n",
    "        new = len(class_i) - len(class_i & L)\n",
    "        if new > 0:\n",
    "            print(f'{i}: {new}')\n",
    "r1 = sorted({0,1} | crv._generate_cyclotomic_class(8, 2))\n",
    "print(f'round 1: {r1}')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "abf98c25",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "33\n",
      "7: 8\n",
      "11: 8\n",
      "13: 8\n",
      "14: 8\n",
      "15: 8\n",
      "17: 4\n",
      "19: 8\n",
      "21: 8\n",
      "22: 8\n",
      "23: 8\n",
      "25: 8\n",
      "26: 8\n",
      "27: 8\n",
      "28: 8\n",
      "29: 8\n",
      "30: 8\n",
      "31: 8\n",
      "34: 4\n",
      "35: 8\n",
      "37: 8\n",
      "38: 8\n",
      "39: 8\n",
      "41: 8\n",
      "42: 8\n",
      "43: 8\n",
      "44: 8\n",
      "45: 8\n",
      "46: 8\n",
      "47: 8\n",
      "49: 8\n",
      "50: 8\n",
      "51: 4\n",
      "52: 8\n",
      "53: 8\n",
      "54: 8\n",
      "55: 8\n",
      "56: 8\n",
      "57: 8\n",
      "58: 8\n",
      "59: 8\n",
      "60: 8\n",
      "61: 8\n",
      "62: 8\n",
      "63: 8\n",
      "67: 8\n",
      "68: 4\n",
      "69: 8\n",
      "70: 8\n",
      "71: 8\n",
      "73: 8\n",
      "74: 8\n",
      "75: 8\n",
      "76: 8\n",
      "77: 8\n",
      "78: 8\n",
      "79: 8\n",
      "81: 8\n",
      "82: 8\n",
      "83: 8\n",
      "84: 8\n",
      "85: 2\n",
      "86: 8\n",
      "87: 8\n",
      "88: 8\n",
      "89: 8\n",
      "90: 8\n",
      "91: 8\n",
      "92: 8\n",
      "93: 8\n",
      "94: 8\n",
      "95: 8\n",
      "97: 8\n",
      "98: 8\n",
      "99: 8\n",
      "100: 8\n",
      "101: 8\n",
      "102: 4\n",
      "103: 8\n",
      "104: 8\n",
      "105: 8\n",
      "106: 8\n",
      "107: 8\n",
      "108: 8\n",
      "109: 8\n",
      "110: 8\n",
      "111: 8\n",
      "112: 8\n",
      "113: 8\n",
      "114: 8\n",
      "115: 8\n",
      "116: 8\n",
      "117: 8\n",
      "118: 8\n",
      "119: 4\n",
      "120: 8\n",
      "121: 8\n",
      "122: 8\n",
      "123: 8\n",
      "124: 8\n",
      "125: 8\n",
      "126: 8\n",
      "127: 8\n",
      "131: 8\n",
      "133: 8\n",
      "134: 8\n",
      "135: 8\n",
      "136: 4\n",
      "137: 8\n",
      "138: 8\n",
      "139: 8\n",
      "140: 8\n",
      "141: 8\n",
      "142: 8\n",
      "143: 8\n",
      "145: 8\n",
      "146: 8\n",
      "147: 8\n",
      "148: 8\n",
      "149: 8\n",
      "150: 8\n",
      "151: 8\n",
      "152: 8\n",
      "153: 4\n",
      "154: 8\n",
      "155: 8\n",
      "156: 8\n",
      "157: 8\n",
      "158: 8\n",
      "159: 8\n",
      "161: 8\n",
      "162: 8\n",
      "163: 8\n",
      "164: 8\n",
      "165: 8\n",
      "166: 8\n",
      "167: 8\n",
      "168: 8\n",
      "169: 8\n",
      "170: 2\n",
      "171: 8\n",
      "172: 8\n",
      "173: 8\n",
      "174: 8\n",
      "175: 8\n",
      "176: 8\n",
      "177: 8\n",
      "178: 8\n",
      "179: 8\n",
      "180: 8\n",
      "181: 8\n",
      "182: 8\n",
      "183: 8\n",
      "184: 8\n",
      "185: 8\n",
      "186: 8\n",
      "187: 4\n",
      "188: 8\n",
      "189: 8\n",
      "190: 8\n",
      "191: 8\n",
      "193: 8\n",
      "194: 8\n",
      "195: 8\n",
      "196: 8\n",
      "197: 8\n",
      "198: 8\n",
      "199: 8\n",
      "200: 8\n",
      "201: 8\n",
      "202: 8\n",
      "203: 8\n",
      "204: 4\n",
      "205: 8\n",
      "206: 8\n",
      "207: 8\n",
      "208: 8\n",
      "209: 8\n",
      "210: 8\n",
      "211: 8\n",
      "212: 8\n",
      "213: 8\n",
      "214: 8\n",
      "215: 8\n",
      "216: 8\n",
      "217: 8\n",
      "218: 8\n",
      "219: 8\n",
      "220: 8\n",
      "221: 4\n",
      "222: 8\n",
      "223: 8\n",
      "224: 8\n",
      "225: 8\n",
      "226: 8\n",
      "227: 8\n",
      "228: 8\n",
      "229: 8\n",
      "230: 8\n",
      "231: 8\n",
      "232: 8\n",
      "233: 8\n",
      "234: 8\n",
      "235: 8\n",
      "236: 8\n",
      "237: 8\n",
      "238: 4\n",
      "239: 8\n",
      "240: 8\n",
      "241: 8\n",
      "242: 8\n",
      "243: 8\n",
      "244: 8\n",
      "245: 8\n",
      "246: 8\n",
      "247: 8\n",
      "248: 8\n",
      "249: 8\n",
      "250: 8\n",
      "251: 8\n",
      "252: 8\n",
      "253: 8\n",
      "254: 8\n",
      "round 2: [0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 32, 33, 36, 40, 48, 64, 65, 66, 72, 80, 96, 128, 129, 130, 132, 144, 160, 192]\n"
     ]
    }
   ],
   "source": [
    "r2 = sorted(set(r1) | crv._generate_cyclotomic_class(8, 3) \n",
    "            | crv._generate_cyclotomic_class(8, 5) \n",
    "            | crv._generate_cyclotomic_class(8, 9))\n",
    "print(len(r2))\n",
    "next_cyclotomic_classes(r2,4)\n",
    "print(f'round 2: {r2}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "4a99d49c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "41\n",
      "11: 8\n",
      "13: 8\n",
      "15: 8\n",
      "17: 4\n",
      "19: 8\n",
      "21: 8\n",
      "22: 8\n",
      "23: 8\n",
      "25: 8\n",
      "26: 8\n",
      "27: 8\n",
      "29: 8\n",
      "30: 8\n",
      "31: 8\n",
      "34: 4\n",
      "35: 8\n",
      "37: 8\n",
      "38: 8\n",
      "39: 8\n",
      "41: 8\n",
      "42: 8\n",
      "43: 8\n",
      "44: 8\n",
      "45: 8\n",
      "46: 8\n",
      "47: 8\n",
      "49: 8\n",
      "50: 8\n",
      "51: 4\n",
      "52: 8\n",
      "53: 8\n",
      "54: 8\n",
      "55: 8\n",
      "57: 8\n",
      "58: 8\n",
      "59: 8\n",
      "60: 8\n",
      "61: 8\n",
      "62: 8\n",
      "63: 8\n",
      "67: 8\n",
      "68: 4\n",
      "69: 8\n",
      "70: 8\n",
      "71: 8\n",
      "73: 8\n",
      "74: 8\n",
      "75: 8\n",
      "76: 8\n",
      "77: 8\n",
      "78: 8\n",
      "79: 8\n",
      "81: 8\n",
      "82: 8\n",
      "83: 8\n",
      "84: 8\n",
      "85: 2\n",
      "86: 8\n",
      "87: 8\n",
      "88: 8\n",
      "89: 8\n",
      "90: 8\n",
      "91: 8\n",
      "92: 8\n",
      "93: 8\n",
      "94: 8\n",
      "95: 8\n",
      "97: 8\n",
      "98: 8\n",
      "99: 8\n",
      "100: 8\n",
      "101: 8\n",
      "102: 4\n",
      "103: 8\n",
      "104: 8\n",
      "105: 8\n",
      "106: 8\n",
      "107: 8\n",
      "108: 8\n",
      "109: 8\n",
      "110: 8\n",
      "111: 8\n",
      "113: 8\n",
      "114: 8\n",
      "115: 8\n",
      "116: 8\n",
      "117: 8\n",
      "118: 8\n",
      "119: 4\n",
      "120: 8\n",
      "121: 8\n",
      "122: 8\n",
      "123: 8\n",
      "124: 8\n",
      "125: 8\n",
      "126: 8\n",
      "127: 8\n",
      "133: 8\n",
      "134: 8\n",
      "135: 8\n",
      "136: 4\n",
      "137: 8\n",
      "138: 8\n",
      "139: 8\n",
      "140: 8\n",
      "141: 8\n",
      "142: 8\n",
      "143: 8\n",
      "145: 8\n",
      "146: 8\n",
      "147: 8\n",
      "148: 8\n",
      "149: 8\n",
      "150: 8\n",
      "151: 8\n",
      "152: 8\n",
      "153: 4\n",
      "154: 8\n",
      "155: 8\n",
      "156: 8\n",
      "157: 8\n",
      "158: 8\n",
      "159: 8\n",
      "161: 8\n",
      "162: 8\n",
      "163: 8\n",
      "164: 8\n",
      "165: 8\n",
      "166: 8\n",
      "167: 8\n",
      "168: 8\n",
      "169: 8\n",
      "170: 2\n",
      "171: 8\n",
      "172: 8\n",
      "173: 8\n",
      "174: 8\n",
      "175: 8\n",
      "176: 8\n",
      "177: 8\n",
      "178: 8\n",
      "179: 8\n",
      "180: 8\n",
      "181: 8\n",
      "182: 8\n",
      "183: 8\n",
      "184: 8\n",
      "185: 8\n",
      "186: 8\n",
      "187: 4\n",
      "188: 8\n",
      "189: 8\n",
      "190: 8\n",
      "191: 8\n",
      "194: 8\n",
      "195: 8\n",
      "196: 8\n",
      "197: 8\n",
      "198: 8\n",
      "199: 8\n",
      "200: 8\n",
      "201: 8\n",
      "202: 8\n",
      "203: 8\n",
      "204: 4\n",
      "205: 8\n",
      "206: 8\n",
      "207: 8\n",
      "208: 8\n",
      "209: 8\n",
      "210: 8\n",
      "211: 8\n",
      "212: 8\n",
      "213: 8\n",
      "214: 8\n",
      "215: 8\n",
      "216: 8\n",
      "217: 8\n",
      "218: 8\n",
      "219: 8\n",
      "220: 8\n",
      "221: 4\n",
      "222: 8\n",
      "223: 8\n",
      "225: 8\n",
      "226: 8\n",
      "227: 8\n",
      "228: 8\n",
      "229: 8\n",
      "230: 8\n",
      "231: 8\n",
      "232: 8\n",
      "233: 8\n",
      "234: 8\n",
      "235: 8\n",
      "236: 8\n",
      "237: 8\n",
      "238: 4\n",
      "239: 8\n",
      "240: 8\n",
      "241: 8\n",
      "242: 8\n",
      "243: 8\n",
      "244: 8\n",
      "245: 8\n",
      "246: 8\n",
      "247: 8\n",
      "248: 8\n",
      "249: 8\n",
      "250: 8\n",
      "251: 8\n",
      "252: 8\n",
      "253: 8\n",
      "254: 8\n"
     ]
    }
   ],
   "source": [
    "r3 = sorted(set(r2) | crv._generate_cyclotomic_class(8, 7) \n",
    "            )\n",
    "print(len(r3))\n",
    "next_cyclotomic_classes(r3,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "0946f4aa",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, 1, 2, 3, 5, 9, 17, 23]: found\n",
      "[0, 1, 2, 3, 5, 9, 17, 27]: found\n",
      "[0, 1, 2, 3, 5, 9, 17, 39]: found\n"
     ]
    }
   ],
   "source": [
    "for alphas in [\n",
    "    [0,1,2,3, 5, 23],\n",
    "    [0,1,2,3, 5, 27],\n",
    "    [0,1,2,3, 5, 39],\n",
    "    [0,1,2,3, 5, 9, 23],\n",
    "    [0,1,2,3, 5, 9, 27],\n",
    "    [0,1,2,3, 5, 9, 39],\n",
    "    [0,1,2,3, 5, 9, 17, 23],\n",
    "    [0,1,2,3, 5, 9, 17, 27],\n",
    "    [0,1,2,3, 5, 9, 17, 39],\n",
    "]:\n",
    "    i = crv.CRV(F8, alphas)\n",
    "    try:\n",
    "        i.find_q_polynomials(tries=3, check_full_rank=False, sbox=[spnutils.natural_encoding(F8, x) for x in SKINNY_S8]) \n",
    "        print(f'{alphas}: found')\n",
    "    except ValueError:\n",
    "        pass"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "e8d42d2e",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "41\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[(x^5 + x^4)*z^224 + (x^4 + x^3)*z^194 + (x^4 + x^2 + x + 1)*z^193 + (x^7 + x^6 + x^4 + x^3)*z^192 + (x^4 + x^2 + x + 1)*z^176 + (x^6 + x^3 + 1)*z^160 + (x^7 + x^5 + 1)*z^133 + (x^7 + x^5 + x^2)*z^131 + (x^6 + x^4 + 1)*z^130 + (x^6 + x^5 + x^4 + x)*z^129 + (x^7 + x^6 + x^5 + x^3 + x^2 + 1)*z^128 + (x^6 + x^4 + x)*z^112 + (x^6 + x^4 + x^2 + 1)*z^97 + (x^7 + x^5 + x^4 + x^3 + x^2 + 1)*z^96 + (x^7 + x^4 + x^3 + x^2 + x)*z^88 + (x^7 + x^6 + x^2 + x)*z^80 + (x^7 + x^3 + x^2 + x)*z^65 + (x^7 + x^4 + x^2 + x)*z^64 + (x^3 + 1)*z^56 + (x^6 + x^5 + x^4 + x^3 + 1)*z^48 + (x^6 + x^5 + x^2 + x)*z^44 + (x^5 + x^4 + x^2)*z^40 + (x^7 + x^5 + x^4 + x^3 + x^2 + x)*z^32 + (x^5 + x^2 + 1)*z^28 + (x^7 + x^5 + x^3 + x^2 + x + 1)*z^24 + (x^6 + x^2 + x)*z^22 + (x^6 + x^5 + x^4 + x^3)*z^20 + x^4*z^16 + (x^7 + x^5 + x^4 + x^2 + x)*z^14 + (x^7 + x^5 + x^4)*z^12 + (x^7 + x^6 + x^5 + x^2)*z^11 + (x^7 + x^6 + x^5 + x^4 + x)*z^10 + (x^7 + x^3 + x)*z^8 + (x^5 + x^3 + x^2 + 1)*z^7 + (x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*z^6 + (x^6 + x^4 + x^2 + x + 1)*z^5 + (x^7 + x^5 + x^4 + x)*z^4 + (x^6 + x^3 + 1)*z^3 + (x^6 + x^5 + x^4 + x^3 + x^2)*z^2 + (x^7 + x^4 + x)*z + x^7 + x^6 + x^5 + x^4 + x + 1,\n",
       " (x^7 + x^5 + x^3 + x^2 + x)*z^224 + (x^6 + x^4 + x^3 + x)*z^194 + (x^6 + x^4 + x^3 + x^2)*z^193 + (x^7 + x^3 + 1)*z^192 + (x^6 + x^5 + x^3 + x^2 + x + 1)*z^176 + (x^6 + x^5 + x^3 + x + 1)*z^160 + (x^5 + x^2)*z^133 + (x^7 + x^6 + x^5 + x^4 + x^3 + 1)*z^131 + (x^7 + x^6 + x^4 + x^3 + x^2 + x + 1)*z^130 + z^129 + (x^5 + x^4 + x^3 + x^2)*z^128 + (x^7 + x^5 + x^4 + x^3 + x + 1)*z^112 + (x^6 + x^5 + x^2 + x)*z^97 + (x^6 + x^4 + x^2 + x + 1)*z^96 + (x^7 + x^6 + 1)*z^88 + (x^5 + x^4 + x^3 + x^2)*z^80 + (x^5 + x^4 + x^2 + x + 1)*z^65 + (x^7 + x^6 + x^5 + x^3 + x)*z^64 + (x^7 + x^6 + x^4 + 1)*z^56 + (x^5 + x^3 + x^2)*z^48 + (x^4 + x^3 + x^2 + 1)*z^44 + (x^4 + x^2 + 1)*z^40 + (x^7 + x^5 + x^3 + x^2 + 1)*z^32 + (x^6 + x^5 + x^4 + x^2 + 1)*z^28 + (x^6 + x^3 + x^2 + x)*z^24 + (x^3 + x)*z^22 + (x^7 + x^6 + x^3 + 1)*z^20 + (x^7 + x^6 + x^5 + x^4 + x^2 + x)*z^16 + (x^6 + x^4 + x^3 + 1)*z^14 + (x^5 + x^4 + 1)*z^12 + (x^7 + x^4 + x^3 + x^2)*z^11 + (x^5 + x^3 + x^2)*z^10 + x^3*z^8 + (x^5 + x^4)*z^7 + z^6 + (x^5 + x^3 + x^2 + x + 1)*z^5 + (x^4 + x^2 + x)*z^4 + (x^7 + x^6 + x)*z^3 + (x^3 + x^2 + x)*z^2 + (x^5 + x^4 + x^3 + x)*z + x^5 + x^4 + x^3,\n",
       " (x^7 + x^6 + x^5 + x^3 + x^2 + 1)*z^224 + (x^7 + x^4 + x^3 + 1)*z^194 + (x^7 + x^6 + x^3 + x^2 + x + 1)*z^193 + (x^5 + x^3 + x^2 + x + 1)*z^192 + (x^6 + x^3 + 1)*z^176 + (x^7 + x^5 + x^3)*z^160 + (x^7 + x^6 + x^5 + x + 1)*z^133 + (x^7 + x^6 + x^4 + x^3 + x + 1)*z^131 + (x^5 + x^4 + x^2 + x)*z^130 + (x^7 + x^6 + x^3 + 1)*z^129 + (x^6 + x^5 + x^4 + x^2 + x + 1)*z^128 + (x^6 + x^5 + x + 1)*z^112 + (x^7 + x^6 + x^2 + x)*z^97 + (x^6 + x^5 + x^3 + 1)*z^96 + (x^5 + x^4 + x^3 + x)*z^88 + (x^7 + x^5)*z^80 + (x^7 + x^6 + x^5 + x^4 + x^3 + x^2)*z^65 + (x^7 + x^2 + x + 1)*z^64 + (x^7 + x^5 + x^4)*z^56 + (x^7 + x^5 + x^4 + x^3)*z^48 + (x^7 + x^5 + x^2 + x)*z^44 + (x^7 + x^5 + x^2)*z^40 + (x^7 + x^6 + x^5 + x^4 + x^3)*z^32 + x^4*z^28 + (x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + 1)*z^24 + (x^7 + x^2)*z^22 + (x^6 + x^5 + x^4 + x^3 + x + 1)*z^20 + (x^7 + x^5 + x^4 + x^3)*z^16 + (x^7 + x^6 + x^5 + x^4 + x^2 + 1)*z^14 + (x^7 + x^6 + x^5 + x^4 + x)*z^12 + (x^7 + x^4 + x^3 + x)*z^11 + (x^6 + x^5 + x^4 + x^2)*z^10 + (x^6 + x^5 + x^4 + x^2 + x + 1)*z^8 + (x^3 + x^2)*z^7 + (x^6 + x^5 + x^2)*z^6 + (x^6 + x^4 + x^3 + x + 1)*z^5 + (x^7 + x^5 + x^4 + x^3 + x^2 + x + 1)*z^4 + (x^7 + x^6 + 1)*z^3 + (x^6 + x^3 + x^2 + 1)*z^2 + (x^5 + x^4 + x^2 + x + 1)*z + x^7 + x^6 + x^5 + x^3 + x^2 + x,\n",
       " (x^7 + x^3 + x + 1)*z^224 + (x^5 + x^4 + x^3 + x^2 + x)*z^194 + (x^6 + x^5 + x^2)*z^193 + (x^7 + x^6 + x^3 + x^2 + x)*z^192 + (x^7 + x^6 + x^5 + x^3 + x^2 + x + 1)*z^176 + (x^7 + x^5 + x^3 + x + 1)*z^160 + (x^7 + x^6 + x)*z^133 + (x^7 + x^6 + x^5 + x^4 + x)*z^131 + (x^7 + x^6 + x^5 + x^2 + x)*z^130 + (x^7 + x^5 + x^4 + x^3 + x^2 + x)*z^129 + (x^7 + x^5 + x^3 + x^2 + x)*z^128 + (x^6 + x^5 + x^3 + x^2 + x + 1)*z^112 + (x^7 + x^3)*z^97 + (x^7 + x^4 + x^3 + x + 1)*z^96 + (x^6 + x^5 + x^4 + x^3 + x^2)*z^88 + (x^5 + x^4 + x^2 + 1)*z^80 + (x^5 + x^3 + x^2 + x)*z^65 + (x^6 + x)*z^64 + (x^7 + x^3)*z^56 + (x^3 + x + 1)*z^48 + (x^4 + x^2 + x)*z^44 + (x^6 + x^5 + x^4)*z^40 + (x^7 + x^5 + x^4 + x^3 + 1)*z^32 + (x^5 + x + 1)*z^24 + (x^7 + x^4 + x + 1)*z^22 + (x^6 + x^4 + x^2 + 1)*z^20 + (x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + 1)*z^16 + (x^6 + x^5 + x^4 + x + 1)*z^14 + (x^7 + x^6)*z^12 + (x^7 + x^6 + x^5 + x^2 + x + 1)*z^11 + (x^7 + x^6 + x^4 + x^3 + x^2 + x)*z^10 + (x^7 + x^6 + x^4 + x^3 + x + 1)*z^8 + (x^7 + x^6 + x)*z^7 + (x^6 + x^5 + x^3 + x)*z^6 + (x^7 + x^6 + x^5 + x^4 + x)*z^5 + (x^5 + x^2 + 1)*z^4 + (x^7 + x^6 + x^4 + x^2 + x + 1)*z^3 + (x^5 + x^4 + x^3 + x + 1)*z^2 + (x^7 + x^6 + x^3 + x^2)*z + x^7 + x^6 + x^3,\n",
       " (x^6 + x^3 + x^2 + x + 1)*z^224 + (x^6 + x^4 + x^2 + x + 1)*z^194 + (x^7 + x^6 + x^4 + x^3 + 1)*z^193 + (x^5 + x^2 + 1)*z^192 + (x^7 + x^4 + x^3 + x^2 + 1)*z^176 + (x^7 + x^6 + x^3 + x^2)*z^160 + (x^7 + x^5 + x^3 + x + 1)*z^133 + (x^7 + x^4 + x^2)*z^131 + (x^6 + x^5 + x^2 + x + 1)*z^130 + (x^5 + x^2 + x + 1)*z^129 + (x^4 + x^3)*z^128 + (x^7 + x^3 + x^2 + x)*z^112 + (x^4 + x^3 + x + 1)*z^97 + (x^6 + x^2)*z^96 + (x^7 + x^6 + 1)*z^88 + (x^7 + x^2 + x)*z^80 + (x^5 + x^2 + x)*z^65 + (x^7 + x^5 + x^3 + x^2)*z^64 + (x^7 + x^5 + x^3)*z^56 + (x^6 + x^5 + x^2 + x)*z^48 + (x^7 + x^6 + x^4 + x^3 + x^2 + x + 1)*z^44 + (x^6 + x^5 + x^3 + x^2 + x + 1)*z^32 + (x^7 + x^4 + x^3)*z^28 + (x^6 + x^4 + x^3 + x^2 + 1)*z^24 + (x^7 + x^2 + x + 1)*z^22 + (x^3 + x^2)*z^20 + (x^7 + x^4)*z^16 + (x^4 + x^3 + x + 1)*z^14 + (x^7 + x^6 + x^5 + x + 1)*z^12 + (x^4 + x^2 + x + 1)*z^11 + (x^7 + x^5 + x^3 + x^2 + x)*z^10 + (x^6 + 1)*z^8 + (x^7 + x^6 + x)*z^7 + (x^5 + x^4 + x + 1)*z^6 + (x^7 + x^6 + x^4 + x^2)*z^5 + (x^7 + x^5 + x^3 + x)*z^4 + (x^6 + x^3 + x + 1)*z^3 + (x^6 + x^5 + x^4 + x^3 + x)*z^2 + (x^6 + x^3 + x)*z + x^5 + x^4 + x^3 + x^2 + 1,\n",
       " (x^7 + x^3 + x + 1)*z^224 + (x^6 + x^4 + x^3 + 1)*z^194 + (x^3 + x^2 + x + 1)*z^193 + (x^7 + x^6 + x^5 + x^3 + 1)*z^192 + (x^5 + x^4 + 1)*z^176 + (x^6 + x^4 + x^2)*z^160 + (x^5 + x^3 + x^2 + 1)*z^133 + (x^6 + x^3 + x + 1)*z^131 + (x^5 + x^4 + x + 1)*z^130 + (x^7 + x^5 + x^4 + x^2 + x + 1)*z^129 + (x^7 + x^4 + x^2 + 1)*z^128 + (x^7 + x^6 + x^5 + x)*z^112 + (x^7 + x^5 + x^4 + x^3 + x^2 + x + 1)*z^97 + (x^5 + x^2 + 1)*z^96 + (x^7 + x^5 + x^4 + x^3 + x)*z^88 + (x^7 + x^6 + x^5 + x^4 + x^3 + x + 1)*z^80 + (x^7 + x^6 + x^2 + x)*z^65 + (x^7 + x^6 + x^4)*z^64 + (x^6 + x^3 + 1)*z^56 + (x^7 + x^6 + x^5 + x + 1)*z^48 + (x^6 + x^3 + x)*z^44 + (x^7 + x^4 + x + 1)*z^40 + (x^7 + x^6 + x^4 + x^3 + x^2 + x)*z^32 + (x^6 + x^5 + 1)*z^28 + (x^6 + x^5 + x^4 + x^3 + x^2 + 1)*z^24 + (x^6 + x^3 + x^2)*z^22 + (x^7 + x^6 + x^2 + x)*z^20 + (x^6 + x^5 + x^2)*z^16 + (x^7 + x^6 + x^3 + x^2 + x + 1)*z^14 + (x^4 + x^2 + 1)*z^12 + (x^4 + x^2 + x)*z^11 + x^3*z^10 + (x^7 + x^6 + x^5 + x^4 + x^2 + x)*z^8 + (x^7 + x^5 + x^2 + x + 1)*z^7 + (x^5 + x^4 + 1)*z^6 + (x^7 + x^3 + x^2 + x + 1)*z^5 + (x^7 + x^6 + x^2 + x + 1)*z^4 + (x^7 + x^4 + x^3 + 1)*z^3 + (x^7 + x^6 + x^4)*z^2 + (x^7 + x^6 + x^5 + x^2)*z + x^6 + x^2 + x]"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "instance = crv.CRV(F8, [0,1,3,5,7,11])\n",
    "print(len(instance.L))\n",
    "instance.find_q_polynomials(t=7, tries=5, check_full_rank=False, sbox=[spnutils.natural_encoding(F8, x) for x in SKINNY_S8])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "81f3bc55",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "t = 7\n",
      "L = [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 16, 20, 22, 24, 28, 32, 40, 44, 48, 56, 64, 65, 80, 88, 96, 97, 112, 128, 129, 130, 131, 133, 160, 176, 192, 193, 194, 224]\n"
     ]
    }
   ],
   "source": [
    "res = instance.polynomial_decomposition(SKINNY_S8)\n",
    "print(f't = {res.t}')\n",
    "print(f'L = {res.L}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "418f974a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "q_0: 0xf3 z^0 + 0x92 z^1 + 0x7c z^2 + 0x49 z^3 + 0xb2 z^4 + 0x57 z^5 + 0xff z^6 + 0x2d z^7 + 0x8a z^8 + 0xf2 z^10 + 0xe4 z^11 + 0xb0 z^12 + 0xb6 z^14 + 0x10 z^16 + 0x78 z^20 + 0x46 z^22 + 0xaf z^24 + 0x25 z^28 + 0xbe z^32 + 0x34 z^40 + 0x66 z^44 + 0x79 z^48 + 0x9 z^56 + 0x96 z^64 + 0x8e z^65 + 0xc6 z^80 + 0x9e z^88 + 0xbd z^96 + 0x55 z^97 + 0x52 z^112 + 0xed z^128 + 0x72 z^129 + 0x51 z^130 + 0xa4 z^131 + 0xa1 z^133 + 0x49 z^160 + 0x17 z^176 + 0xd8 z^192 + 0x17 z^193 + 0x18 z^194 + 0x30 z^224\n",
      "\n",
      "q_1: 0x38 z^0 + 0x3a z^1 + 0xe z^2 + 0xc2 z^3 + 0x16 z^4 + 0x2f z^5 + 0x1 z^6 + 0x30 z^7 + 0x8 z^8 + 0x2c z^10 + 0x9c z^11 + 0x31 z^12 + 0x59 z^14 + 0xf6 z^16 + 0xc9 z^20 + 0xa z^22 + 0x4e z^24 + 0x75 z^28 + 0xad z^32 + 0x15 z^40 + 0x1d z^44 + 0x2c z^48 + 0xd1 z^56 + 0xea z^64 + 0x37 z^65 + 0x3c z^80 + 0xc1 z^88 + 0x57 z^96 + 0x66 z^97 + 0xbb z^112 + 0x3c z^128 + 0x1 z^129 + 0xdf z^130 + 0xf9 z^131 + 0x24 z^133 + 0x6b z^160 + 0x6f z^176 + 0x89 z^192 + 0x5c z^193 + 0x5a z^194 + 0xae z^224\n",
      "\n",
      "q_2: 0xee z^0 + 0x37 z^1 + 0x4d z^2 + 0xc1 z^3 + 0xbf z^4 + 0x5b z^5 + 0x64 z^6 + 0xc z^7 + 0x77 z^8 + 0x74 z^10 + 0x9a z^11 + 0xf2 z^12 + 0xf5 z^14 + 0xb8 z^16 + 0x7b z^20 + 0x84 z^22 + 0xfd z^24 + 0x10 z^28 + 0xf8 z^32 + 0xa4 z^40 + 0xa6 z^44 + 0xb8 z^48 + 0xb0 z^56 + 0x87 z^64 + 0xfc z^65 + 0xa0 z^80 + 0x3a z^88 + 0x69 z^96 + 0xc6 z^97 + 0x63 z^112 + 0x77 z^128 + 0xc9 z^129 + 0x36 z^130 + 0xdb z^131 + 0xe3 z^133 + 0xa8 z^160 + 0x49 z^176 + 0x2f z^192 + 0xcf z^193 + 0x99 z^194 + 0xed z^224\n",
      "\n",
      "q_3: 0xc8 z^0 + 0xcc z^1 + 0x3b z^2 + 0xd7 z^3 + 0x25 z^4 + 0xf2 z^5 + 0x6a z^6 + 0xc2 z^7 + 0xdb z^8 + 0xde z^10 + 0xe7 z^11 + 0xc0 z^12 + 0x73 z^14 + 0xfd z^16 + 0x55 z^20 + 0x93 z^22 + 0x23 z^24 + 0xb9 z^32 + 0x70 z^40 + 0x16 z^44 + 0xb z^48 + 0x88 z^56 + 0x42 z^64 + 0x2e z^65 + 0x35 z^80 + 0x7c z^88 + 0x9b z^96 + 0x88 z^97 + 0x6f z^112 + 0xae z^128 + 0xbe z^129 + 0xe6 z^130 + 0xf2 z^131 + 0xc2 z^133 + 0xab z^160 + 0xef z^176 + 0xce z^192 + 0x64 z^193 + 0x3e z^194 + 0x8b z^224\n",
      "\n",
      "q_4: 0x3d z^0 + 0x4a z^1 + 0x7a z^2 + 0x4b z^3 + 0xaa z^4 + 0xd4 z^5 + 0x33 z^6 + 0xc2 z^7 + 0x41 z^8 + 0xae z^10 + 0x17 z^11 + 0xe3 z^12 + 0x1b z^14 + 0x90 z^16 + 0xc z^20 + 0x87 z^22 + 0x5d z^24 + 0x98 z^28 + 0x6f z^32 + 0xdf z^44 + 0x66 z^48 + 0xa8 z^56 + 0xac z^64 + 0x26 z^65 + 0x86 z^80 + 0xc1 z^88 + 0x44 z^96 + 0x1b z^97 + 0x8e z^112 + 0x18 z^128 + 0x27 z^129 + 0x67 z^130 + 0x94 z^131 + 0xab z^133 + 0xcc z^160 + 0x9d z^176 + 0x25 z^192 + 0xd9 z^193 + 0x57 z^194 + 0x4f z^224\n",
      "\n",
      "q_5: 0x46 z^0 + 0xe4 z^1 + 0xd0 z^2 + 0x99 z^3 + 0xc7 z^4 + 0x8f z^5 + 0x31 z^6 + 0xa7 z^7 + 0xf6 z^8 + 0x8 z^10 + 0x16 z^11 + 0x15 z^12 + 0xcf z^14 + 0x64 z^16 + 0xc6 z^20 + 0x4c z^22 + 0x7d z^24 + 0x61 z^28 + 0xde z^32 + 0x93 z^40 + 0x4a z^44 + 0xe3 z^48 + 0x49 z^56 + 0xd0 z^64 + 0xc6 z^65 + 0xfb z^80 + 0xba z^88 + 0x25 z^96 + 0xbf z^97 + 0xe2 z^112 + 0x95 z^128 + 0xb7 z^129 + 0x33 z^130 + 0x4b z^131 + 0x2d z^133 + 0x54 z^160 + 0x31 z^176 + 0xe9 z^192 + 0xf z^193 + 0x59 z^194 + 0x8b z^224\n",
      "\n"
     ]
    }
   ],
   "source": [
    "for j,q in enumerate(res.q):\n",
    "    print(f'q_{j}: ' + ' + '.join(f'0x{spnutils.natural_encoding_to_int(c):x} z^{i}' for i,c in enumerate(q.coefficients(sparse=False)) if c != 0))\n",
    "    print()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "e38665b5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p_0: 0x1e z^0 + 0x4f z^2 + 0x52 z^3 + 0x64 z^4 + 0xfe z^5 + 0xab z^6 + 0x5b z^7 + 0x97 z^8 + 0xa5 z^9 + 0xbd z^10 + 0xe8 z^11 + 0x6f z^12 + 0x19 z^13 + 0x87 z^14 + 0xef z^15 + 0xc1 z^16 + 0xea z^17 + 0x92 z^18 + 0x8f z^19 + 0xe1 z^20 + 0x6f z^21 + 0x23 z^22 + 0x5d z^23 + 0xbb z^24 + 0xc7 z^25 + 0xae z^26 + 0x13 z^27 + 0xd7 z^28 + 0x5b z^29 + 0x14 z^30 + 0x2a z^31 + 0xbe z^32 + 0xd0 z^33 + 0x5c z^34 + 0xb4 z^35 + 0x4b z^36 + 0xcd z^37 + 0xb3 z^38 + 0x58 z^39 + 0xbc z^40\n",
      "p_1: 0x88 z^0 + 0x57 z^1 + 0xc4 z^2 + 0x4e z^3 + 0x85 z^4 + 0x5f z^5 + 0xef z^6 + 0x21 z^7 + 0xc6 z^8 + 0xfd z^9 + 0xb4 z^10 + 0x6c z^11 + 0xb9 z^12 + 0xe2 z^13 + 0x8d z^14 + 0xa7 z^15 + 0x19 z^16 + 0x44 z^17 + 0xd7 z^18 + 0xbf z^19 + 0xb0 z^20 + 0xeb z^21 + 0xad z^22 + 0x88 z^23 + 0xaf z^24 + 0xc0 z^25 + 0xa3 z^26 + 0xcd z^27 + 0xd4 z^29 + 0xf2 z^30 + 0x8 z^31 + 0x7 z^32 + 0x16 z^33 + 0xd1 z^34 + 0x56 z^35 + 0x1f z^36 + 0xf4 z^37 + 0xaf z^38 + 0x89 z^39\n",
      "p_2: 0xf4 z^0 + 0x4c z^1 + 0x99 z^2 + 0xd6 z^3 + 0xc9 z^4 + 0x15 z^5 + 0xf8 z^6 + 0xa0 z^7 + 0xeb z^8 + 0x91 z^9 + 0x1b z^10 + 0xff z^11 + 0x39 z^12 + 0x72 z^13 + 0x76 z^14 + 0x8 z^15 + 0xba z^16 + 0x70 z^17 + 0x18 z^18 + 0xf8 z^19 + 0x44 z^20 + 0x83 z^21 + 0xbf z^22 + 0xd7 z^23 + 0xdd z^24 + 0x6 z^25 + 0x19 z^26 + 0x59 z^27 + 0xd2 z^28 + 0xd2 z^29 + 0x5c z^30 + 0x5a z^31 + 0xdb z^32 + 0x96 z^33 + 0xa5 z^34 + 0x4c z^35 + 0x32 z^36 + 0x88 z^37 + 0xaf z^38\n",
      "p_3: 0x2a z^0 + 0xa7 z^1 + 0xf3 z^2 + 0xc1 z^3 + 0x56 z^4 + 0xfe z^5 + 0x79 z^6 + 0xd4 z^7 + 0x25 z^8 + 0x72 z^9 + 0x6f z^10 + 0xa6 z^11 + 0x38 z^12 + 0x6e z^13 + 0x28 z^14 + 0xdf z^15 + 0xca z^16 + 0x3a z^17 + 0xce z^18 + 0x9d z^19 + 0xdc z^20 + 0x38 z^21 + 0x74 z^22 + 0x58 z^23 + 0x44 z^24 + 0xdb z^25 + 0x91 z^26 + 0x4d z^27 + 0x29 z^28 + 0x3e z^29 + 0x84 z^30 + 0xbd z^31 + 0xc z^32 + 0xdb z^33 + 0x4e z^34 + 0x43 z^35 + 0xbc z^36 + 0x20 z^37\n",
      "p_4: 0x17 z^0 + 0x5e z^1 + 0x90 z^2 + 0x55 z^3 + 0x92 z^4 + 0xc9 z^5 + 0x3f z^6 + 0x5d z^7 + 0xd z^8 + 0x32 z^9 + 0x30 z^10 + 0xff z^11 + 0xd3 z^12 + 0xef z^13 + 0x1d z^14 + 0xf6 z^15 + 0x78 z^16 + 0x79 z^17 + 0x26 z^18 + 0xd2 z^19 + 0x20 z^20 + 0x42 z^21 + 0xf2 z^22 + 0xc2 z^23 + 0xfe z^24 + 0xb4 z^25 + 0xd9 z^26 + 0xfb z^27 + 0x97 z^28 + 0x79 z^29 + 0x79 z^30 + 0xc3 z^31 + 0xbc z^32 + 0x3f z^33 + 0x7b z^34 + 0x83 z^35 + 0x3c z^36\n",
      "p_5: 0x9f z^0 + 0x41 z^1 + 0xfd z^2 + 0xa4 z^3 + 0x1f z^4 + 0x4 z^5 + 0xa1 z^6 + 0x47 z^7 + 0x4e z^8 + 0x8c z^9 + 0x11 z^10 + 0xbe z^11 + 0x8d z^12 + 0xb9 z^13 + 0x39 z^14 + 0xe z^15 + 0x4c z^16 + 0xb8 z^17 + 0x7f z^18 + 0xd6 z^19 + 0x75 z^20 + 0xb9 z^21 + 0xb2 z^22 + 0xe8 z^23 + 0xb z^24 + 0x27 z^25 + 0xd9 z^26 + 0x7f z^27 + 0xe0 z^28 + 0x66 z^29 + 0xc8 z^30 + 0x98 z^31 + 0xba z^32 + 0xe2 z^33 + 0x50 z^34 + 0x64 z^35\n",
      "p_6: 0xe6 z^0 + 0x45 z^1 + 0xe2 z^2 + 0x7d z^3 + 0x48 z^4 + 0x96 z^5 + 0xa1 z^6 + 0x14 z^7 + 0xda z^8 + 0x6b z^9 + 0xef z^10 + 0x91 z^11 + 0x2a z^12 + 0xba z^13 + 0x49 z^14 + 0x2c z^15\n"
     ]
    }
   ],
   "source": [
    "for j,p in enumerate(res.p):\n",
    "    pi = [F8(list(c)[0]) for c in p]\n",
    "    print(f'p_{j}: ' + ' + '.join(f'0x{spnutils.natural_encoding_to_int(c):x} z^{i}' for i,c in enumerate(pi) if c != 0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "ba408309",
   "metadata": {},
   "outputs": [],
   "source": [
    "# check correctness of decomposition\n",
    "for x,y in enumerate(SKINNY_S8):\n",
    "    x_f = spnutils.natural_encoding(F8, x)\n",
    "    y_f = res.compute(x_f)[0]\n",
    "    a = spnutils.natural_encoding_to_int(y_f)\n",
    "    assert y == a, f'x=0x{x:02x} expected 0x{y:02x}, got 0x{a:02x}'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "faaa389b",
   "metadata": {},
   "outputs": [],
   "source": [
    "res_inv = instance.polynomial_decomposition(SKINNY_S8_inv)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "777cdd11",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "t = 7\n",
      "L = [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 16, 20, 22, 24, 28, 32, 40, 44, 48, 56, 64, 65, 80, 88, 96, 97, 112, 128, 129, 130, 131, 133, 160, 176, 192, 193, 194, 224]\n"
     ]
    }
   ],
   "source": [
    "print(f't = {res_inv.t}')\n",
    "print(f'L = {res_inv.L}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "fff57fe4",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p_0: 0x9d z^0 + 0x54 z^1 + 0x1d z^2 + 0x29 z^3 + 0x40 z^4 + 0xf3 z^5 + 0xea z^6 + 0x6f z^7 + 0x93 z^8 + 0x55 z^9 + 0xc0 z^10 + 0x58 z^11 + 0x71 z^12 + 0x83 z^13 + 0x22 z^14 + 0x62 z^15 + 0xf7 z^16 + 0xa2 z^17 + 0x95 z^18 + 0xd5 z^19 + 0xd0 z^20 + 0x6c z^21 + 0x50 z^22 + 0x26 z^23 + 0x3 z^24 + 0x7d z^25 + 0xce z^26 + 0x88 z^27 + 0x8c z^28 + 0x8f z^29 + 0x73 z^30 + 0x82 z^31 + 0x3b z^32 + 0xa2 z^33 + 0x1 z^34 + 0x48 z^35 + 0xe2 z^36 + 0x36 z^37 + 0xfe z^38 + 0xe9 z^39 + 0x63 z^40\n",
      "p_1: 0x1f z^0 + 0x44 z^1 + 0xf2 z^2 + 0x52 z^3 + 0x9f z^4 + 0xc3 z^5 + 0x72 z^6 + 0x5a z^7 + 0xbc z^8 + 0x43 z^9 + 0x67 z^10 + 0x6a z^11 + 0xe3 z^12 + 0xad z^13 + 0xf4 z^14 + 0xb9 z^15 + 0x2 z^16 + 0x61 z^17 + 0xb9 z^18 + 0xdf z^19 + 0xdd z^20 + 0x2a z^21 + 0xa9 z^22 + 0x3c z^23 + 0x91 z^24 + 0x91 z^25 + 0x23 z^26 + 0xdc z^27 + 0xb2 z^28 + 0x93 z^29 + 0xad z^30 + 0x30 z^31 + 0x2e z^32 + 0x44 z^33 + 0xc6 z^34 + 0x58 z^35 + 0xa3 z^36 + 0x3c z^37 + 0x14 z^38 + 0x80 z^39\n",
      "p_2: 0xe9 z^0 + 0x17 z^1 + 0xcf z^2 + 0xa5 z^3 + 0xf0 z^4 + 0x23 z^5 + 0x63 z^6 + 0x2e z^7 + 0x1f z^8 + 0xb3 z^9 + 0x47 z^10 + 0xe0 z^11 + 0x6c z^12 + 0xf8 z^13 + 0x7b z^14 + 0x27 z^15 + 0x35 z^16 + 0xe8 z^17 + 0xa9 z^18 + 0xfa z^19 + 0xfb z^20 + 0x6 z^21 + 0xdb z^22 + 0x5a z^23 + 0x26 z^24 + 0x6e z^25 + 0x22 z^26 + 0xc6 z^27 + 0xc3 z^28 + 0x20 z^29 + 0xc4 z^30 + 0x84 z^31 + 0x27 z^32 + 0xa8 z^33 + 0xb1 z^34 + 0xd z^35 + 0x8c z^36 + 0x3e z^37 + 0x1b z^38\n",
      "p_3: 0x83 z^0 + 0x85 z^1 + 0x50 z^2 + 0xa1 z^3 + 0xd6 z^4 + 0x35 z^5 + 0x66 z^6 + 0xe9 z^7 + 0x98 z^8 + 0xdb z^9 + 0xe3 z^10 + 0xe0 z^11 + 0x56 z^12 + 0xf3 z^13 + 0xda z^14 + 0x7c z^15 + 0x4a z^16 + 0x22 z^17 + 0xe6 z^18 + 0x85 z^19 + 0xe0 z^20 + 0xaf z^21 + 0x75 z^22 + 0x79 z^23 + 0x8c z^24 + 0x6a z^25 + 0x9a z^26 + 0x29 z^27 + 0x7d z^28 + 0xbf z^29 + 0xe0 z^30 + 0x53 z^31 + 0xd9 z^32 + 0x83 z^33 + 0x8d z^34 + 0x5c z^35 + 0xf z^36 + 0xad z^37\n",
      "p_4: 0x1e z^0 + 0xf2 z^1 + 0x50 z^2 + 0x58 z^3 + 0xb2 z^4 + 0xd6 z^5 + 0xd1 z^6 + 0xc6 z^7 + 0x5 z^8 + 0x15 z^9 + 0x53 z^10 + 0x47 z^11 + 0xc6 z^12 + 0xb4 z^13 + 0xb8 z^14 + 0x9b z^15 + 0xb4 z^16 + 0x8b z^17 + 0xed z^18 + 0x8 z^19 + 0x31 z^20 + 0x65 z^21 + 0x6 z^22 + 0xf0 z^23 + 0x2d z^24 + 0xb6 z^25 + 0x50 z^26 + 0x58 z^27 + 0x7f z^28 + 0x8e z^29 + 0x14 z^30 + 0x39 z^31 + 0x8f z^32 + 0x21 z^33 + 0x94 z^34 + 0xbe z^35 + 0xa7 z^36\n",
      "p_5: 0xec z^0 + 0x36 z^1 + 0x34 z^2 + 0x99 z^3 + 0x83 z^4 + 0x52 z^5 + 0x7 z^6 + 0xed z^7 + 0x41 z^8 + 0x92 z^9 + 0x93 z^10 + 0x98 z^11 + 0xec z^12 + 0x33 z^13 + 0xe9 z^14 + 0x8a z^15 + 0x7d z^16 + 0xd5 z^17 + 0x13 z^18 + 0x27 z^19 + 0xd1 z^20 + 0xd3 z^21 + 0x7f z^22 + 0x48 z^23 + 0x53 z^24 + 0xba z^25 + 0x46 z^26 + 0x5b z^27 + 0x57 z^28 + 0x41 z^29 + 0x23 z^30 + 0x47 z^31 + 0x9b z^32 + 0x9a z^33 + 0x52 z^35\n",
      "p_6: 0x80 z^0 + 0x9d z^1 + 0x8b z^2 + 0xc7 z^3 + 0x4c z^4 + 0x4b z^5 + 0xf7 z^6 + 0x12 z^7 + 0x17 z^8 + 0xdc z^9 + 0xd4 z^11 + 0xe0 z^12 + 0x82 z^13 + 0xa9 z^14 + 0x8d z^15\n"
     ]
    }
   ],
   "source": [
    "for j,p in enumerate(res_inv.p):\n",
    "    pi = [F8(list(c)[0]) for c in p]\n",
    "    print(f'p_{j}: ' + ' + '.join(f'0x{spnutils.natural_encoding_to_int(c):x} z^{i}' for i,c in enumerate(pi) if c != 0))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3ffb53a3",
   "metadata": {},
   "source": [
    "### SKINNY 8-bit addition chains"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "2efe3ca7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 182, 183, 184, 185, 186, 188, 189, 190, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 222, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 240, 241, 242, 243, 244, 245, 246, 248, 249, 250, 252]\n",
      "241\n"
     ]
    }
   ],
   "source": [
    "c = spnutils.interpolate(F8, [spnutils.natural_encoding(F8, i) for i in range(256)], [spnutils.natural_encoding(F8, x) for x in SKINNY_S8_inv])\n",
    "powers = [i for i in range(256) if c[i] != 0]\n",
    "print(powers)\n",
    "print(len(powers))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "ad98cd76",
   "metadata": {},
   "outputs": [],
   "source": [
    "res = spnutils.shallow_dense_addition_chain(powers, 8, doubles=[])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "f90a80cd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Round 1 [free]: \n",
      "Round 1: 2 = 1 + 1\n",
      "Round 2 [free]: \n",
      "Round 2: 3 = 1 + 2, 4 = 2 + 2\n",
      "Round 3 [free]: \n",
      "Round 3: 5 = 1 + 4, 6 = 2 + 4, 7 = 3 + 4, 8 = 4 + 4\n",
      "Round 4 [free]: \n",
      "Round 4: 9 = 1 + 8, 10 = 2 + 8, 11 = 3 + 8, 12 = 4 + 8, 14 = 6 + 8, 15 = 7 + 8, 16 = 8 + 8\n",
      "Round 5 [free]: \n",
      "Round 5: 17 = 1 + 16, 18 = 2 + 16, 19 = 3 + 16, 20 = 4 + 16, 21 = 5 + 16, 22 = 6 + 16, 23 = 7 + 16, 24 = 8 + 16, 25 = 9 + 16, 26 = 10 + 16, 27 = 11 + 16, 28 = 12 + 16, 29 = 14 + 15, 30 = 14 + 16, 31 = 15 + 16, 32 = 16 + 16\n",
      "Round 6 [free]: \n",
      "Round 6: 33 = 1 + 32, 34 = 2 + 32, 35 = 3 + 32, 36 = 4 + 32, 37 = 5 + 32, 38 = 6 + 32, 39 = 7 + 32, 40 = 8 + 32, 41 = 9 + 32, 42 = 10 + 32, 43 = 11 + 32, 44 = 12 + 32, 45 = 14 + 31, 46 = 14 + 32, 47 = 15 + 32, 48 = 16 + 32, 49 = 17 + 32, 50 = 18 + 32, 51 = 19 + 32, 52 = 20 + 32, 53 = 21 + 32, 54 = 22 + 32, 55 = 23 + 32, 56 = 24 + 32, 57 = 25 + 32, 58 = 26 + 32, 59 = 27 + 32, 60 = 28 + 32, 61 = 29 + 32, 62 = 30 + 32, 63 = 31 + 32, 64 = 32 + 32\n",
      "Round 7 [free]: \n",
      "Round 7: 65 = 1 + 64, 66 = 2 + 64, 67 = 3 + 64, 68 = 4 + 64, 69 = 5 + 64, 70 = 6 + 64, 71 = 7 + 64, 72 = 8 + 64, 73 = 9 + 64, 74 = 10 + 64, 75 = 11 + 64, 76 = 12 + 64, 77 = 14 + 63, 78 = 14 + 64, 79 = 15 + 64, 80 = 16 + 64, 81 = 17 + 64, 82 = 18 + 64, 83 = 19 + 64, 84 = 20 + 64, 85 = 21 + 64, 86 = 22 + 64, 87 = 23 + 64, 88 = 24 + 64, 89 = 25 + 64, 90 = 26 + 64, 91 = 27 + 64, 92 = 28 + 64, 93 = 29 + 64, 94 = 30 + 64, 95 = 31 + 64, 96 = 32 + 64, 97 = 33 + 64, 98 = 34 + 64, 99 = 35 + 64, 100 = 36 + 64, 101 = 37 + 64, 102 = 38 + 64, 103 = 39 + 64, 104 = 40 + 64, 105 = 41 + 64, 106 = 42 + 64, 107 = 43 + 64, 108 = 44 + 64, 109 = 45 + 64, 110 = 46 + 64, 111 = 47 + 64, 112 = 48 + 64, 113 = 49 + 64, 114 = 50 + 64, 115 = 51 + 64, 116 = 52 + 64, 117 = 53 + 64, 118 = 54 + 64, 120 = 56 + 64, 121 = 57 + 64, 122 = 58 + 64, 123 = 59 + 64, 124 = 60 + 64, 125 = 61 + 64, 126 = 62 + 64, 128 = 64 + 64\n",
      "Round 8 [free]: \n",
      "Round 8: 129 = 1 + 128, 130 = 2 + 128, 131 = 3 + 128, 132 = 4 + 128, 133 = 5 + 128, 134 = 6 + 128, 135 = 7 + 128, 136 = 8 + 128, 137 = 9 + 128, 138 = 10 + 128, 139 = 11 + 128, 140 = 12 + 128, 141 = 15 + 126, 142 = 14 + 128, 143 = 15 + 128, 144 = 16 + 128, 145 = 17 + 128, 146 = 18 + 128, 147 = 19 + 128, 148 = 20 + 128, 149 = 21 + 128, 150 = 22 + 128, 151 = 23 + 128, 152 = 24 + 128, 153 = 25 + 128, 154 = 26 + 128, 155 = 27 + 128, 156 = 28 + 128, 157 = 29 + 128, 158 = 30 + 128, 159 = 31 + 128, 160 = 32 + 128, 161 = 33 + 128, 162 = 34 + 128, 163 = 35 + 128, 164 = 36 + 128, 165 = 37 + 128, 166 = 38 + 128, 167 = 39 + 128, 168 = 40 + 128, 169 = 41 + 128, 170 = 42 + 128, 171 = 43 + 128, 172 = 44 + 128, 173 = 45 + 128, 174 = 46 + 128, 175 = 47 + 128, 176 = 48 + 128, 177 = 49 + 128, 178 = 50 + 128, 179 = 51 + 128, 180 = 52 + 128, 182 = 54 + 128, 183 = 55 + 128, 184 = 56 + 128, 185 = 57 + 128, 186 = 58 + 128, 188 = 60 + 128, 189 = 61 + 128, 190 = 62 + 128, 192 = 64 + 128, 193 = 65 + 128, 194 = 66 + 128, 195 = 67 + 128, 196 = 68 + 128, 197 = 69 + 128, 198 = 70 + 128, 199 = 71 + 128, 200 = 72 + 128, 201 = 73 + 128, 202 = 74 + 128, 203 = 75 + 128, 204 = 76 + 128, 205 = 77 + 128, 206 = 78 + 128, 207 = 79 + 128, 208 = 80 + 128, 209 = 81 + 128, 210 = 82 + 128, 211 = 83 + 128, 212 = 84 + 128, 213 = 85 + 128, 214 = 86 + 128, 215 = 87 + 128, 216 = 88 + 128, 217 = 89 + 128, 218 = 90 + 128, 219 = 91 + 128, 220 = 92 + 128, 222 = 94 + 128, 224 = 96 + 128, 225 = 97 + 128, 226 = 98 + 128, 227 = 99 + 128, 228 = 100 + 128, 229 = 101 + 128, 230 = 102 + 128, 231 = 103 + 128, 232 = 104 + 128, 233 = 105 + 128, 234 = 106 + 128, 235 = 107 + 128, 236 = 108 + 128, 237 = 109 + 128, 240 = 112 + 128, 241 = 113 + 128, 242 = 114 + 128, 243 = 115 + 128, 244 = 116 + 128, 245 = 117 + 128, 246 = 118 + 128, 248 = 120 + 128, 249 = 121 + 128, 250 = 122 + 128, 252 = 124 + 128\n"
     ]
    }
   ],
   "source": [
    "res.print_levels()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "0e9ead0e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "muls: 239, free squares: 0, bits: 0 in 8 rounds\n"
     ]
    }
   ],
   "source": [
    "muls, squares, bd = res.required_additions()\n",
    "print(f'muls: {muls+squares}, free squares: {len(res._frees)}, bits: {8 * bd} in {res.required_levels()} rounds')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "609bedce",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "adding 2 as double\n",
      "adding 3 as double\n",
      "adding 5 as double\n",
      "adding 9 as double\n",
      "adding 17 as double\n",
      "adding 7 as double\n",
      "adding 11 as double\n",
      "adding 15 as double\n",
      "adding 19 as double\n",
      "adding 21 as double\n",
      "adding 23 as double\n",
      "adding 25 as double\n",
      "adding 26 as double\n",
      "adding 27 as double\n",
      "adding 29 as double\n",
      "adding 37 as double\n",
      "adding 39 as double\n",
      "adding 43 as double\n",
      "adding 45 as double\n",
      "adding 51 as double\n",
      "adding 53 as double\n",
      "adding 85 as double\n",
      "adding 31 as double\n",
      "adding 47 as double\n",
      "adding 55 as double\n",
      "adding 59 as double\n",
      "adding 61 as double\n",
      "adding 63 as double\n",
      "adding 87 as double\n",
      "adding 91 as double\n",
      "adding 95 as double\n",
      "adding 111 as double\n"
     ]
    }
   ],
   "source": [
    "# gradually add more and more free squares to understand the trade-off\n",
    "doubles = []\n",
    "res = spnutils.shallow_dense_addition_chain(powers, 8, doubles)\n",
    "l = [res]\n",
    "while True:\n",
    "    # find next value to add to doubles\n",
    "    new_value = None\n",
    "    for level in res._levels:\n",
    "        for x,y,z in level:\n",
    "            if new_value == None and z not in res._frees and z not in doubles:\n",
    "                new_value = z\n",
    "    if new_value == None:\n",
    "        # done\n",
    "        break\n",
    "    print(f'adding {new_value} as double')\n",
    "    doubles.append(new_value)\n",
    "    res = spnutils.shallow_dense_addition_chain(powers, 8, doubles)\n",
    "    l.append(res)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "929ced80",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "33"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(l)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "17d6ac72",
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "2d940114",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[239, 233, 226, 219, 212, 209, 202, 195, 188, 181, 174, 167, 160, 154, 147, 140, 133, 126, 119, 112, 109, 102, 101, 94, 87, 80, 73, 66, 59, 52, 46, 39, 32]\n",
      "[0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256]\n",
      "[8, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5]\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "muls = [sum(r.required_additions()[:2]) for r in l]\n",
    "print(muls)\n",
    "bits = [8*r.required_additions()[2] for r in l]\n",
    "print(bits)\n",
    "rounds = [r.required_levels() for r in l]\n",
    "print(rounds)\n",
    "scatter = plt.scatter(bits, muls, c=rounds, norm='linear')\n",
    "legend = plt.legend(*scatter.legend_elements(), title='Rounds')\n",
    "plt.ylabel('# Multiplications')\n",
    "plt.xlabel('# Bits')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "da90add9",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 10.1",
   "language": "sage",
   "name": "sagemath"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}