1{
  2 "cells": [
  3  {
  4   "cell_type": "code",
  5   "execution_count": 1,
  6   "id": "f890e69f",
  7   "metadata": {},
  8   "outputs": [],
  9   "source": [
 10    "q = pow(2,255) - 19"
 11   ]
 12  },
 13  {
 14   "cell_type": "code",
 15   "execution_count": 2,
 16   "id": "d90a7f0b",
 17   "metadata": {},
 18   "outputs": [
 19    {
 20     "data": {
 21      "text/plain": [
 22       "37095705934669439343138083508754565189542113879843219016388785533085940283555"
 23      ]
 24     },
 25     "execution_count": 2,
 26     "metadata": {},
 27     "output_type": "execute_result"
 28    }
 29   ],
 30   "source": [
 31    "d = ((q - 1) * (121665 * pow(121666, -1, q)) % q)\n",
 32    "d"
 33   ]
 34  },
 35  {
 36   "cell_type": "code",
 37   "execution_count": 3,
 38   "id": "1b2aebc5",
 39   "metadata": {},
 40   "outputs": [
 41    {
 42     "data": {
 43      "text/plain": [
 44       "46316835694926478169428394003475163141307993866256225615783033603165251855960"
 45      ]
 46     },
 47     "execution_count": 3,
 48     "metadata": {},
 49     "output_type": "execute_result"
 50    }
 51   ],
 52   "source": [
 53    "y = 4 * pow(5, -1, q) % q\n",
 54    "y"
 55   ]
 56  },
 57  {
 58   "cell_type": "code",
 59   "execution_count": 4,
 60   "id": "aae2f8bf",
 61   "metadata": {},
 62   "outputs": [
 63    {
 64     "data": {
 65      "text/plain": [
 66       "15112221349535400772501151409588531511454012693041857206046113283949847762202"
 67      ]
 68     },
 69     "execution_count": 4,
 70     "metadata": {},
 71     "output_type": "execute_result"
 72    }
 73   ],
 74   "source": [
 75    "x = 0x216936D3CD6E53FEC0A4E231FDD6DC5C692CC7609525A7B2C9562D608F25D51A\n",
 76    "x"
 77   ]
 78  },
 79  {
 80   "cell_type": "code",
 81   "execution_count": 5,
 82   "id": "ea9a4d90",
 83   "metadata": {},
 84   "outputs": [
 85    {
 86     "data": {
 87      "text/plain": [
 88       "26187595835145689230469591415084376402084551887632582719101735842039498021991"
 89      ]
 90     },
 91     "execution_count": 5,
 92     "metadata": {},
 93     "output_type": "execute_result"
 94    }
 95   ],
 96   "source": [
 97    "x_squared = x^2 % q\n",
 98    "x_squared"
 99   ]
100  },
101  {
102   "cell_type": "code",
103   "execution_count": 6,
104   "id": "3499c817",
105   "metadata": {},
106   "outputs": [
107    {
108     "data": {
109      "text/plain": [
110       "37053468555941182535542715202780130513046395093004980492626426882532201484768"
111      ]
112     },
113     "execution_count": 6,
114     "metadata": {},
115     "output_type": "execute_result"
116    }
117   ],
118   "source": [
119    "y_squared = y^2 % q\n",
120    "y_squared"
121   ]
122  },
123  {
124   "cell_type": "code",
125   "execution_count": 7,
126   "id": "d9617d2d",
127   "metadata": {},
128   "outputs": [
129    {
130     "data": {
131      "text/plain": [
132       "10865872720795493305073123787695754110961843205372397773524691040492703462777"
133      ]
134     },
135     "execution_count": 7,
136     "metadata": {},
137     "output_type": "execute_result"
138    }
139   ],
140   "source": [
141    "((q - 1) * x_squared + y_squared) % q"
142   ]
143  },
144  {
145   "cell_type": "code",
146   "execution_count": 8,
147   "id": "ece46a74",
148   "metadata": {},
149   "outputs": [
150    {
151     "data": {
152      "text/plain": [
153       "10865872720795493305073123787695754110961843205372397773524691040492703462777"
154      ]
155     },
156     "execution_count": 8,
157     "metadata": {},
158     "output_type": "execute_result"
159    }
160   ],
161   "source": [
162    "((d * x_squared * y_squared) % q + 1) % q"
163   ]
164  },
165  {
166   "cell_type": "code",
167   "execution_count": 9,
168   "id": "b430f926",
169   "metadata": {},
170   "outputs": [
171    {
172     "data": {
173      "text/plain": [
174       "21711016731996786641919559689128982722488122124807605757398297001483711807481"
175      ]
176     },
177     "execution_count": 9,
178     "metadata": {},
179     "output_type": "execute_result"
180    }
181   ],
182   "source": [
183    "pow(8, -1, q)"
184   ]
185  },
186  {
187   "cell_type": "code",
188   "execution_count": 10,
189   "id": "d8575e54",
190   "metadata": {},
191   "outputs": [
192    {
193     "data": {
194      "text/plain": [
195       "486662"
196      ]
197     },
198     "execution_count": 10,
199     "metadata": {},
200     "output_type": "execute_result"
201    }
202   ],
203   "source": [
204    "A = 2 * (-1 + d) / (-1 - d)\n",
205    "A"
206   ]
207  },
208  {
209   "cell_type": "code",
210   "execution_count": 11,
211   "id": "2ee580a3",
212   "metadata": {},
213   "outputs": [
214    {
215     "data": {
216      "text/plain": [
217       "57896044618658097711785492504343953926634992332820282019728792003956564333285"
218      ]
219     },
220     "execution_count": 11,
221     "metadata": {},
222     "output_type": "execute_result"
223    }
224   ],
225   "source": [
226    "B = 4 / (-1 - d)\n",
227    "B"
228   ]
229  },
230  {
231   "cell_type": "code",
232   "execution_count": null,
233   "id": "ec089e21",
234   "metadata": {},
235   "outputs": [],
236   "source": []
237  }
238 ],
239 "metadata": {
240  "kernelspec": {
241   "display_name": "SageMath 9.2",
242   "language": "sage",
243   "name": "sagemath"
244  },
245  "language_info": {
246   "codemirror_mode": {
247    "name": "ipython",
248    "version": 3
249   },
250   "file_extension": ".py",
251   "mimetype": "text/x-python",
252   "name": "python",
253   "nbconvert_exporter": "python",
254   "pygments_lexer": "ipython3",
255   "version": "3.9.7"
256  }
257 },
258 "nbformat": 4,
259 "nbformat_minor": 5
260}