{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "f890e69f",
   "metadata": {},
   "outputs": [],
   "source": [
    "q = pow(2,255) - 19"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "d90a7f0b",
   "metadata": {},
   "outputs": [],
   "source": [
    "A = 486662\n",
    "B = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "1b2aebc5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "486664\n",
      "486660\n"
     ]
    }
   ],
   "source": [
    "a = (A + 2) * 1\n",
    "d = (A - 2) * 1\n",
    "print(a)\n",
    "print(d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "aae2f8bf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = 9\n",
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "ea9a4d90",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "14781619447589544791020593568409986887264606134616475288964881837755586237401"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = mod(u^3 + A * u^2 + u, q).sqrt()\n",
    "v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "95895004",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "38213832894368730265794714087330135568483813637251082400757400312561599933396"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u * pow(v, -1, q) % q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "1134cf74",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "46316835694926478169428394003475163141307993866256225615783033603165251855960"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(u - 1) * pow(u + 1, -1, q) % q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ec089e21",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 9.2",
   "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": "3.9.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}