Notebook version of https://kel.bz/post/sage-p256/

sage-p256.ipynb

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 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This notebook demonstrates how to create a NIST P-256 curve ([aka secp256r1](https://tools.ietf.org/search/rfc4492#appendix-A)) and it's standard base point in [Sagemath](https://www.sagemath.org/).\n",
    "\n",
    "First, we define the parameters that make up the P-256 curve. The parameters are from \"[SEC 2: Recommended Elliptic Curve Domain Parameters](https://www.secg.org/SEC2-Ver-1.0.pdf)\"."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Finite field prime\n",
    "p256 = 0xFFFFFFFF00000001000000000000000000000000FFFFFFFFFFFFFFFFFFFFFFFF\n",
    "\n",
    "# Curve parameters for the curve equation: y^2 = x^3 + a256*x +b256\n",
    "a256 = p256 - 3 \n",
    "b256 = 0x5AC635D8AA3A93E7B3EBBD55769886BC651D06B0CC53B0F63BCE3C3E27D2604B\n",
    "\n",
    "# Base point (x, y)  \n",
    "gx = 0x6B17D1F2E12C4247F8BCE6E563A440F277037D812DEB33A0F4A13945D898C296\n",
    "gy = 0x4FE342E2FE1A7F9B8EE7EB4A7C0F9E162BCE33576B315ECECBB6406837BF51F5\n",
    "\n",
    "# Curve order\n",
    "qq = 0xFFFFFFFF00000000FFFFFFFFFFFFFFFFBCE6FAADA7179E84F3B9CAC2FC632551"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then we can create a EllipticCurve sage object over a finite field."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create a finite field of order p256\n",
    "FF = GF(p256) \n",
    "\n",
    "# Define a curve over that field with specified Weierstrass a and b parameters\n",
    "EC = EllipticCurve([FF(a256), FF(b256)]) \n",
    "\n",
    "# Since we know P-256's order we can skip computing it and set it explicitly\n",
    "EC.set_order(qq) \n",
    "\n",
    "# Create a variable for the base point\n",
    "G = EC(FF(gx), FF(gy))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can compare results to a few [public test vectors](http://point-at-infinity.org/ecc/nisttv) to make sure everything is working as intended.\n",
    "\n",
    "These test vectors are defined as three-tuples: (scalar `k`, x-coordinate of `k*G`, y-coordinate of `k*G`)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "test_vectors = [\n",
    "    (1, \n",
    "     0x6B17D1F2E12C4247F8BCE6E563A440F277037D812DEB33A0F4A13945D898C296, \n",
    "     0x4FE342E2FE1A7F9B8EE7EB4A7C0F9E162BCE33576B315ECECBB6406837BF51F5),\n",
    "    (2,\n",
    "     0x7CF27B188D034F7E8A52380304B51AC3C08969E277F21B35A60B48FC47669978,\n",
    "     0x07775510DB8ED040293D9AC69F7430DBBA7DADE63CE982299E04B79D227873D1),\n",
    "    (3,\n",
    "     0x5ECBE4D1A6330A44C8F7EF951D4BF165E6C6B721EFADA985FB41661BC6E7FD6C,\n",
    "     0x8734640C4998FF7E374B06CE1A64A2ECD82AB036384FB83D9A79B127A27D5032),\n",
    "    (4,\n",
    "     0xE2534A3532D08FBBA02DDE659EE62BD0031FE2DB785596EF509302446B030852,\n",
    "     0xE0F1575A4C633CC719DFEE5FDA862D764EFC96C3F30EE0055C42C23F184ED8C6),\n",
    "    (5,\n",
    "     0x51590B7A515140D2D784C85608668FDFEF8C82FD1F5BE52421554A0DC3D033ED,\n",
    "     0xE0C17DA8904A727D8AE1BF36BF8A79260D012F00D4D80888D1D0BB44FDA16DA4)\n",
    "]\n",
    "\n",
    "for k, x, y in test_vectors:\n",
    "    P = k*G\n",
    "    Px, Py = P.xy()\n",
    "    assert Px == x\n",
    "    assert Py == y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That seems alright. We can do a simple ECDH to double check as well."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "for _ in range(100):\n",
    "    alice_private = randint(0, qq-1)\n",
    "    alice_public = alice_private*G\n",
    "\n",
    "    bob_private = randint(0, qq-1)\n",
    "    bob_public = bob_private*G\n",
    "\n",
    "    assert alice_private*bob_public == bob_private*alice_public"
   ]
  }
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