More latex please

TTM4135.ipynb

@@ -10,7 +10,7 @@
  },
  {
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- "execution_count": 1,
+ "execution_count": 14,
  "metadata": {},
  "outputs": [
  {
@@ -19,7 +19,7 @@
  "5"
  ]
  },
- "execution_count": 1,
+ "execution_count": 14,
  "metadata": {},
  "output_type": "execute_result"
  }
@@ -46,7 +46,7 @@
  },
  {
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- "execution_count": 2,
+ "execution_count": 17,
  "metadata": {},
  "outputs": [
  {
@@ -72,7 +72,7 @@
  " x = y1 - (b//a) * x1 \n",
  " y = x1 \n",
  "\n",
- " return gcd,x,y \n",
+ " return gcd, x, y \n",
  "\n",
  "a, b = 35,10\n",
  "g, x, y = gcdExtended(a, b)\n",
@@ -92,7 +92,7 @@
  },
  {
  "cell_type": "code",
- "execution_count": 3,
+ "execution_count": 16,
  "metadata": {},
  "outputs": [
  {
@@ -130,7 +130,7 @@
  },
  {
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- "execution_count": 16,
+ "execution_count": 4,
  "metadata": {},
  "outputs": [
  {
@@ -171,6 +171,57 @@
  "cell_type": "markdown",
  "metadata": {},
  "source": [
+ "# Integer factorisation\n",
+ "Given an integer, find its prime factors"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "The factors of 641361 are [3, 7, 7, 4363], verified\n",
+ "Manual test: True\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Brute force solution\n",
+ "def prime_factors(n):\n",
+ " i = 2\n",
+ " factors = []\n",
+ " while i * i <= n:\n",
+ " if n % i:\n",
+ " i += 1\n",
+ " else:\n",
+ " n //= i\n",
+ " factors.append(i)\n",
+ " if n > 1:\n",
+ " factors.append(n)\n",
+ " return factors\n",
+ "\n",
+ "# Check if the numbers in array a is the factors of n\n",
+ "def factor_test(a, n):\n",
+ " k = 1\n",
+ " for i in factors:\n",
+ " k=k*i\n",
+ " return k==n\n",
+ "\n",
+ "n = 641361\n",
+ "factors = prime_factors(n)\n",
+ "print(f\"The factors of {n} are {factors}, {'verified' if factor_test(factors, n) else 'ERROR'}\")\n",
+ "\n",
+ "print(\"Manual test:\", factor_test([3, 7, 7, 4363], 641361))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
  "# Euler function φ and Z*n\n",
  "From slides 2020-4135-l07 - Number Theory for Public Key Cryptography\n",
  "\n",
@@ -179,7 +230,7 @@
  },
  {
  "cell_type": "code",
- "execution_count": 14,
+ "execution_count": 6,
  "metadata": {},
  "outputs": [
  {
@@ -233,7 +284,7 @@
  },
  {
  "cell_type": "code",
- "execution_count": 13,
+ "execution_count": 7,
  "metadata": {},
  "outputs": [
  {
@@ -286,16 +337,16 @@
  "- https://crypto.stanford.edu/pbc/notes/numbertheory/gen.html\n",
  "- Lecture 2, page 19\n",
  "\n",
- "A generator of $Z^∗p$ is an element of order $p − 1$\n",
+ "A generator of $Z_p^∗$ is an element of order $p − 1$\n",
  "\n",
- "To find a generator of Z∗p we can choose a value g and test it as follows:\n",
- "1. compute all the distinct prime factors of p − 1 and call them f1,f2,...,fr\n",
- "2. then g is a generator as long as $g^{\\frac{p−1}{fi}} \\neq 1 \\mod(p)$ for $i = 1,2,,...,r$"
+ "To find a generator of $Z_p^∗$ we can choose a value g and test it as follows:\n",
+ "1. compute all the distinct prime factors of $p − 1$ and call them $f_1, f_2, ..., f_r$\n",
+ "2. then $g$ is a generator as long as $g^{\\frac{p−1}{f_i}} \\neq 1 \\mod(p)$ for $i = 1,2,,...,r$"
  ]
  },
  {
  "cell_type": "code",
- "execution_count": 52,
+ "execution_count": 8,
  "metadata": {},
  "outputs": [
  {
@@ -331,7 +382,7 @@
  },
  {
  "cell_type": "code",
- "execution_count": 7,
+ "execution_count": 9,
  "metadata": {},
  "outputs": [
  {
@@ -355,7 +406,7 @@
  },
  {
  "cell_type": "code",
- "execution_count": 8,
+ "execution_count": 10,
  "metadata": {},
  "outputs": [
  {
@@ -409,7 +460,7 @@
  },
  {
  "cell_type": "code",
- "execution_count": 9,
+ "execution_count": 11,
  "metadata": {},
  "outputs": [
  {
@@ -486,57 +537,6 @@
  "cell_type": "markdown",
  "metadata": {},
  "source": [
- "# Integer factorisation\n",
- "Given an integer, find its prime factors"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 10,
- "metadata": {},
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "The factors of 641361 are [3, 7, 7, 4363], verified\n",
- "Manual test: True\n"
- ]
- }
- ],
- "source": [
- "# Brute force solution\n",
-  "def prime_factors(n):\n",
- " i = 2\n",
- " factors = []\n",
- " while i * i <= n:\n",
- " if n % i:\n",
- " i += 1\n",
- " else:\n",
- " n //= i\n",
- " factors.append(i)\n",
- " if n > 1:\n",
- " factors.append(n)\n",
- " return factors\n",
- "\n",
- "# Check if the numbers in array a is the factors of n\n",
- "def factor_test(a, n):\n",
- " k = 1\n",
- " for i in factors:\n",
- " k=k*i\n",
- " return k==n\n",
- "\n",
- "n = 641361\n",
- "factors = prime_factors(n)\n",
- "print(f\"The factors of {n} are {factors}, {'verified' if factor_test(factors, n) else 'ERROR'}\")\n",
- "\n",
- "print(\"Manual test:\", factor_test([3, 7, 7, 4363], 641361))"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
  "# Discrete logarithm problem\n",
  "Given a prime `p` and an integer `y` with `0 < y < p`, find `x` such that \n",
  "\n",
@@ -552,7 +552,7 @@
  },
  {
  "cell_type": "code",
- "execution_count": 11,
+ "execution_count": 12,
  "metadata": {},
  "outputs": [
  {
@@ -629,7 +629,7 @@
  },
  {
  "cell_type": "code",
- "execution_count": 12,
+ "execution_count": 13,
  "metadata": {},
  "outputs": [
  {