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@ -130,7 +130,7 @@ |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": 4, |
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"execution_count": 16, |
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"metadata": {}, |
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"outputs": [ |
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{ |
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@ -179,15 +179,15 @@ |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": 5, |
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"execution_count": 14, |
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"metadata": {}, |
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"outputs": [ |
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{ |
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"name": "stdout", |
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"output_type": "stream", |
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"text": [ |
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"φ(16) = 8 since the 8 items in Z*16 are each relatively prime to 16.\n", |
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"Z*16 = [1, 3, 5, 7, 9, 11, 13, 15]\n" |
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"φ(21) = 12 since the 12 items in Z*21 are each relatively prime to 21.\n", |
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"Z*21 = [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20]\n" |
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] |
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} |
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], |
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@ -206,7 +206,7 @@ |
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" r.append(i)\n", |
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" return r\n", |
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"\n", |
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"n = 16\n", |
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"n = 21\n", |
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"ar = z_star(n)\n", |
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"e = len(ar)\n", |
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"print(f\"φ({n}) = {e} since the {e} items in Z*{n} are each relatively prime to {n}.\")\n", |
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@ -233,7 +233,7 @@ |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": 6, |
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"execution_count": 13, |
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"metadata": {}, |
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"outputs": [ |
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{ |
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@ -275,7 +275,50 @@ |
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" return (int)(result)\n", |
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"\n", |
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"n = 1125\n", |
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"print(f\"𝜑({n}) = {totient(1125)}\")" |
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"print(f\"𝜑({n}) = {totient(n)}\")" |
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] |
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}, |
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{ |
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"cell_type": "markdown", |
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"metadata": {}, |
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"source": [ |
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"## Finding a generator of Z*p\n", |
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"- https://crypto.stanford.edu/pbc/notes/numbertheory/gen.html\n", |
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"- Lecture 2, page 19\n", |
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"\n", |
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"A generator of $Z^∗p$ is an element of order $p − 1$\n", |
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"\n", |
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"To find a generator of Z∗p we can choose a value g and test it as follows:\n", |
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"1. compute all the distinct prime factors of p − 1 and call them f1,f2,...,fr\n", |
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"2. then g is a generator as long as $g^{\\frac{p−1}{fi}} \\neq 1 \\mod(p)$ for $i = 1,2,,...,r$" |
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] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": 52, |
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"metadata": {}, |
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"outputs": [ |
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{ |
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"name": "stdout", |
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"output_type": "stream", |
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"text": [ |
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"Is 3 a generator for Z*4? True\n" |
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] |
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} |
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], |
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"source": [ |
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"# Finding a generator g of Z∗p\n", |
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"def is_generator(p, g):\n", |
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" pf = prime_factors(p-1)\n", |
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" for f in pf:\n", |
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" if (g**((p-1)/f))%p == 1:\n", |
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" return False\n", |
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" return True\n", |
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"\n", |
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"g = 3 # Generator\n", |
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"p = 4 # Z*p\n", |
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"\n", |
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"print(f\"Is {g} a generator for Z*{p}? {is_generator(p, g)}\")" |
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] |
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}, |
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{ |
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