| 1 | <html> |
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| 2 | <head> |
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| 3 | <title> |
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| 4 | A Tour of NTL: Examples: Extension Rings and Fields </title> |
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| 5 | </head> |
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| 6 | |
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| 7 | <body bgcolor="#fff9e6"> |
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| 12 | </center> |
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| 13 | |
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| 14 | <h1> |
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| 15 | <p align=center> |
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| 16 | A Tour of NTL: Examples: Extension Rings and Fields |
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| 17 | </p> |
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| 18 | </h1> |
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| 19 | |
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| 20 | <p> <hr> <p> |
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| 21 | |
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| 22 | NTL also supports extension rings and fields over finite fields, |
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| 23 | and polynomial arithmetic over such extensions. |
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| 24 | Here is a little program that illustrates this. |
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| 25 | |
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| 26 | <p> |
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| 27 | <pre> |
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| 28 | #include <NTL/ZZ_pXFactoring.h> |
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| 29 | #include <NTL/ZZ_pEX.h> |
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| 30 | |
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| 31 | int main() |
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| 32 | { |
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| 33 | ZZ_p::init(to_ZZ(17)); // define GF(17) |
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| 34 | |
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| 35 | ZZ_pX P; |
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| 36 | BuildIrred(P, 10); // generate an irreducible polynomial P |
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| 37 | // of degree 10 over GF(17) |
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| 38 | |
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| 39 | ZZ_pE::init(P); // define GF(17^10) |
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| 40 | |
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| 41 | ZZ_pEX f, g, h; // declare polynomials over GF(17^10) |
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| 42 | |
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| 43 | random(f, 20); // f is a random, monic polynomial of degree 20 |
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| 44 | SetCoeff(f, 20); |
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| 45 | |
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| 46 | random(h, 20); // h is a random polynomial of degree less than 20 |
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| 47 | |
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| 48 | g = MinPolyMod(h, f); // compute the minimum polynomial of h modulo f |
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| 49 | |
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| 50 | if (g == 0) Error("oops (1)"); // check that g != 0 |
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| 51 | |
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| 52 | if (CompMod(g, h, f) != 0) // check that g(h) = 0 mod f |
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| 53 | Error("oops (2)"); |
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| 54 | } |
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| 55 | </pre> |
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| 56 | |
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| 57 | <p> |
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| 58 | This example illustrates building extension rings over <tt>ZZ_p</tt>. |
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| 59 | One can also use <tt>zz_p</tt> and <tt>GF2</tt> as base classes; |
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| 60 | the syntax is exactly the same. |
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| 61 | |
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| 62 | <p> |
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| 63 | See <a href="ZZ_pE.txt"><tt>ZZ_pE.txt</tt></a> for the basics of the extension |
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| 64 | ring <tt>ZZ_pE</tt> over <tt>ZZ_p</tt>. |
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| 65 | Also see <a href="ZZ_pEX.txt"><tt>ZZ_pEX.txt</tt></a> for polynomial |
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| 66 | arithmetic over <tt>ZZ_pE</tt>, and |
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| 67 | <a href="ZZ_pEXFactoring.txt"><tt>ZZ_pEXFactoring.txt</tt></a> for factoring |
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| 68 | routines over <tt>ZZ_pE</tt>. |
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| 69 | See <a href="vec_ZZ_pE.txt"><tt>vec_ZZ_pE.txt</tt></a> for vectors over <tt>ZZ_pE</tt>, |
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| 70 | and <a href="mat_ZZ_pE.txt"><tt>mat_ZZ_pE.txt</tt></a> for matrices over <tt>ZZ_pE</tt>. |
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| 71 | |
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| 72 | <p> |
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| 73 | See <a href="lzz_pE.txt"><tt>lzz_pE.txt</tt></a> for the basics of the extension |
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| 74 | ring <tt>zz_pE</tt> over <tt>zz_p</tt>. |
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| 75 | Also see <a href="lzz_pEX.txt"><tt>lzz_pEX.txt</tt></a> for polynomial |
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| 76 | arithmetic over <tt>zz_pE</tt>, and |
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| 77 | <a href="lzz_pEXFactoring.txt"><tt>lzz_pEXFactoring.txt</tt></a> for factoring |
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| 78 | routines over <tt>zz_pE</tt>. |
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| 79 | See <a href="vec_lzz_pE.txt"><tt>vec_lzz_pE.txt</tt></a> for vectors over <tt>zz_pE</tt>, |
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| 80 | and <a href="mat_lzz_pE.txt"><tt>mat_lzz_pE.txt</tt></a> for matrices over <tt>zz_pE</tt>. |
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| 81 | |
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| 82 | <p> |
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| 83 | See <a href="GF2E.txt"><tt>GF2E.txt</tt></a> for the basics of the extension |
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| 84 | ring <tt>GF2E</tt> over <tt>GF2</tt>. |
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| 85 | Also see <a href="GF2EX.txt"><tt>GF2EX.txt</tt></a> for polynomial |
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| 86 | arithmetic over <tt>GF2E</tt>, and |
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| 87 | <a href="GF2EXFactoring.txt"><tt>GF2EXFactoring.txt</tt></a> for factoring |
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| 88 | routines over <tt>GF2E</tt>. |
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| 89 | See <a href="vec_GF2E.txt"><tt>vec_GF2E.txt</tt></a> for vectors over <tt>GF2E</tt>, |
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| 90 | and <a href="mat_GF2E.txt"><tt>mat_GF2E.txt</tt></a> for matrices over <tt>GF2E</tt>. |
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| 91 | |
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| 92 | |
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| 93 | <center> |
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