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1<html>
2<head>
3<title>
4A Tour of NTL: Summary of NTL's Main Modules  </title>
5</head>
6
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8<center>
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13
14<h1> 
15<p align=center>
16A Tour of NTL: Summary of NTL's Main Modules
17</p>
18</h1>
19
20<p> <hr> <p>
21
22<p>
23
24NTL consists of a number of software modules.
25Generally speaking, for each module <tt>foo</tt>, there is
26<ul>
27<li>
28a header
29file <tt>&lt;NTL/foo.h&gt;</tt>, found in subdirectory <tt>include</tt>,
30<li>
31a documentation file <tt>foo.txt</tt>,
32found in subdirectory <tt>doc</tt>,
33and
34<li>
35a source file <tt>foo.c</tt>, found in subdirectory <tt>src</tt>.
36</ul>
37
38<p>
39Note that all of the header files for NTL modules include the header file
40<tt>&lt;NTL/tools.h&gt;</tt>, and by default, this header file includes
41the standard headers
42<ul>
43<li>
44<tt>&lt;cstdlib&gt;</tt>,
45<li>
46<tt>&lt;cmath&gt;</tt>, and
47<li>
48<tt>&lt;iostream&gt</tt>.
49</ul>
50Moreover,
51the standard library is wrapped in namespace <tt>std</tt>
52and NTL is wrapped in namespace <tt>NTL</tt>.
53
54
55<p>
56
57However, if you compiled NTL in
58<a href="tour-stdcxx.html">traditional (non-ISO mode)</a>,
59then <tt>&lt;NTL/tools.h&gt;</tt>
60includes the traditional (pre-ISO) headers
61<ul>
62<li>
63<tt>&lt;stdlib.h&gt;</tt>,
64<li>
65<tt>&lt;math.h&gt;</tt>, and
66<li>
67<tt>&lt;iostream.h&gt</tt>.
68</ul>
69Moreover,
70all of names of the standard library and NTL are defined
71in the global namespace.
72
73
74<p>
75
76The documentation  file takes the form of a header file,
77but stripped of implementation
78details and declarations of some of the more esoteric
79routines and data structures, and it contains more complete
80and usually clearer documentation than in the header file.
81Also, note that the documentation files do not explicitly
82mention anything about namespaces <tt>std</tt> and <tt>NTL</tt>.
83
84
85<p>
86
87There is a plethora of conversion routines.
88These are not documented in any of the individual
89documentation files, but rather,
90they are all briefly summarized in
91<a href="conversions.txt"><tt>conversions.txt</tt></a>.
92
93<p>
94
95The following is a summary of the main NTL modules.
96The corresponding documentation file can be obtained by clicking on
97the module name.
98
99<p>
100
101<table >
102
103
104
105<!-- ----------- GF2.txt ----------- -->
106<p><tr valign=top> <td> <b>
107<a href="GF2.txt"><tt>GF2</tt></a>
108
109</b> <td>
110
111class <tt>GF2</tt>: integers mod 2
112
113
114<!-- ----------- GF2X.txt ----------- -->
115<p><tr valign=top> <td> <b>
116<a href="GF2X.txt"><tt>GF2X</tt></a>
117
118</b> <td>
119
120class <tt>GF2X</tt>: polynomials over <tt>GF(2)</tt> (much more efficient
121than using <tt>zz_pX</tt> with <tt>p=2</tt>);
122includes routines for GCDs and minimal polynomials
123
124
125<!-- ----------- GF2XFactoring.txt ----------- -->
126<p><tr valign=top> <td> <b>
127<a href="GF2XFactoring.txt"><tt>GF2XFactoring</tt></a>
128
129</b> <td>
130routines for factoring polynomials over <tt>GF(2)</tt>;
131also includes routines for testing for and constructing
132irreducible polynomials
133
134
135<!-- ----------- GF2XVec.txt ----------- -->
136<p><tr valign=top> <td> <b>
137<a href="GF2XVec.txt"><tt>GF2XVec</tt></a>
138
139</b> <td>
140class <tt>GF2XVec</tt>: fixed-length vectors of fixed-length <tt>GF2X</tt>s;
141less flexible, but more efficient than <tt>vec_GF2X</tt>
142
143
144
145
146<!-- ----------- GF2E.txt ----------- -->
147<p><tr valign=top> <td> <b>
148<a href="GF2E.txt"><tt>GF2E</tt></a>
149
150</b> <td>
151class <tt>GF2E</tt>: polynomial extension field/ring over <tt>GF(2)</tt>,
152implemented as GF(2)[X]/(P).
153
154
155<!-- ----------- GF2EX.txt ----------- -->
156<p><tr valign=top> <td> <b>
157<a href="GF2EX.txt"><tt>GF2EX</tt></a>
158
159</b> <td>
160class <tt>GF2EX</tt>
161class <tt>GF2EX</tt>: polynomials over <tt>GF2E</tt>;
162includes routines for modular polynomials arithmetic,
163modular composition, minimal and characteristic
164polynomials, and interpolation.
165
166
167
168<!-- ----------- GF2EXFactoring.txt ----------- -->
169<p><tr valign=top> <td> <b>
170<a href="GF2EXFactoring.txt"><tt>GF2EXFactoring</tt></a>
171
172</b> <td>
173routines for factoring polynomials over <tt>GF2E</tt>;
174also includes routines for testing for and constructing
175irreducible polynomials
176
177
178<!-- ----------- HNF.txt ----------- -->
179<p><tr valign=top> <td> <b>
180<a href="HNF.txt"><tt>HNF</tt></a>
181
182</b> <td>
183routines for computing the Hermite Normal Form
184of a lattice
185
186
187
188<!-- ----------- LLL.txt ----------- -->
189<p><tr valign=top> <td> <b>
190<a href="LLL.txt"><tt>LLL</tt></a>
191
192</b> <td>
193routines for performing lattice basis reduction,
194including very fast and robust implementations of the Schnorr-Euchner LLL
195and Block Korkin Zolotarev reduction algorithm,
196as well as an integer-only reduction algorithm.
197Also, there are routines here for computing the kernel
198and image of an integer matrix, as well as finding integer
199solutions to linear systems of equations over the integers.
200
201
202
203<!-- ----------- RR.txt ----------- -->
204<p><tr valign=top> <td> <b>
205<a href="RR.txt"><tt>RR</tt></a>
206
207</b> <td>
208class <tt>RR</tt>: arbitrary-precision floating point numbers.
209
210
211
212<!-- ----------- ZZ.txt ----------- -->
213<p><tr valign=top> <td> <b>
214<a href="ZZ.txt"><tt>ZZ</tt></a>
215
216</b> <td>
217class <tt>ZZ</tt>: arbitrary length integers;
218includes routines for GCDs, Jacobi symbols,
219modular arithmetic, and primality testing;
220also includes small prime generation routines
221and in-line routines for single-precision
222modular arithmetic
223
224
225
226<!-- ----------- ZZVec.txt ----------- -->
227<p><tr valign=top> <td> <b>
228<a href="ZZVec.txt"><tt>ZZVec</tt></a>
229
230</b> <td>
231class <tt>ZZVec</tt>: fixed-length vectors of fixed-length <tt>ZZ</tt>s;
232less flexible, but more efficient than <tt>vec_ZZ</tt>
233
234
235
236<!-- ----------- ZZX.txt ----------- -->
237<p><tr valign=top> <td> <b>
238<a href="ZZX.txt"><tt>ZZX</tt></a>
239
240</b> <td>
241class <tt>ZZX</tt>: polynomials over <tt>ZZ</tt>;
242includes routines for GCDs, minimal and characteristic
243polynomials, norms and traces
244
245
246
247<!-- ----------- ZZXFactoring.txt ----------- -->
248<p><tr valign=top> <td> <b>
249<a href="ZZXFactoring.txt"><tt>ZZXFactoring</tt></a>
250
251</b> <td>
252routines for factoring univariate polynomials over <tt>ZZ</tt>
253
254
255
256<!-- ----------- ZZ_p.txt ----------- -->
257<p><tr valign=top> <td> <b>
258<a href="ZZ_p.txt"><tt>ZZ_p</tt></a>
259
260</b> <td>
261class <tt>ZZ_p</tt>: integers mod <tt>p</tt>
262
263<!-- ----------- ZZ_pE.txt ----------- -->
264<p><tr valign=top> <td> <b>
265<a href="ZZ_pE.txt"><tt>ZZ_pE</tt></a>
266
267</b> <td>
268class <tt>ZZ_pE</tt>: ring/field extension of <tt>ZZ_p</tt>
269
270
271
272<!-- ----------- ZZ_pEX.txt ----------- -->
273<p><tr valign=top> <td> <b>
274<a href="ZZ_pEX.txt"><tt>ZZ_pEX</tt></a>
275
276</b> <td>
277class <tt>ZZ_pEX</tt>: polynomials over <tt>ZZ_pE</tt>;
278includes routines for modular polynomials arithmetic,
279modular composition, minimal and characteristic
280polynomials, and interpolation.
281
282<!-- ----------- ZZ_pEXFactoring.txt ----------- -->
283<p><tr valign=top> <td> <b>
284<a href="ZZ_pEXFactoring.txt"><tt>ZZ_pEXFactoring</tt></a>
285
286</b> <td>
287routines for factoring polynomials over <tt>ZZ_pE</tt>;
288also includes routines for testing for and constructing
289irreducible polynomials
290
291
292
293<!-- ----------- ZZ_pX.txt ----------- -->
294<p><tr valign=top> <td> <b>
295<a href="ZZ_pX.txt"><tt>ZZ_pX</tt></a>
296
297</b> <td>
298class <tt>ZZ_pX</tt>: polynomials over <tt>ZZ_p</tt>;
299includes routines for modular polynomials arithmetic,
300modular composition, minimal and characteristic
301polynomials, and interpolation.
302
303
304
305<!-- ----------- ZZ_pXFactoring.txt ----------- -->
306<p><tr valign=top> <td> <b>
307<a href="ZZ_pXFactoring.txt"><tt>ZZ_pXFactoring</tt></a>
308
309</b> <td>
310routines for factoring polynomials over <tt>ZZ_p</tt>;
311also includes routines for testing for and constructing
312irreducible polynomials
313
314
315
316<!-- ----------- lzz_p.txt ----------- -->
317<p><tr valign=top> <td> <b>
318<a href="lzz_p.txt"><tt>lzz_p</tt></a>
319
320</b> <td>
321class <tt>zz_p</tt>:
322integers mod <tt>p</tt>, where <tt>p</tt> is single-precision
323
324
325<!-- ----------- lzz_pE.txt ----------- -->
326<p><tr valign=top> <td> <b>
327<a href="lzz_pE.txt"><tt>lzz_pE</tt></a>
328
329</b> <td>
330class <tt>zz_pE</tt>:
331ring/field extension of zz_p
332
333
334<!-- ----------- lzz_pEX.txt ----------- -->
335<p><tr valign=top> <td> <b>
336<a href="lzz_pEX.txt"><tt>lzz_pEX</tt></a>
337
338</b> <td>
339class <tt>zz_pEX</tt>: polynomials over <tt>zz_pE</tt>;
340provides the same functionality as class <tt>ZZ_pEX</tt>,
341but for single-precision <tt>p</tt>
342
343<!-- ----------- lzz_pEXFactoring.txt ----------- -->
344<p><tr valign=top> <td> <b>
345<a href="lzz_pEXFactoring.txt"><tt>lzz_pEXFactoring</tt></a>
346
347</b> <td>
348routines for factoring polynomials over <tt>zz_pE</tt>;
349provides the same functionality as class <tt>ZZ_pEX</tt>,
350but for single-precision <tt>p</tt>
351
352
353<!-- ----------- lzz_pX.txt ----------- -->
354<p><tr valign=top> <td> <b>
355<a href="lzz_pX.txt"><tt>lzz_pX</tt></a>
356
357</b> <td>
358class <tt>zz_pX</tt>: polynomials over <tt>zz_p</tt>;
359provides the same functionality as class <tt>ZZ_pX</tt>,
360but for single-precision <tt>p</tt>
361
362
363<!-- ----------- lzz_pXFactoring.txt ----------- -->
364<p><tr valign=top> <td> <b>
365<a href="lzz_pXFactoring.txt"><tt>lzz_pXFactoring</tt></a>
366
367</b> <td>
368routines for factoring polynomials over <tt>zz_p</tt>;
369provides the same functionality as class <tt>ZZ_pX</tt>,
370but for single-precision <tt>p</tt>
371
372
373<!-- ----------- matrix.txt ----------- -->
374<p><tr valign=top> <td> <b>
375<a href="matrix.txt"><tt>matrix</tt></a>
376
377</b> <td>
378template-like macros for
379dynamic-size 2-dimensional arrays
380
381<!-- ----------- mat_GF2.txt ----------- -->
382<p><tr valign=top> <td> <b>
383<a href="mat_GF2.txt"><tt>mat_GF2</tt></a>
384
385</b> <td>
386class <tt>mat_GF2</tt>: matrices over <tt>GF(2)</tt>;
387includes basic matrix arithmetic operations,
388including determinant calculation, matrix inversion,
389solving nonsingular systems of linear equations,
390and Gaussian elimination
391
392
393<!-- ----------- mat_GF2E.txt ----------- -->
394<p><tr valign=top> <td> <b>
395<a href="mat_GF2E.txt"><tt>mat_GF2E</tt></a>
396
397</b> <td>
398class <tt>mat_GF2E</tt>: matrices over <tt>GF2E</tt>;
399includes basic matrix arithmetic operations,
400including determinant calculation, matrix inversion,
401solving nonsingular systems of linear equations,
402and Gaussian elimination
403
404
405
406<!-- ----------- mat_RR.txt ----------- -->
407<p><tr valign=top> <td> <b>
408<a href="mat_RR.txt"><tt>mat_RR</tt></a>
409
410</b> <td>
411class <tt>mat_RR</tt>: matrices over <tt>RR</tt>;
412includes basic matrix arithmetic operations,
413including determinant calculation, matrix inversion,
414and solving nonsingular systems of linear equations.
415
416
417
418<!-- ----------- mat_ZZ.txt ----------- -->
419<p><tr valign=top> <td> <b>
420<a href="mat_ZZ.txt"><tt>mat_ZZ</tt></a>
421
422</b> <td>
423class <tt>mat_ZZ</tt>: matrices over <tt>ZZ</tt>;
424includes basic matrix arithmetic operations,
425including determinant calculation, matrix inversion,
426and solving nonsingular systems of linear equations.
427See also the LLL module for additional routines.
428
429
430
431<!-- ----------- mat_ZZ_p.txt ----------- -->
432<p><tr valign=top> <td> <b>
433<a href="mat_ZZ_p.txt"><tt>mat_ZZ_p</tt></a>
434
435</b> <td>
436class <tt>mat_ZZ_p</tt>: matrices over <tt>ZZ_p</tt>;
437includes basic matrix arithmetic operations,
438including determinant calculation, matrix inversion,
439solving nonsingular systems of linear equations,
440and Gaussian elimination
441
442
443<!-- ----------- mat_ZZ_pE.txt ----------- -->
444<p><tr valign=top> <td> <b>
445<a href="mat_ZZ_pE.txt"><tt>mat_ZZ_pE</tt></a>
446
447</b> <td>
448class <tt>mat_ZZ_pE</tt>: matrices over <tt>ZZ_pE</tt>;
449includes basic matrix arithmetic operations,
450including determinant calculation, matrix inversion,
451solving nonsingular systems of linear equations,
452and Gaussian elimination
453
454
455
456
457<!-- ----------- mat_lzz_p.txt ----------- -->
458<p><tr valign=top> <td> <b>
459<a href="mat_lzz_p.txt"><tt>mat_lzz_p</tt></a>
460
461</b> <td>
462class <tt>mat_zz_p</tt>: matrices over <tt>zz_p</tt>;
463includes basic matrix arithmetic operations,
464including determinant calculation, matrix inversion,
465solving nonsingular systems of linear equations,
466and Gaussian elimination
467
468<!-- ----------- mat_lzz_pE.txt ----------- -->
469<p><tr valign=top> <td> <b>
470<a href="mat_lzz_pE.txt"><tt>mat_lzz_pE</tt></a>
471
472</b> <td>
473class <tt>mat_zz_pE</tt>: matrices over <tt>zz_pE</tt>;
474includes basic matrix arithmetic operations,
475including determinant calculation, matrix inversion,
476solving nonsingular systems of linear equations,
477and Gaussian elimination
478
479
480
481<!-- ----------- mat_poly_ZZ.txt ----------- -->
482<p><tr valign=top> <td> <b>
483<a href="mat_poly_ZZ.txt"><tt>mat_poly_ZZ</tt></a>
484
485</b> <td>
486routine for computing the characteristic
487polynomial of a <tt>mat_ZZ</tt>
488
489
490
491<!-- ----------- mat_poly_ZZ_p.txt ----------- -->
492<p><tr valign=top> <td> <b>
493<a href="mat_poly_ZZ_p.txt"><tt>mat_poly_ZZ_p</tt></a>
494
495</b> <td>
496routine for computing the characteristic
497polynomial of a <tt>mat_ZZ_p</tt>
498
499
500
501<!-- ----------- mat_poly_lzz_p.txt ----------- -->
502<p><tr valign=top> <td> <b>
503<a href="mat_poly_lzz_p.txt"><tt>mat_poly_lzz_p</tt></a>
504
505</b> <td>
506routine for computing the characteristic
507polynomial of a <tt>mat_zz_p</tt>
508
509
510
511
512
513<!-- ----------- pair.txt ----------- -->
514<p><tr valign=top> <td> <b>
515<a href="pair.txt"><tt>pair</tt></a>
516
517</b> <td>
518template-like macros for
519pairs
520
521
522
523<!-- ----------- quad_float.txt ----------- -->
524<p><tr valign=top> <td> <b>
525<a href="quad_float.txt"><tt>quad_float</tt></a>
526
527</b> <td>
528class <tt>quad_float</tt>: quadruple-precision floating point numbers.
529
530
531
532<!-- ----------- tools.txt ----------- -->
533<p><tr valign=top> <td> <b>
534<a href="tools.txt"><tt>tools</tt></a>
535
536</b> <td>
537some basic types and utility routines, including the
538timing function <tt>GetTime()</tt>, and several overloaded
539versions of <tt>min()</tt> and <tt>max()</tt>
540
541
542<!-- ----------- vector.txt ----------- -->
543<p><tr valign=top> <td> <b>
544<a href="vector.txt"><tt>vector</tt></a>
545
546</b> <td>
547template-like macros for
548dynamic-size vectors
549
550<!-- ----------- vec_GF2.txt ----------- -->
551<p><tr valign=top> <td> <b>
552<a href="vec_GF2.txt"><tt>vec_GF2</tt></a>
553
554</b> <td>
555class <tt>vec_GF2</tt>: vectors over <tt>GF(2)</tt>, with arithmetic
556
557<!-- ----------- vec_GF2E.txt ----------- -->
558<p><tr valign=top> <td> <b>
559<a href="vec_GF2E.txt"><tt>vec_GF2E</tt></a>
560
561</b> <td>
562class <tt>vec_GF2E</tt>: vectors over <tt>GF2E</tt>, with arithmetic
563
564
565<!-- ----------- vec_RR.txt ----------- -->
566<p><tr valign=top> <td> <b>
567<a href="vec_RR.txt"><tt>vec_RR</tt></a>
568
569</b> <td>
570class <tt>vec_RR</tt>: vectors over <tt>RR</tt>, with arithmetic
571
572
573
574<!-- ----------- vec_ZZ.txt ----------- -->
575<p><tr valign=top> <td> <b>
576<a href="vec_ZZ.txt"><tt>vec_ZZ</tt></a>
577
578</b> <td>
579class <tt>vec_ZZ</tt>: vectors over <tt>ZZ</tt>, with arithmetic
580
581
582
583<!-- ----------- vec_ZZ_p.txt ----------- -->
584<p><tr valign=top> <td> <b>
585<a href="vec_ZZ_p.txt"><tt>vec_ZZ_p</tt></a>
586
587</b> <td>
588class <tt>vec_ZZ_p</tt>: vectors over <tt>ZZ_p</tt>, with arithmetic
589
590<!-- ----------- vec_ZZ_pE.txt ----------- -->
591<p><tr valign=top> <td> <b>
592<a href="vec_ZZ_pE.txt"><tt>vec_ZZ_pE</tt></a>
593
594</b> <td>
595class <tt>vec_ZZ_pE</tt>: vectors over <tt>ZZ_pE</tt>, with arithmetic
596
597
598
599<!-- ----------- vec_lzz_p.txt ----------- -->
600<p><tr valign=top> <td> <b>
601<a href="vec_lzz_p.txt"><tt>vec_lzz_p</tt></a>
602
603</b> <td>
604class <tt>vec_zz_p</tt>: vectors over <tt>zz_p</tt>, with arithmetic
605
606
607<!-- ----------- vec_lzz_pE.txt ----------- -->
608<p><tr valign=top> <td> <b>
609<a href="vec_lzz_pE.txt"><tt>vec_lzz_pE</tt></a>
610
611</b> <td>
612class <tt>vec_zz_pE</tt>: vectors over <tt>zz_pE</tt>, with arithmetic
613
614
615<!-- ----------- version.txt ----------- -->
616<p><tr valign=top> <td> <b>
617<a href="version.txt"><tt>version</tt></a>
618
619</b> <td>
620macros defining the NTL version number
621
622
623
624<!-- ----------- xdouble.txt ----------- -->
625<p><tr valign=top> <td> <b>
626<a href="xdouble.txt"><tt>xdouble</tt></a>
627
628</b> <td>
629class <tt>xdouble</tt>: double-precision floating point numbers with
630extended exponent range.
631
632</table>
633
634<p>
635
636<h3>
637Some other types
638</h3>
639
640
641<p>
642In addition to the above, other generic vectors are declared,
643not explicitly documented elsewhere:
644<p>
645
646<ul>
647<li> <tt>vec_GF2XVec</tt>
648<li> <tt>vec_ZZVec</tt>
649<li> <tt>vec_double</tt>
650<li> <tt>vec_long</tt>
651<li> <tt>vec_quad_float</tt>
652<li> <tt>vec_ulong</tt>
653<li> <tt>vec_vec_GF2</tt>
654<li> <tt>vec_vec_GF2E</tt>
655<li> <tt>vec_vec_RR</tt>
656<li> <tt>vec_vec_ZZ</tt>
657<li> <tt>vec_vec_ZZ_p</tt>
658<li> <tt>vec_vec_ZZ_pE</tt>
659<li> <tt>vec_vec_long</tt>
660<li> <tt>vec_vec_lzz_p</tt>
661<li> <tt>vec_vec_lzz_pE</tt>
662<li> <tt>vec_vec_ulong</tt>
663<li> <tt>vec_xdouble</tt>
664</ul>
665
666<p>
667
668These decalarations are found in "<tt>.h</tt>" files with
669corresponding names.
670No additional functionality is provided.
671
672<p>
673All of the header files for polynomial classes <tt>ZZ_pX</tt>,
674<tt>ZZX</tt>, etc., declare generic vectors <tt>vec_ZZ_pX</tt>,
675<tt>vec_ZZX</tt>, etc.
676
677<p>
678There are also a number of generic pair classes defined,
679not explicitly documented elsewhere:
680
681<p>
682
683<ul>
684<li> <tt>pair_GF2EX_long</tt>
685<li> <tt>pair_GF2X_long</tt>
686<li> <tt>pair_ZZX_long</tt>
687<li> <tt>pair_ZZ_pEX_long</tt>
688<li> <tt>pair_ZZ_pX_long</tt>
689<li> <tt>pair_lzz_pEX_long</tt>
690<li> <tt>pair_lzz_pX_long</tt>
691</ul>
692
693<p>
694
695These decalarations are found in "<tt>.h</tt>" files with
696corresponding names.
697These files also declare corresponding generic vector types
698<tt>vec_pair_GF2EX_long</tt>, etc.
699No additional functionality is provided.
700
701
702<p>
703
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