A Tour of NTL: Summary of NTL's Main Modules

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A Tour of NTL: Summary of NTL's Main Modules

NTL consists of a number of software modules. Generally speaking, for each module `foo`, there is

- a header file `<NTL/foo.h>`, found in subdirectory `include`,
- a documentation file `foo.txt`, found in subdirectory `doc`, and
- a source file `foo.c`, found in subdirectory `src`.

Note that all of the header files for NTL modules include the header file `<NTL/tools.h>`, and by default, this header file includes the standard headers

- `<cstdlib>`,
- `<cmath>`, and
- `<iostream>`.

Moreover, the standard library is wrapped in namespace `std` and NTL is wrapped in namespace `NTL`.

However, if you compiled NTL in traditional (non-ISO mode), then `<NTL/tools.h>` includes the traditional (pre-ISO) headers

- `<stdlib.h>`,
- `<math.h>`, and
- `<iostream.h>`.

Moreover, all of names of the standard library and NTL are defined in the global namespace.

The documentation file takes the form of a header file, but stripped of implementation details and declarations of some of the more esoteric routines and data structures, and it contains more complete and usually clearer documentation than in the header file. Also, note that the documentation files do not explicitly mention anything about namespaces `std` and `NTL`.

There is a plethora of conversion routines. These are not documented in any of the individual documentation files, but rather, they are all briefly summarized in `conversions.txt`.

The following is a summary of the main NTL modules. The corresponding documentation file can be obtained by clicking on the module name.

`GF2` | class `GF2`: integers mod 2
`GF2X` | class `GF2X`: polynomials over `GF(2)` (much more efficient than using `zz_pX` with `p=2`); includes routines for GCDs and minimal polynomials
`GF2XFactoring` | routines for factoring polynomials over `GF(2)`; also includes routines for testing for and constructing irreducible polynomials
`GF2XVec` | class `GF2XVec`: fixed-length vectors of fixed-length `GF2X`s; less flexible, but more efficient than `vec_GF2X`
`GF2E` | class `GF2E`: polynomial extension field/ring over `GF(2)`, implemented as GF(2)[X]/(P).
`GF2EX` | class `GF2EX` class `GF2EX`: polynomials over `GF2E`; includes routines for modular polynomials arithmetic, modular composition, minimal and characteristic polynomials, and interpolation.
`GF2EXFactoring` | routines for factoring polynomials over `GF2E`; also includes routines for testing for and constructing irreducible polynomials
`HNF` | routines for computing the Hermite Normal Form of a lattice
`LLL` | routines for performing lattice basis reduction, including very fast and robust implementations of the Schnorr-Euchner LLL and Block Korkin Zolotarev reduction algorithm, as well as an integer-only reduction algorithm. Also, there are routines here for computing the kernel and image of an integer matrix, as well as finding integer solutions to linear systems of equations over the integers.
`RR` | class `RR`: arbitrary-precision floating point numbers.
`ZZ` | class `ZZ`: arbitrary length integers; includes routines for GCDs, Jacobi symbols, modular arithmetic, and primality testing; also includes small prime generation routines and in-line routines for single-precision modular arithmetic
`ZZVec` | class `ZZVec`: fixed-length vectors of fixed-length `ZZ`s; less flexible, but more efficient than `vec_ZZ`
`ZZX` | class `ZZX`: polynomials over `ZZ`; includes routines for GCDs, minimal and characteristic polynomials, norms and traces
`ZZXFactoring` | routines for factoring univariate polynomials over `ZZ`
`ZZ_p` | class `ZZ_p`: integers mod `p`
`ZZ_pE` | class `ZZ_pE`: ring/field extension of `ZZ_p`
`ZZ_pEX` | class `ZZ_pEX`: polynomials over `ZZ_pE`; includes routines for modular polynomials arithmetic, modular composition, minimal and characteristic polynomials, and interpolation.
`ZZ_pEXFactoring` | routines for factoring polynomials over `ZZ_pE`; also includes routines for testing for and constructing irreducible polynomials
`ZZ_pX` | class `ZZ_pX`: polynomials over `ZZ_p`; includes routines for modular polynomials arithmetic, modular composition, minimal and characteristic polynomials, and interpolation.
`ZZ_pXFactoring` | routines for factoring polynomials over `ZZ_p`; also includes routines for testing for and constructing irreducible polynomials
`lzz_p` | class `zz_p`: integers mod `p`, where `p` is single-precision
`lzz_pE` | class `zz_pE`: ring/field extension of zz_p
`lzz_pEX` | class `zz_pEX`: polynomials over `zz_pE`; provides the same functionality as class `ZZ_pEX`, but for single-precision `p`
`lzz_pEXFactoring` | routines for factoring polynomials over `zz_pE`; provides the same functionality as class `ZZ_pEX`, but for single-precision `p`
`lzz_pX` | class `zz_pX`: polynomials over `zz_p`; provides the same functionality as class `ZZ_pX`, but for single-precision `p`
`lzz_pXFactoring` | routines for factoring polynomials over `zz_p`; provides the same functionality as class `ZZ_pX`, but for single-precision `p`
`matrix` | template-like macros for dynamic-size 2-dimensional arrays
`mat_GF2` | class `mat_GF2`: matrices over `GF(2)`; includes basic matrix arithmetic operations, including determinant calculation, matrix inversion, solving nonsingular systems of linear equations, and Gaussian elimination
`mat_GF2E` | class `mat_GF2E`: matrices over `GF2E`; includes basic matrix arithmetic operations, including determinant calculation, matrix inversion, solving nonsingular systems of linear equations, and Gaussian elimination
`mat_RR` | class `mat_RR`: matrices over `RR`; includes basic matrix arithmetic operations, including determinant calculation, matrix inversion, and solving nonsingular systems of linear equations.
`mat_ZZ` | class `mat_ZZ`: matrices over `ZZ`; includes basic matrix arithmetic operations, including determinant calculation, matrix inversion, and solving nonsingular systems of linear equations. See also the LLL module for additional routines.
`mat_ZZ_p` | class `mat_ZZ_p`: matrices over `ZZ_p`; includes basic matrix arithmetic operations, including determinant calculation, matrix inversion, solving nonsingular systems of linear equations, and Gaussian elimination
`mat_ZZ_pE` | class `mat_ZZ_pE`: matrices over `ZZ_pE`; includes basic matrix arithmetic operations, including determinant calculation, matrix inversion, solving nonsingular systems of linear equations, and Gaussian elimination
`mat_lzz_p` | class `mat_zz_p`: matrices over `zz_p`; includes basic matrix arithmetic operations, including determinant calculation, matrix inversion, solving nonsingular systems of linear equations, and Gaussian elimination
`mat_lzz_pE` | class `mat_zz_pE`: matrices over `zz_pE`; includes basic matrix arithmetic operations, including determinant calculation, matrix inversion, solving nonsingular systems of linear equations, and Gaussian elimination
`mat_poly_ZZ` | routine for computing the characteristic polynomial of a `mat_ZZ`
`mat_poly_ZZ_p` | routine for computing the characteristic polynomial of a `mat_ZZ_p`
`mat_poly_lzz_p` | routine for computing the characteristic polynomial of a `mat_zz_p`
`pair` | template-like macros for pairs
`quad_float` | class `quad_float`: quadruple-precision floating point numbers.
`tools` | some basic types and utility routines, including the timing function `GetTime()`, and several overloaded versions of `min()` and `max()`
`vector` | template-like macros for dynamic-size vectors
`vec_GF2` | class `vec_GF2`: vectors over `GF(2)`, with arithmetic
`vec_GF2E` | class `vec_GF2E`: vectors over `GF2E`, with arithmetic
`vec_RR` | class `vec_RR`: vectors over `RR`, with arithmetic
`vec_ZZ` | class `vec_ZZ`: vectors over `ZZ`, with arithmetic
`vec_ZZ_p` | class `vec_ZZ_p`: vectors over `ZZ_p`, with arithmetic
`vec_ZZ_pE` | class `vec_ZZ_pE`: vectors over `ZZ_pE`, with arithmetic
`vec_lzz_p` | class `vec_zz_p`: vectors over `zz_p`, with arithmetic
`vec_lzz_pE` | class `vec_zz_pE`: vectors over `zz_pE`, with arithmetic
`version` | macros defining the NTL version number
`xdouble` | class `xdouble`: double-precision floating point numbers with extended exponent range.

### Some other types

In addition to the above, other generic vectors are declared, not explicitly documented elsewhere:

- `vec_GF2XVec`
- `vec_ZZVec`
- `vec_double`
- `vec_long`
- `vec_quad_float`
- `vec_ulong`
- `vec_vec_GF2`
- `vec_vec_GF2E`
- `vec_vec_RR`
- `vec_vec_ZZ`
- `vec_vec_ZZ_p`
- `vec_vec_ZZ_pE`
- `vec_vec_long`
- `vec_vec_lzz_p`
- `vec_vec_lzz_pE`
- `vec_vec_ulong`
- `vec_xdouble`

These decalarations are found in "`.h`" files with corresponding names. No additional functionality is provided.

All of the header files for polynomial classes `ZZ_pX`, `ZZX`, etc., declare generic vectors `vec_ZZ_pX`, `vec_ZZX`, etc.

There are also a number of generic pair classes defined, not explicitly documented elsewhere:

- `pair_GF2EX_long`
- `pair_GF2X_long`
- `pair_ZZX_long`
- `pair_ZZ_pEX_long`
- `pair_ZZ_pX_long`
- `pair_lzz_pEX_long`
- `pair_lzz_pX_long`

These decalarations are found in "`.h`" files with corresponding names. These files also declare corresponding generic vector types `vec_pair_GF2EX_long`, etc. No additional functionality is provided.