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A Tour of NTL: Programming Interface </title>
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<h1>
<p align=center>
A Tour of NTL: Programming Interface
</p>
</h1>
<p> <hr> <p>
In this section, we give a general overview of the
NTL's programming interface.
<p>
<p>
<h3>
Basic Ring Classes
</h3>
<p>
The basic ring classes are:
<ul>
<li>
<tt>ZZ</tt>: big integers
<li>
<tt>ZZ_p</tt>: big integers modulo <tt>p</tt>
<li>
<tt>zz_p</tt>: integers mod "single precision" <tt>p</tt>
<li>
<tt>GF2</tt>: integers mod 2
<li>
<tt>ZZX</tt>: univariate polynomials over <tt>ZZ</tt>
<li>
<tt>ZZ_pX</tt>: univariate polynomials over <tt>ZZ_p</tt>
<li>
<tt>zz_pX</tt>: univariate polynomials over <tt>zz_p</tt>
<li>
<tt>GF2X</tt>: polynomials over GF2
<li>
<tt>ZZ_pE</tt>: ring/field extension over ZZ_p
<li>
<tt>zz_pE</tt>: ring/field extension over zz_p
<li>
<tt>GF2E</tt>: ring/field extension over GF2
<li>
<tt>ZZ_pEX</tt>: univariate polynomials over <tt>ZZ_pE</tt>
<li>
<tt>zz_pEX</tt>: univariate polynomials over <tt>zz_pE</tt>
<li>
<tt>GF2EX</tt>: univariate polynomials over <tt>GF2E</tt>
</ul>
<p>
All these classes all support basic
arithmetic operators
<pre>
+, -, (unary) -, +=, -=, ++, --,
*, *=, /, /=, %, %=.
</pre>
<p>
However, the operations
<pre>
%, %=
</pre>
only exist for integer and polynomial classes, and
do not exist
for classes
<pre>
ZZ_p, zz_p, GF2, ZZ_pE, zz_pE, GF2E.
</pre>
<p>
The standard equality operators (<tt>==</tt> and <tt>!=</tt>)
are provided for each class.
In addition, the class <tt>ZZ</tt>
supports the usual inequality
operators.
<p>
The integer and polynomial classes also support "shift operators"
for left and right shifting.
For polynomial classes, this means multiplication or division
by a power of <tt>X</tt>.
<p>
<p>
<h3>
Floating Point Classes
</h3>
<p>
In addition to the above ring classes, NTL also provides three
different floating point classes:
<ul>
<li>
<tt>xdouble</tt>: "double precision" floating point with
extended exponent range (for very large numbers);
<li>
<tt>quad_float</tt>: "quasi" quadruple-precision floating point;
<li>
<tt>RR</tt>: aribitrary precision floating point.
</ul>
<p>
<p>
<h3>
Vectors and Matrices
</h3>
<p>
There are also vectors and matrices over
<pre>
ZZ ZZ_p zz_p GF2 ZZ_pE zz_pE GF2E RR
</pre>
which support the usual arithmetic operations.
<p>
<p>
<h3>
Functional and Procedural forms
</h3>
<p>
Generally, for any function defined by NTL, there is
a functional form, and a procedural form.
For example:
<pre>
ZZ x, a, n;
x = InvMod(a, n); // functional form
InvMod(x, a, n); // procedural form
</pre>
<p>
This example illustrates the normal way these two forms differ
syntactically.
However, there are exceptions.
First, if there is a operator that can play the role of the
functional form, that is the notation used:
<pre>
ZZ x, a, b;
x = a + b; // functional form
add(x, a, b); // procedural form
</pre>
Second, if the functional form's name would be ambiguous,
the return type is simply appended to its name:
<pre>
ZZ_p x;
x = random_ZZ_p(); // functional form
random(x); // procedural form
</pre>
Third, there are a number of conversion functions (see below), whose name
in procedural form is <tt>conv</tt>, but whose name in
functioanl form is <tt>to_T</tt>, where <tt>T</tt> is the return type:
<pre>
ZZ x;
double a;
x = to_ZZ(a); // functional form
conv(x, a); // procedural form
</pre>
<p>
The use of the procedural form may be more efficient,
since it will generally avoid the creation of a temporary object
to store its result.
However, it is generally silly to get too worked up about
such efficiencies, and the functional form is usually preferable
because the resulting code is usually easier to understand.
<p>
The above rules converning procedural and functional forms apply
to essentially all of the arithmetic classes supported by NTL,
with the exception of
<tt>xdouble</tt> and <tt>quad_float</tt>.
These two classes only support the functional/operator notation
for arithmetic operations (but do support both forms for conversion).
<p>
<p>
<h3>
Conversions and Promotions
</h3>
<p>
NTL does not provide automatic conversions from, say,
<tt>int</tt> to <tt>ZZ</tt>.
Most <tt>C++</tt> experts consider such automatic conversions
bad form in library design, and I would agree with them.
Some earlier versions of NTL had automatic conversions,
but they caused too much trouble, so I took them out.
Indeed, combining function overloading and automatic conversions
is generally considered by programming language experts
to be a bad idea (but that did not stop
the designers of <tt>C++</tt> from doing it).
It makes it very difficult to figure out which function
ought to be called.
<tt>C++</tt> has an incredibly complex set of rules for doing this;
moreover, these rules have been changing over time,
and no two compilers seem to implement exactly the same
set of rules.
And if a compiler has a hard time doing this, imagine what it
is like for a programmer.
In fact, the rules have become so complicated, that the latest
edition of Stroustrup's <tt>C++</tt> book does not even explain them,
although
earlier verisons did.
Possible explanations:
<em>(a)</em> Stroustrup thinks his readers are
too stupid to understand the rules, or
<em>(b)</em> Stroustrup does not understand the rules, or
<em>(c)</em> the rules are so complicated that Stroustrup finds it embarassing
to talk about them.
<p>
Now it should be more clear why I didn't just implement,
say, the <tt>int</tt> to <tt>ZZ</tt> conversion function
as a <tt>ZZ</tt> constructor taking an argument of type <tt>int</tt>,
instead of calling it <tt>to_ZZ</tt>.
This would have introduced an automatic conversion, which I
wanted to avoid for the reasons explained above.
"OK. But why not make the constructor <tt>explict</tt>?" you ask.
The main reason is that this is a fairly recently introduced
language feature that is not universally available.
And even if it were, what about, say, the <tt>ZZ</tt> to <tt>int</tt>
conversion routine?
How would you name <em>that</em>?
The strategy I chose is simple, consistent, and portable.
<p>
As mentioned above, there are numerous explicit conversion routines,
which come in both functional and procedural forms.
A complete list of these can be found in
<a href="conversions.txt">conversions.txt</a>.
This is the only place these are documented; they do not appear
in the ".txt" files.
<p>
Even though there are no automatic conversions, users
of NTL can still have most of their benefits, while
avoiding their pitfalls.
This is because all of the basic arithmetic operations
(in both their functional and procedural forms),
comparison operators, and assignment are overloaded
to get the effect of automatic "promotions".
For example:
<pre>
ZZ x, a;
x = a + 1;
if (x < 0)
mul(x, 2, a);
else
x = -1;
</pre>
<p>
These promotions are documented in the ".txt" files,
usually using a kind of "short hand" notation.
For example:
<pre>
ZZ operator+(const ZZ& a, const ZZ& b);
// PROMOTIONS: operator + promotes long to ZZ on (a, b).
</pre>
This means that in addition to the declared function, there
are two other functions that are logically equivalent to the following:
<pre>
ZZ operator+(long a, const ZZ& b) { return to_ZZ(a) + b; }
ZZ operator+(const ZZ& a, long b) { return a + to_ZZ(b); }
</pre>
<p>
Note that this is not how NTL actually implements these functions.
It is in generally more efficient to write
<pre>
x = y + 2;
</pre>
than it is to write
<pre>
x = y + to_ZZ(2);
</pre>
The former notation avoids the creation and destruction
of a temporary <tt>ZZ</tt>
object to hold the value 2.
<p>
Also, don't have any inhibitions about writing tests like
<pre>
if (x == 0) ...
</pre>
and assignments like
<pre>
x = 1;
</pre>
These are all optimized, and do not execute significaltly slower
than the "lower level" (and much less natural)
<pre>
if (IsZero(x)) ...
</pre>
and
<pre>
set(x);
</pre>
<p>
Some types have even more promotions.
For example, the type <tt>ZZ_pX</tt> has promotions
from <tt>long</tt> and <tt>ZZ_p</tt>.
Thus, the <tt>add</tt> function for <tt>ZZ_pX</tt> takes the following
argument types:
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