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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:
<pre>
(ZZ_pX, ZZ_pX), (ZZ_pX, ZZ_p), (ZZ_pX, long), (ZZ_p, ZZ_pX), (long, ZZ_pX)
</pre>
Each of these functions effectively converts the argument to be promoted
to a <tt>ZZ_pX</tt>.
<p>
Note that when promoting a pair of arguments, at least one
of the arguments must be of the target type.
<p>
I have tried to be very consistent with these promotions so
that one usually won't need to hunt through the documentation.
For a given type, there is a natural, fixed set of types
that promote to it.
Here is the complete list:
<pre>
destination: source
xdouble: double
quad_float: double
RR: double
ZZ: long
ZZ_p: long
ZZ_pX: long, ZZ_p
zz_p: long
ZZ_pX: long, zz_p
ZZX: long, ZZ
GF2: long
GF2X: long, GF2
GF2E: long, GF2
GF2EX: long, GF2, GF2E
ZZ_pE: long, ZZ_p
ZZ_pEX: long, ZZ_p, ZZ_pE
zz_pE: long, zz_p
zz_pEX: long, zz_p, zz_pE
</pre>
<p>
All the promotions are documented, but here
are a few general rules describing the available promotions:
<ul>
<li>
Promotions apply uniformly to both procedural and functional
forms, as well as to the corresponding assignment operator forms.
E.g.,
<pre>
x = x + 2;
add(x, x, 2);
x += 2;
</pre>
<li>
The addition, subtraction, multiplication, equality and comparison
routines always promote both arguments. E.g.,
<pre>
x = 2 + y;
add(x, 2, y);
if (3 > x || y == 5) ...
</pre>
<li>
The assignment operator always promotes the right-hand side.
E.g.,
<pre>
x = 2;
</pre>
<li>
For non-integer, non-polynomial types, the division routine
promotes both arguments.
E.g.,
<pre>
RR x, y, z;
...
x = 1.0/y;
z = y/2.0;
</pre>
For integer or polynomial types, the division routine
promotes the denominator only. E.g.,
<pre>
ZZ x, y;
...
y = x/2;
</pre>
<li>
Matrix by scalar and vector by scalar multiplication promote the scalar.
E.g.,
<pre>
vec_ZZ v, w;
...
v = w*2;
v = 2*w;
v *= 2;
</pre>
<li>
The monomial constructors for polynomials
and the corresponding <tt>SetCoeff</tt> routines
promote the coefficient argument.
E.g.,
<pre>
ZZX f;
f = ZZX(3, 5); // f == 5*X^3
SetCoeff(f, 0, 2); // f == 5*x^3 + 2;
</pre>
<li>
In module <tt>ZZ</tt>, the modular arithmetic routines, as well as
the bit-wise <i>and</i>, <i>or</i>, and <i>xor</i> routines promote their arguments.
There are also several other routines in module <tt>ZZ</tt>
that have both <tt>ZZ</tt> and <tt>long</tt> versions, e.g.,
<tt>NumBits</tt>, <tt>bit</tt>, <tt>weight</tt>.
Check the documentation in <a href="ZZ.txt"><tt>ZZ.txt</tt></a>
for complete details.
</ul>
<p>
<p>
<p>
<h3>
Some Conversion and Promotion Technicalities
</h3>
<p>
<p>
Usually, conversions and promotions are semantically equivalent.
There are three exceptions, however.
<p>
One exception
is conversion of floating point <tt>double</tt> to
<tt>ZZ</tt>.
The safest way to do this is to apply an explicit conversion operator,
and not to rely on promotions.
For example, consider
<pre>
ZZ a; double x;
a = a + x;
</pre>
This is equivialent to
<pre>
a = a + long(x);
</pre>
One could also use an explicit conversion function:
<pre>
a = a + to_ZZ(x);
</pre>
The second version guarantees that there is no loss of precision,
and also guarantees that the floor of <tt>x</tt> is computed.
With the first version, one may lose precision when <tt>x</tt>
is converted to a <tt>long</tt>, and also the direction of truncation
for negative numbers is implementation dependent
(usually truncating towards zero, instead of computing the floor).
<p>
The second exception is conversion of <tt>unsigned int</tt>
or <tt>unsigned long</tt> to <tt>ZZ</tt>.
Again, the safest way to do this is with an explicit conversion operator.
As above, if one relies on promotions, the unsigned integer
will be first converted to a <i>signed</i> <tt>long</tt>, which is most
likely not what was intended.
<p>
The third exception can occur
on 64-bit machines when
converting a signed or unsigned <tt>long</tt> to one of NTL's
extended precision floating-point types (<tt>RR</tt> or <tt>quad_float</tt>).
These types only provide promotions from <tt>double</tt>,
and converting a <tt>long</tt> to a <tt>double</tt> on a 64-bit machine
can lead to a loss of precision.
Again, if one uses the appropriate NTL conversion routine,
no loss of precision will occur.
<p>
Another pitfall too avoid is initialzing <tt>ZZ</tt>s
with integer constants that are too big.
Consider the following:
<pre>
ZZ x;
x = 1234567890123456789012;
</pre>
This integer constant is too big, and this overflow
condition may or may not cause your compiler to give
you a warning or an error.
The easiest way to introduce such large constants into your
program is as follows:
<pre>
ZZ x;
x = to_ZZ("1234567890123456789012");
</pre>
Conversion functions are provided for converting <tt>C</tt> character strings
to the types <tt>ZZ</tt>, <tt>RR</tt>, <tt>quad_float</tt>,
and <tt>xdouble</tt>.
<p>
One should also be careful when converting to <tt>RR</tt>.
All of these conversions round to the current working precision, which is
usually, but not always what one wants.
<p>
<p>
<h3>
Aliasing
</h3>
<p>
An important feature of NTL is that aliasing of input and output
parameters is <i>always</i> allowed. For example, if you
write <tt>mul(x, a, b)</tt>, then <tt>a</tt> or <tt>b</tt>
may alias (have the same address as) <tt>x</tt>
(or any object that <tt>x</tt> contains, e.g., scalar/vector
or scalar/polynomial multiplication).
<p>
<p>
<h3>
Constructors, Destructors, and Memory Management
</h3>
<p>
NTL generally takes care of managing the space occupied by large,
dynamically sized objects, like objects of class <tt>ZZ</tt> or any of
NTL's dynamic vectors.
However, it is helpful to understand a little of what is happening behind the scenes.
<p>
Most classes are implemented as a pointer, and the default constructor
just sets this pointer to 0.
Space is allocated for the object as needed, and when the object's
destructor is called, the space is freed.
Exceptions to this are the "modular" classes <tt>ZZ_p</tt>, <tt>ZZ_pE</tt>, <tt>zz_pE</tt>,
and <tt>GF2E</tt>.
Since, for a given modulus, the sizes of these objects are fixed, the default constructor
allocates the appropriate amount of space.
<p>
Copies are "deep" rather than "shallow".
This means the data itself is copied, and not just a pointer to the data.
If the destination object does not have enough space to hold the source data,
then the space held by the destination object is "grown".
This is done using the <tt>C</tt> routine <tt>realloc()</tt>.
Note, however, that if the source object is smaller than the destination
object, the space held by the destination object is retained.
This strategy usually yields reasonable behaviour;
however, one can take explicit control of the situation if necessary, since
almost all NTL classes have a method <tt>kill()</tt>
which frees all space held by the object, and sets its state to
the default initial state (a value 0 or a zero-length vector).
<p>
The only exception to the above are the special classes <tt>ZZ_pBak</tt>,
<tt>ZZ_pContext</tt>, and the analogous classes for <tt>zz_p</tt>,
<tt>ZZ_pE</tt>, <tt>zz_pE</tt>, and <tt>GF2E</tt>.
These objects are implemented as referenced-counted pointers,
and copies are "shallow".
<p>
While we are discussing initialization, there is one technical point
worth mentioning.
It is safe to declare global objects of any NTL type (except modular types),
as long as one uses only the default constructor.
For example, the global declarations
<pre>
ZZ global_integer;
vec_ZZ_p global_vector;
</pre>
should always work, since their initialization only involves
setting a pointer to 0.
However,
one should avoid initializing global objects with
non-default constructors, and should avoid doing anything that would lead to
non-trivial computations with NTL objects
prior to the beginning of the execution of routine <tt>main()</tt>.
The reasons for this are quite esoteric and can only be appreciated
by a true
<tt>C++</tt> afficianado.
Actually, most such initializations and computations probably will work,
but it is somewhat platform dependant.
<p>
Normal people usually do none of these things, so all of this
should not matter too much.
There is, however, one possible exception to this.
A programmer might want to have a global constant initialized like this:
<pre>
const quad_float Pi = to_quad_float("3.1415926535897932384626433832795029");
</pre>
While this probably will work fine on most platforms,
it may not be an entirely portable construction,
since it will involve a non-trivial computation before
execution of <tt>main()</tt> begins.
A more portable strategy
is to define a function returning a read-only
reference:
<pre>
const quad_float& Pi()
{
static quad_float pi =
to_quad_float("3.1415926535897932384626433832795029");
return pi;
}
</pre>
and then call the function <tt>Pi()</tt> to get a read-only reference
to this constant value:
<pre>
area = Pi()*r*r;
</pre>
The initialization will then take place the first time <tt>Pi()</tt>
is called, which is presumably after <tt>main()</tt> starts,
and so everything should work fine.
This is a very simple and general strategy that most <tt>C++</tt>
experts recommend using whenever the initialization of a non-global
object requires non-trivial computation.
<p>
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