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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>
<h2>
Basic Ring Classes
</h2>
<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>
<h2>
Floating Point Classes
</h2>
<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>
<h2>
Vectors and Matrices
</h2>
<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>
<h2>
Functional and Procedural forms
</h2>
<p>

Generally, for any function defined by NTL, there is 
a functional form, and a procedural form.
For example:

<!-- STARTPLAIN
   ZZ x, a, n;
   x = InvMod(a, n);  // functional form
   InvMod(x, a, n);   // procedural form
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; ZZ x, a, n;<br>
&nbsp;&nbsp; x = InvMod(a, n);&nbsp;&nbsp;<font color="#0000ee"><i>// functional form</i></font><br>
&nbsp;&nbsp; InvMod(x, a, n);&nbsp;&nbsp;&nbsp;<font color="#0000ee"><i>// procedural form</i></font><br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->


<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:

<!-- STARTPLAIN
   ZZ x, a, b;
   x = a + b;    // functional form
   add(x, a, b); // procedural form
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; ZZ x, a, b;<br>
&nbsp;&nbsp; x = a + b;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#0000ee"><i>// functional form</i></font><br>
&nbsp;&nbsp; add(x, a, b);&nbsp;<font color="#0000ee"><i>// procedural form</i></font><br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->


Second, if the functional form's name would be ambiguous,
the return type is simply appended to its name:

<!-- STARTPLAIN
   ZZ_p x;
   x = random_ZZ_p();  // functional form
   random(x);          // procedural form
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; ZZ_p x;<br>
&nbsp;&nbsp; x = random_ZZ_p();&nbsp;&nbsp;<font color="#0000ee"><i>// functional form</i></font><br>
&nbsp;&nbsp; random(x);&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#0000ee"><i>// procedural form</i></font><br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->


Third, there are a number of conversion functions (see below), whose name
in procedural form is <tt>conv</tt>, but whose name in 
functional form is <tt>conv&lt;T&gt;</tt>, where <tt>T</tt> is the return type:

<!-- STARTPLAIN
   ZZ x;  
   double a;

   x = conv<ZZ>(a);  // functional form
   conv(x, a);       // procedural form
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; ZZ x;&nbsp;&nbsp;<br>
&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>double</b></font>&nbsp;a;<br>
<br>
&nbsp;&nbsp; x = conv&lt;ZZ&gt;(a);&nbsp;&nbsp;<font color="#0000ee"><i>// functional form</i></font><br>
&nbsp;&nbsp; conv(x, a);&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#0000ee"><i>// procedural form</i></font><br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->




<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 governing 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>
<h2>
Conversions and Promotions
</h2>
<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 other ".txt" files.

<p>
It is worth mentioning here, however, that generic conversion operators
are provided for vectors and matrices, which act component-wise.
For example, since there is a conversion from <tt>ZZ</tt> to <tt>RR</tt>,
there is automatically a conversion from 
<tt>Vec&lt;ZZ&gt;</tt> to <tt>Vec&lt;RR&gt</tt>.





<p>

Even though there are no implicit conversions, users
of NTL can still have most of their benefits.
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:

<!-- STARTPLAIN
   ZZ x, a;

   x = a + 1;
   if (x < 0) 
      mul(x, 2, a);
   else
      x = -1;
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; ZZ x, a;<br>
<br>
&nbsp;&nbsp; x = a +&nbsp;<font color="#ff8c00">1</font>;<br>
&nbsp;&nbsp;&nbsp;<font color="#b03060"><b>if</b></font>&nbsp;(x &lt;&nbsp;<font color="#ff8c00">0</font>)&nbsp;<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;mul(x,&nbsp;<font color="#ff8c00">2</font>, a);<br>
&nbsp;&nbsp;&nbsp;<font color="#b03060"><b>else</b></font><br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;x = -<font color="#ff8c00">1</font>;<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->


<p>

These promotions are documented in the ".txt" files, 
usually using a kind of "short hand" notation.
For example:

<!-- STARTPLAIN
ZZ operator+(const ZZ& a, const ZZ& b);

// PROMOTIONS: operator + promotes long to ZZ on (a, b).
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
ZZ operator+(<font color="#008b00"><b>const</b></font>&nbsp;ZZ&amp; a,&nbsp;<font color="#008b00"><b>const</b></font>&nbsp;ZZ&amp; b);<br>
<br>
<font color="#0000ee"><i>// PROMOTIONS: operator + promotes long to ZZ on (a, b).</i></font><br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->


This means that in addition to the declared function, there
are two other functions that are logically equivalent to the following:
<!-- STARTPLAIN
ZZ operator+(long a, const ZZ& b) { return ZZ(a) + b; }
ZZ operator+(const ZZ& a, long b) { return a + ZZ(b); }
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
ZZ operator+(<font color="#008b00"><b>long</b></font>&nbsp;a,&nbsp;<font color="#008b00"><b>const</b></font>&nbsp;ZZ&amp; b) {&nbsp;<font color="#b03060"><b>return</b></font>&nbsp;ZZ(a) + b; }<br>
ZZ operator+(<font color="#008b00"><b>const</b></font>&nbsp;ZZ&amp; a,&nbsp;<font color="#008b00"><b>long</b></font>&nbsp;b) {&nbsp;<font color="#b03060"><b>return</b></font>&nbsp;a + ZZ(b); }<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->


<p>
Note that this is not how NTL actually implements these functions.
It is in generally more efficient to write
<!-- STARTPLAIN
   x = y + 2;
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; x = y +&nbsp;<font color="#ff8c00">2</font>;<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

than it is to write
<!-- STARTPLAIN
   x = y + ZZ(2);
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; x = y + ZZ(<font color="#ff8c00">2</font>);<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

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
<!-- STARTPLAIN
   if (x == 0) ...
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp;&nbsp;<font color="#b03060"><b>if</b></font>&nbsp;(x ==&nbsp;<font color="#ff8c00">0</font>) ...<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

and assignments like
<!-- STARTPLAIN
   x = 1; 
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; x =&nbsp;<font color="#ff8c00">1</font>;&nbsp;<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

These are all optimized, and  do not execute significaltly slower
than the "lower level"  (and much less natural) 
<!-- STARTPLAIN
   if (IsZero(x)) ...
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp;&nbsp;<font color="#b03060"><b>if</b></font>&nbsp;(IsZero(x)) ...<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

and
<!-- STARTPLAIN
   set(x);
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; set(x);<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->


<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:
<!-- STARTPLAIN
   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
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; destination&nbsp;&nbsp;source<br>
&nbsp;&nbsp;&nbsp;<br>
&nbsp;&nbsp; xdouble&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>double</b></font><br>
&nbsp;&nbsp; quad_float&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>double</b></font><br>
&nbsp;&nbsp; RR&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>double</b></font><br>
&nbsp;&nbsp; ZZ&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>long</b></font><br>
&nbsp;&nbsp; ZZ_p&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>long</b></font><br>
&nbsp;&nbsp; ZZ_pX&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>long</b></font>, ZZ_p<br>
&nbsp;&nbsp; zz_p&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>long</b></font><br>
&nbsp;&nbsp; zz_pX&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>long</b></font>, zz_p<br>
&nbsp;&nbsp; ZZX&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>long</b></font>, ZZ<br>
&nbsp;&nbsp; GF2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>long</b></font><br>
&nbsp;&nbsp; GF2X&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>long</b></font>, GF2<br>
&nbsp;&nbsp; GF2E&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>long</b></font>, GF2<br>
&nbsp;&nbsp; GF2EX&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>long</b></font>, GF2, GF2E<br>
&nbsp;&nbsp; ZZ_pE&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>long</b></font>, ZZ_p<br>
&nbsp;&nbsp; ZZ_pEX&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>long</b></font>, ZZ_p, ZZ_pE<br>
&nbsp;&nbsp; zz_pE&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>long</b></font>, zz_p<br>
&nbsp;&nbsp; zz_pEX&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>long</b></font>, zz_p, zz_pE<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->


<p>
All the promotions are documented, but here
are a few general rules describing the available promotions:

<ul>

<li>
All classes provide explicit constructors for promoted types.
For example,
<!-- STARTPLAIN
   ZZ w = ZZ(1);
   ZZ x(1);  // allowed
   ZZ y{1};  // allowed in C++11
   ZZ z = 1; // not allowed
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; ZZ w = ZZ(<font color="#ff8c00">1</font>);<br>
&nbsp;&nbsp; ZZ x(<font color="#ff8c00">1</font>);&nbsp;&nbsp;<font color="#0000ee"><i>// allowed</i></font><br>
&nbsp;&nbsp; ZZ y{<font color="#ff8c00">1</font>};&nbsp;&nbsp;<font color="#0000ee"><i>// allowed in C++11</i></font><br>
&nbsp;&nbsp; ZZ z =&nbsp;<font color="#ff8c00">1</font>;&nbsp;<font color="#0000ee"><i>// not allowed</i></font><br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->


<li>
Promotions apply uniformly to both procedural and functional 
forms, as well as to the corresponding assignment operator forms.
E.g.,
<!-- STARTPLAIN
   x = x + 2;
   add(x, x, 2);
   x += 2;
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; x = x +&nbsp;<font color="#ff8c00">2</font>;<br>
&nbsp;&nbsp; add(x, x,&nbsp;<font color="#ff8c00">2</font>);<br>
&nbsp;&nbsp; x +=&nbsp;<font color="#ff8c00">2</font>;<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->


<li>
The addition, subtraction, multiplication, equality and comparison
routines always promote both arguments.  E.g.,
<!-- STARTPLAIN
   x = 2 + y;
   add(x, 2, y);
   if (3 > x || y == 5) ...
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; x =&nbsp;<font color="#ff8c00">2</font>&nbsp;+ y;<br>
&nbsp;&nbsp; add(x,&nbsp;<font color="#ff8c00">2</font>, y);<br>
&nbsp;&nbsp;&nbsp;<font color="#b03060"><b>if</b></font>&nbsp;(<font color="#ff8c00">3</font>&nbsp;&gt; x || y ==&nbsp;<font color="#ff8c00">5</font>) ...<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->


<li>
The assignment operator always promotes the right-hand side.
E.g.,
<!-- STARTPLAIN
   x = 2;
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; x =&nbsp;<font color="#ff8c00">2</font>;<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->


<li>
For non-integer,  non-polynomial types, the division routine
promotes both arguments.
E.g.,
<!-- STARTPLAIN
   RR x, y, z;
      ...
   x = 1.0/y;
   z = y/2.0;
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; RR x, y, z;<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;...<br>
&nbsp;&nbsp; x =&nbsp;<font color="#ff8c00">1.0</font>/y;<br>
&nbsp;&nbsp; z = y/<font color="#ff8c00">2.0</font>;<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->


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.,
<!-- STARTPLAIN
   Vec<ZZ> v, w;
      ...
   v = w*2;
   v = 2*w;
   v *= 2;
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; Vec&lt;ZZ&gt; v, w;<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;...<br>
&nbsp;&nbsp; v = w*<font color="#ff8c00">2</font>;<br>
&nbsp;&nbsp; v =&nbsp;<font color="#ff8c00">2</font>*w;<br>
&nbsp;&nbsp; v *=&nbsp;<font color="#ff8c00">2</font>;<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->



<li>
The monomial constructors for polynomials
and the corresponding <tt>SetCoeff</tt> routines 
promote the coefficient argument.
E.g.,
<!-- STARTPLAIN
   ZZX f;
   f = ZZX(INIT_MONO, 3, 5);  // f == 5*X^3
   SetCoeff(f, 0, 2);  // f == 5*x^3 + 2;
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; ZZX f;<br>
&nbsp;&nbsp; f = ZZX(INIT_MONO,&nbsp;<font color="#ff8c00">3</font>,&nbsp;<font color="#ff8c00">5</font>);&nbsp;&nbsp;<font color="#0000ee"><i>// f == 5*X^3</i></font><br>
&nbsp;&nbsp; SetCoeff(f,&nbsp;<font color="#ff8c00">0</font>,&nbsp;<font color="#ff8c00">2</font>);&nbsp;&nbsp;<font color="#0000ee"><i>// f == 5*x^3 + 2;</i></font><br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->


<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.cpp.html"><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
<!-- STARTPLAIN
   ZZ a; double x;

   a = a + x;
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; ZZ a;&nbsp;<font color="#008b00"><b>double</b></font>&nbsp;x;<br>
<br>
&nbsp;&nbsp; a = a + x;<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

This is equivialent to
<!-- STARTPLAIN
   a = a + long(x);
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; a = a +&nbsp;<font color="#008b00"><b>long</b></font>(x);<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

and to 
<!-- STARTPLAIN
   a = a + ZZ(x);
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; a = a + ZZ(x);<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

One could also use an explicit conversion function:
<!-- STARTPLAIN
   a = a + conv<ZZ>(x);
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; a = a + conv&lt;ZZ&gt;(x);<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

This last 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:
<!-- STARTPLAIN
   ZZ x;
   x = 1234567890123456789012;
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; ZZ x;<br>
&nbsp;&nbsp; x =&nbsp;<font color="#ff8c00">1234567890123456789012</font>;<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

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:
<!-- STARTPLAIN
   ZZ x;
   x = conv<ZZ>("1234567890123456789012");
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; ZZ x;<br>
&nbsp;&nbsp; x = conv&lt;ZZ&gt;(<font color="#4a708b">&quot;1234567890123456789012&quot;</font>);<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

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>
<h2>
Input and Output
</h2>
<p>
NTL provides input and output operators for all
types, using the usual conventions for input and output streams.
If an input error occurs, the "fail bit" of the input stream
is set, and the input variable remains unchanged.
<p>
Although conversions from <tt>C</tt>-style character strings
to the types <tt>ZZ</tt>, <tt>xdouble</tt>, <tt>quad_float</tt>,
and <tt>RR</tt> are provided, one can always read and write
to <tt>C++</tt> character streams using the <tt>stringstream</tt>
class from the standard library, in conjunction with the input
and output operators provided by NTL.

<p>
<p>
<h2>
Aliasing
</h2>
<p>

An important feature of NTL is that aliasing of input and output
parameters is generally 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>
One exception to this rule:
the generic conversions provided for vectors and
matrices assume that their inputs do not alias their outputs.


<p>
<p>
<h2>
Constructors, Destructors, and Memory Management
</h2>
<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>
Almost all 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.

<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 is the class
<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 
as long as one uses only the default constructor.
For example, the global declarations
<!-- STARTPLAIN
   ZZ global_integer;
   Vec<ZZ_p> global_vector;
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; ZZ global_integer;<br>
&nbsp;&nbsp; Vec&lt;ZZ_p&gt; global_vector;<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

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:
<!-- STARTPLAIN
   const quad_float Pi = conv<quad_float>("3.1415926535897932384626433832795029");
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>const</b></font>&nbsp;quad_float Pi = conv&lt;quad_float&gt;(<font color="#4a708b">&quot;3.1415926535897932384626433832795029&quot;</font>);<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

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:
<!-- STARTPLAIN
   const quad_float& Pi()
   {
      static quad_float pi = 
         conv<quad_float>("3.1415926535897932384626433832795029");
      return pi;
   }
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>const</b></font>&nbsp;quad_float&amp; Pi()<br>
&nbsp;&nbsp; {<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#008b00"><b>static</b></font>&nbsp;quad_float pi =&nbsp;<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; conv&lt;quad_float&gt;(<font color="#4a708b">&quot;3.1415926535897932384626433832795029&quot;</font>);<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#b03060"><b>return</b></font>&nbsp;pi;<br>
&nbsp;&nbsp; }<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

and then call the function <tt>Pi()</tt> to get a read-only reference
to this constant value:
<!-- STARTPLAIN
   area = Pi()*r*r;
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; area = Pi()*r*r;<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

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>
<p>
<h2>
Residue class rings and modulus switching
</h2>
<p>

NTL provides a number of classes to represent residue class rings:
<pre>
   ZZ_p, zz_p, GF2, ZZ_pE, lzz_pE, GF2E.
</pre>
For each such class, except <tt>GF2</tt>, there is a global, current modulus.

<p>
We focus on the class <tt>ZZ_p</tt>, but similar comments apply to the other
residue class types.
For example, for <tt>ZZ_p</tt>, you can set the current modulus to <tt>p</tt>
as follows:
<!-- STARTPLAIN
   ZZ_p::init(p);
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; ZZ_p::init(p);<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

The current modulus <i>must</i> be initialized before any operations
on <tt>ZZ_p</tt>'s are performed.  The modulus may be changed, and a mechanism is provided
for saving and restoring a modulus.

<p>
Here is what you do to save the current modulus, temporarily
set it to p, and automatically restore it:

<!-- STARTPLAIN
   { 
      ZZ_pPush push(p); 

      ...

   }
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000">
<font face="monospace">
&nbsp;&nbsp; {&nbsp;<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;ZZ_pPush push(p);&nbsp;<br>
<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;...<br>
<br>
&nbsp;&nbsp; }<br>
</font>
</font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->


The constructor for <tt>push</tt> will save the current modulus, and install <tt>p</tt> as the
current modulus.  The destructor for <tt>push</tt> will restore the old modulus when the
scope enclosing it exits.  This is the so-called RAII (resource acquisition is
initialization) paradigm.

<p>
You could also do the following:

<!-- STARTPLAIN
   {
      ZZ_pPush push; // just backup current modulus

        ...

      ZZ_p::init(p1); // install p1 

        ...

      ZZ_p::init(p2); // install p2

      // reinstall original modulus at close of scope
   }
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000"><font face="monospace">
&nbsp;&nbsp; {<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;ZZ_pPush push; <font color="#0000ed"><i>// just backup current modulus</i></font><br>
<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;...<br>
<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;ZZ_p::init(p1); <font color="#0000ed"><i>// install p1 </i></font><br>
<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;...<br>
<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;ZZ_p::init(p2); <font color="#0000ed"><i>// install p2</i></font><br>
<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#0000ed"><i>// reinstall original modulus at close of scope</i></font><br>
&nbsp;&nbsp; }<br>
</font></font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->


<p>
<b>Warning:</b> <tt>C++</tt> syntax can be rather unfriendly sometimes.
When using RAII objects like <tt>ZZ_pPush</tt>, watch out for
the following errors:
<!-- STARTPLAIN
   ZZ_pPush push();  // ERROR: local declaration of a function!!
   ZZ_pPush(p);      // ERROR: temporary RAII-object created and 
                     //        immediately destroyed!!
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000"><font face="monospace">
&nbsp;&nbsp; ZZ_pPush push();&nbsp;&nbsp;<font color="#0000ed"><i>// ERROR: local declaration of a function!!</i></font><br>
&nbsp;&nbsp; ZZ_pPush(p);&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font color="#0000ed"><i>// ERROR: temporary RAII-object created and </i></font><br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <font color="#0000ed"><i>//&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;immediately destroyed!!</i></font><br>
</font></font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

Unfortunately, most compilers do not issue any warnings
in these situations.
I have fallen into both traps myself.
      
<p>
The <tt>ZZ_pPush</tt> interface is good for implementing simple stack-like
"context switching".  For more general context switching,
use the class <tt>ZZ_pContext</tt>:
<!-- STARTPLAIN
   ZZ_p::init(p);     // set current modulus to p

      ...

   ZZ_pContext context;
   context.save();    // save the current modulus p

      ...

   ZZ_p::init(q);     // set current modulus to q

      ...
   
   context.restore(); // restore p as the current modulus
ENDPLAIN -->
<!-- STARTPRETTY {{{ -->
<p><p><table cellPadding=10px><tr><td><font color="#000000"><font face="monospace">
&nbsp;&nbsp; ZZ_p::init(p);&nbsp;&nbsp;&nbsp;&nbsp; <font color="#0000ed"><i>// set current modulus to p</i></font><br>
<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;...<br>
<br>
&nbsp;&nbsp; ZZ_pContext context;<br>
&nbsp;&nbsp; context.save();&nbsp;&nbsp;&nbsp;&nbsp;<font color="#0000ed"><i>// save the current modulus p</i></font><br>
<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;...<br>
<br>
&nbsp;&nbsp; ZZ_p::init(q);&nbsp;&nbsp;&nbsp;&nbsp; <font color="#0000ed"><i>// set current modulus to q</i></font><br>
<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;...<br>
&nbsp;&nbsp; <br>
&nbsp;&nbsp; context.restore(); <font color="#0000ed"><i>// restore p as the current modulus</i></font><br>
</font></font></td></tr></table><p><p>
<!-- }}} ENDPRETTY -->

Note that <tt>ZZ_pContext</tt>'s are essentially "smart pointers",
and they may be copied.
Generally speaking, saving, restoring, and copying <tt>ZZ_pContext</tt>'s
are very cheap operations.
Likewise, saving and restoring contexts using <tt>ZZ_pPush</tt>
objects are very cheap operations.


<p>
It is critical that <tt>ZZ_p</tt> objects created under one <tt>ZZ_p</tt> modulus are not used in
any non-trivial way "out of context", i.e., under a different (or undefined)
<tt>ZZ_p</tt> modulus.  However, for ease-of-use, some operations may be safely
performed out of context.  These safe operations include: the default and copy
constructor, the destructor, and the assignment operator.  In addition it is
generally safe to read any <tt>ZZ_p</tt> object out of context (i.e., printing it out, or
fetching its underlying representive using the rep() function).

<p>
Any unsafe uses out of context are not in general checked, and may 
lead to unpredictable behavior.



<p>
The implementations of <tt>Vec&lt;ZZ_p&gt;</tt>, <tt>Vec&lt;GF2E&gt;</tt>, and <tt>Vec&lt;GF2&gt;</tt> 
are specialized to manage memory more
efficiently than in the default implementation of <tt>Vec&lt;T&gt;</tt>:
<ul>
<p><li>
Contiguous elements in a <tt>Vec&lt;ZZ_p&gt;</tt> are allocated in a contiguous region of
memory.  This reduces the number of calls to the memory allocator, and  
leads to greater locality of reference.  A consequence of
this implementation is that any calls to SetLength on a <tt>Vec&lt;ZZ_p&gt;</tt> object will
need to use information about the current modulus, and so such calls should
only be done "in context".  That said, it is still safe to construct a
<tt>Vec&lt;ZZ_p&gt;</tt> using the default or copy contructor, and to assign or append one
<tt>Vec&lt;ZZ_p&gt;</tt> to another "out of context".

<p><li>
The same strategy is used for <tt>Vec&lt;GF2E&gt;</tt>'s.

<p><li>
In any case, the above restrictions adhere to the general rules
for safely using residue class ring objects "out of context".

<p><li>
<tt>Vec&lt;GF2&gt;</tt>'s are implemented by packing coefficients (which are just bits)
into words.  A mechanism is provided to make indexing these vectors
behave like normal vectors, via a class the mimics ordinary references
to <tt>GF2</tt>'s.  
</ul>


<p>
<p>
<h2>
<a name="except">Error Handling and Exceptions</a>
</h2>
<p>
Prior to version 8.0 of NTL, errors were dealt with in a simlple way:
print an error message and abort.

As of version 8.0, NTL provides error handling with exceptions.
To use this feature, you will need to configure NTL with the
<tt>NTL_EXCEPTIONS</tt> flag turned on.
You will also need a <tt>C++11</tt> compiler.

<p>
The exceptions thrown by NTL are either a <tt>std::bad_alloc</tt>
exception (in case of memory allocation error),
or a class (defined in namespace NTL) 
derived from <tt>std::runtime_error</tt>:
<ul>
<li> <tt>ErrorObject</tt> &#8594; <tt>std::runtime_error</tt>
<ul> <li> base class  </ul>
<li> <tt>LogicErrorObject</tt> &#8594; <tt>ErrorObject</tt>
<ul> <li> used to indicate a logic error, such as incorrect
          function parameters, index out of range, etc. </ul>
<li> <tt>ArithmeticErrorObject</tt> &#8594; <tt>ErrorObject</tt>
<ul> <li> used to indicate an arithmetic error, such as divide by zero </ul>
<li> <tt>ResourceErrorObject</tt> &#8594; <tt>ErrorObject</tt>
<ul> <li> used to indicate an overflow error (e.g., when a number cannot be stored as a <tt>long</tt>) </ul>
<li> <tt>FileErrorObject</tt> &#8594; <tt>ErrorObject</tt>
<ul> <li> used to indicate a problem opening or closing a file</ul>
<li> <tt>InputErrorObject</tt> &#8594; <tt>ErrorObject</tt>
<ul> <li> used to indicate a problem reading from a stream</ul>
</ul>

<p>
All of these error objects override the <tt>what()</tt>
method of <tt>std::exception</tt> with an appropriate
error message.

<p>
There is also a special exception class <tt>InvModErrorObject</tt>,
which is derived from <tt>ArithmeticErrorObject</tt>,
and is thrown when a modular inverse computation over <tt>ZZ</tt> fails
(either directly, or indirectly through <tt>PowerMod</tt>
computation, or via an inverse computation in <tt>ZZ_p</tt>).
The  <tt>InvModErrorObject</tt> provides two methods, 
<tt>get_a()</tt> and <tt>get_n()</tt>, which provide read-only
references to the offending objects <tt>a</tt> and <tt>n</tt>
(so <tt>GCD(a, n) != 1</tt>).

<p>
The generic class <tt>ErrorObject</tt> is not thrown directly
by any NTL routines, except for the legacy function <tt>Error</tt>,
which is no longer called by any NTL routines.
New functions 
<pre>
   MemoryError, LogicError, ArithmeticError, ResourceError, FileError, InputError
</pre>
are used to throw exceptions derived from <tt>ErrorObject</tt>.

<p>
<b>Issues with GMP:</b>
As of this writing (December 2014),
GMP itself provides only the very crude print-message-then-abort
error handling.
Note that NTL only uses low-level GMP routines (the <tt>mpn</tt>-level
routines),
and these routines should only abort if they cannot allocate space
for temporary big integers.
So this should only be an issue of you are working with some
very large integers.
The GMP developers are working on improving their error handling.
When that happens, NTL will inherit these improvements.
If you really need proper error handling, and are willing to pay
a certain performance penalty, then you should configure
and build NTL <i>without</i> GMP.

<p>
<b>Issues with gf2x:</b>
Similar comments apply to NTL builds that use the <tt>gf2x</tt>
library.
I have no idea how they handle errors.
If you really need proper error handling, you should not use <tt>gf2x</tt>.


<p>
<p>
<b>Exception safety:</b>
I have tried to carefully document exception safety characterstics
for just a few, critical, low-level classes: 
vectors and matrices 
(<a href="vector.cpp.html">vector.txt</a> and
<a href="matrix.cpp.html">matrix.txt</a>),
smart pointer classes (<a href="SmartPtr.cpp.html">SmartPtr.txt</a>),
thread-safe lazy initialization classes
(<a href="Lazy.cpp.html">Lazy.txt</a> and 
<a href="LazyTable.cpp.html">LazyTable.txt</a>).
Otherwise, it is only safe to assume that NTL functions
provide a weak exception-safety guarantee:
if an exception is thrown, the stack unwinding process will
will not leak any resources and will leave all modified objects
in a reasonable state: at least, such objects may be safely 
destroyed, and may also be assigned to or reset; 
however, they may not necessarily
be safely used as inputs to other functions.
When stronger exception safety is required, you can always 
compute results into dynamically allocated objects
pointed to by "smart pointers",
and then move or swap these pointers into place after all computations
have succeeded.
<p>
As NTL provides <tt>swap</tt> functions for all its major classes,
and as <tt>swap</tt> functions have evolved to play a critical role
in writing exception-safe code, they deserve a special mention here:
<ul>
<p><li> 
For all classes except <tt>ZZ</tt>, <tt>ZZ_p</tt>, <tt>GF2X</tt>,
<tt>GF2E</tt>, and <tt>Vec&lt;T&gt;</tt>, the swap function is guaranteed to not throw
any exceptions.
<p><li>
For <tt>ZZ</tt> objects that are not elements of a <tt>ZZVec</tt>,
<tt>ZZ_p</tt> objects that are not elements of a <tt>Vec&lt;ZZ_p&gt;</tt>,
<tt>GF2X</tt> objects that are not elements of a <tt>GF2XVec</tt>,
and
<tt>GF2E</tt> objects that are not elements of a <tt>Vec&lt;GF2E&gt;</tt>,
the swap function is guaranteed to not throw any exceptions.
<p><li>
For  <tt>Vec&lt;T&gt;</tt> objects whose length has not been fixed,
the swap function is guaranteed to not throw any exceptions.
<p><li>
For the remaining cases, the swap function provides a strong exception-safety
guarantee (the operation either succeeds, or throws and leaves data unchanged).
</ul>
These rules are unfortunatley a bit complicated, due to NTL's historical
legacy, and to its special memory management of
<tt>ZZVec</tt>,
<tt>Vec&lt;ZZ_p&gt;</tt>,
<tt>GF2XVec</tt>,
and
<tt>Vec&lt;GF2E&gt;</tt>
types.







<p>

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