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<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i>MODULE: zz_pE</i></font><br>
<br>
<font color="#0000ed"><i>SUMMARY:</i></font><br>
<br>
<font color="#0000ed"><i>The class zz_pE is used to represent polynomials in Z_p[X] modulo a</i></font><br>
<font color="#0000ed"><i>polynomial P. The modulus P may be any polynomial with deg(P) > 0,</i></font><br>
<font color="#0000ed"><i>not necessarily irreducible. The modulus p defining Z_p need</i></font><br>
<font color="#0000ed"><i>not be prime either.</i></font><br>
<br>
<font color="#0000ed"><i>Objects of the class zz_pE are represented as a zz_pX of degree < deg(P).</i></font><br>
<br>
<font color="#0000ed"><i>An executing program maintains a "current modulus", which is set to P</i></font><br>
<font color="#0000ed"><i>using zz_pE::init(P). The current modulus for zz_pE (as well as for zz_p)</i></font><br>
<font color="#0000ed"><i>*must* be initialized before an operations on zz_pE's are performed.</i></font><br>
<br>
<font color="#0000ed"><i>The modulus may be changed, and a mechanism is provided for saving and</i></font><br>
<font color="#0000ed"><i>restoring a modulus (see classes zz_pEPush and zz_pEContext below).</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<font color="#1773cc">#include </font><font color="#4a6f8b"><NTL/lzz_pX.h></font><br>
<br>
<font color="#008b00"><b>class</b></font> zz_pE {<br>
<font color="#b02f60"><b>public</b></font>:<br>
<br>
zz_pE(); <font color="#0000ed"><i>// initial value 0</i></font><br>
<br>
zz_pE(<font color="#008b00"><b>const</b></font> zz_pE& a); <font color="#0000ed"><i>// copy constructor</i></font><br>
<font color="#008b00"><b>explicit</b></font> zz_pE(<font color="#008b00"><b>const</b></font> zz_p& a); <font color="#0000ed"><i>// promotion </i></font><br>
<font color="#008b00"><b>explicit</b></font> zz_pE(<font color="#008b00"><b>long</b></font> a); <font color="#0000ed"><i>// promotion </i></font><br>
<br>
zz_pE& <font color="#b02f60"><b>operator</b></font>=(<font color="#008b00"><b>const</b></font> zz_pE& a); <font color="#0000ed"><i>// assignment</i></font><br>
zz_pE& <font color="#b02f60"><b>operator</b></font>=(<font color="#008b00"><b>const</b></font> zz_p& a); <font color="#0000ed"><i>// assignment</i></font><br>
zz_pE& <font color="#b02f60"><b>operator</b></font>=(<font color="#008b00"><b>long</b></font> a); <font color="#0000ed"><i>// assignment</i></font><br>
<br>
~zz_pE(); <font color="#0000ed"><i>// destructor</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> init(<font color="#008b00"><b>const</b></font> zz_pX& P);<br>
<font color="#0000ed"><i>// zz_pE::init(P) initializes the current modulus to P;</i></font><br>
<font color="#0000ed"><i>// required: deg(P) >= 1.</i></font><br>
<br>
<font color="#008b00"><b>static</b></font> <font color="#008b00"><b>const</b></font> zz_pXModulus& modulus();<br>
<font color="#0000ed"><i>// zz_pE::modulus() yields read-only reference to the current modulus </i></font><br>
<br>
<font color="#008b00"><b>static</b></font> <font color="#008b00"><b>long</b></font> degree();<br>
<font color="#0000ed"><i>// zz_pE::degree() returns deg(P)</i></font><br>
<br>
<font color="#0000ed"><i>// typedefs to aid generic programming</i></font><br>
<font color="#008b00"><b>typedef</b></font> zz_pX rep_type;<br>
<font color="#008b00"><b>typedef</b></font> zz_pEContext context_type;<br>
<font color="#008b00"><b>typedef</b></font> zz_pEBak bak_type;<br>
<font color="#008b00"><b>typedef</b></font> zz_pEPush push_type;<br>
<font color="#008b00"><b>typedef</b></font> zz_pEX poly_type;<br>
<br>
};<br>
<br>
<br>
<font color="#008b00"><b>const</b></font> zz_pX& rep(<font color="#008b00"><b>const</b></font> zz_pE& a); <font color="#0000ed"><i>// read-only access to representation of a</i></font><br>
<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Comparison</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<font color="#008b00"><b>long</b></font> <font color="#b02f60"><b>operator</b></font>==(<font color="#008b00"><b>const</b></font> zz_pE& a, <font color="#008b00"><b>const</b></font> zz_pE& b);<br>
<font color="#008b00"><b>long</b></font> <font color="#b02f60"><b>operator</b></font>!=(<font color="#008b00"><b>const</b></font> zz_pE& a, <font color="#008b00"><b>const</b></font> zz_pE& b);<br>
<br>
<font color="#008b00"><b>long</b></font> IsZero(<font color="#008b00"><b>const</b></font> zz_pE& a); <font color="#0000ed"><i>// test for 0</i></font><br>
<font color="#008b00"><b>long</b></font> IsOne(<font color="#008b00"><b>const</b></font> zz_pE& a); <font color="#0000ed"><i>// test for 1</i></font><br>
<br>
<font color="#0000ed"><i>// PROMOTIONS: ==, != promote {long, zz_p} to zz_pE on (a, b).</i></font><br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Addition </i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<font color="#0000ed"><i>// operator notation:</i></font><br>
<br>
zz_pE <font color="#b02f60"><b>operator</b></font>+(<font color="#008b00"><b>const</b></font> zz_pE& a, <font color="#008b00"><b>const</b></font> zz_pE& b);<br>
<br>
zz_pE <font color="#b02f60"><b>operator</b></font>-(<font color="#008b00"><b>const</b></font> zz_pE& a, <font color="#008b00"><b>const</b></font> zz_pE& b);<br>
zz_pE <font color="#b02f60"><b>operator</b></font>-(<font color="#008b00"><b>const</b></font> zz_pE& a);<br>
<br>
zz_pE& <font color="#b02f60"><b>operator</b></font>+=(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_pE& a);<br>
zz_pE& <font color="#b02f60"><b>operator</b></font>+=(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_p& a);<br>
zz_pE& <font color="#b02f60"><b>operator</b></font>+=(zz_pE& x, <font color="#008b00"><b>long</b></font> a);<br>
<br>
zz_pE& <font color="#b02f60"><b>operator</b></font>++(zz_pE& x); <font color="#0000ed"><i>// prefix</i></font><br>
<font color="#008b00"><b>void</b></font> <font color="#b02f60"><b>operator</b></font>++(zz_pE& x, <font color="#008b00"><b>int</b></font>); <font color="#0000ed"><i>// postfix</i></font><br>
<br>
zz_pE& <font color="#b02f60"><b>operator</b></font>-=(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_pE& a);<br>
zz_pE& <font color="#b02f60"><b>operator</b></font>-=(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_p& a);<br>
zz_pE& <font color="#b02f60"><b>operator</b></font>-=(zz_pE& x, <font color="#008b00"><b>long</b></font> a);<br>
<br>
zz_pE& <font color="#b02f60"><b>operator</b></font>--(zz_pE& x); <font color="#0000ed"><i>// prefix</i></font><br>
<font color="#008b00"><b>void</b></font> <font color="#b02f60"><b>operator</b></font>--(zz_pE& x, <font color="#008b00"><b>int</b></font>); <font color="#0000ed"><i>// postfix</i></font><br>
<br>
<font color="#0000ed"><i>// procedural versions:</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> add(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_pE& a, <font color="#008b00"><b>const</b></font> zz_pE& b); <font color="#0000ed"><i>// x = a + b</i></font><br>
<font color="#008b00"><b>void</b></font> sub(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_pE& a, <font color="#008b00"><b>const</b></font> zz_pE& b); <font color="#0000ed"><i>// x = a - b </i></font><br>
<font color="#008b00"><b>void</b></font> negate(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_pE& a); <font color="#0000ed"><i>// x = - a </i></font><br>
<br>
<font color="#0000ed"><i>// PROMOTIONS: +, -, add, sub promote {long, zz_p} to zz_pE on (a, b).</i></font><br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Multiplication </i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<font color="#0000ed"><i>// operator notation:</i></font><br>
<br>
zz_pE <font color="#b02f60"><b>operator</b></font>*(<font color="#008b00"><b>const</b></font> zz_pE& a, <font color="#008b00"><b>const</b></font> zz_pE& b);<br>
<br>
zz_pE& <font color="#b02f60"><b>operator</b></font>*=(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_pE& a);<br>
zz_pE& <font color="#b02f60"><b>operator</b></font>*=(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_p& a);<br>
zz_pE& <font color="#b02f60"><b>operator</b></font>*=(zz_pE& x, <font color="#008b00"><b>long</b></font> a);<br>
<br>
<font color="#0000ed"><i>// procedural versions:</i></font><br>
<br>
<br>
<font color="#008b00"><b>void</b></font> mul(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_pE& a, <font color="#008b00"><b>const</b></font> zz_pE& b); <font color="#0000ed"><i>// x = a * b</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> sqr(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_pE& a); <font color="#0000ed"><i>// x = a^2</i></font><br>
zz_pE sqr(<font color="#008b00"><b>const</b></font> zz_pE& a); <br>
<br>
<font color="#0000ed"><i>// PROMOTIONS: *, mul promote {long, zz_p} to zz_pE on (a, b).</i></font><br>
<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Division</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<font color="#0000ed"><i>// operator notation:</i></font><br>
<br>
zz_pE <font color="#b02f60"><b>operator</b></font>/(<font color="#008b00"><b>const</b></font> zz_pE& a, <font color="#008b00"><b>const</b></font> zz_pE& b);<br>
<br>
zz_pE& <font color="#b02f60"><b>operator</b></font>/=(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_pE& a);<br>
zz_pE& <font color="#b02f60"><b>operator</b></font>/=(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_p& a);<br>
zz_pE& <font color="#b02f60"><b>operator</b></font>/=(zz_pE& x, <font color="#008b00"><b>long</b></font> a);<br>
<br>
<br>
<font color="#0000ed"><i>// procedural versions:</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> div(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_pE& a, <font color="#008b00"><b>const</b></font> zz_pE& b);<br>
<font color="#0000ed"><i>// x = a/b. If b is not invertible, an error is raised.</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> inv(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_pE& a);<br>
zz_pE inv(<font color="#008b00"><b>const</b></font> zz_pE& a);<br>
<font color="#0000ed"><i>// x = 1/a</i></font><br>
<br>
<font color="#b02f60"><b>PROMOTIONS</b></font>: /, div promote {<font color="#008b00"><b>long</b></font>, zz_p} to zz_pE on (a, b).<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Exponentiation</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<br>
<font color="#008b00"><b>void</b></font> power(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_pE& a, <font color="#008b00"><b>const</b></font> ZZ& e); <br>
zz_pE power(<font color="#008b00"><b>const</b></font> zz_pE& a, <font color="#008b00"><b>const</b></font> ZZ& e);<br>
<br>
<font color="#008b00"><b>void</b></font> power(zz_pE& x, <font color="#008b00"><b>const</b></font> zz_pE& a, <font color="#008b00"><b>long</b></font> e); <br>
zz_pE power(<font color="#008b00"><b>const</b></font> zz_pE& a, <font color="#008b00"><b>long</b></font> e);<br>
<br>
<font color="#0000ed"><i>// x = a^e (e may be negative)</i></font><br>
<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Random Elements</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<font color="#008b00"><b>void</b></font> random(zz_pE& x);<br>
zz_pE random_zz_pE();<br>
<font color="#0000ed"><i>// x = random element in zz_pE.</i></font><br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Norms and Traces</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<br>
<font color="#008b00"><b>void</b></font> trace(zz_p& x, <font color="#008b00"><b>const</b></font> zz_pE& a); <font color="#0000ed"><i>// x = trace of a</i></font><br>
zz_p trace(<font color="#008b00"><b>const</b></font> zz_pE& a);<br>
<br>
<font color="#008b00"><b>void</b></font> norm(zz_p& x, <font color="#008b00"><b>const</b></font> zz_pE& a); <font color="#0000ed"><i>// x = norm of a</i></font><br>
zz_p norm(<font color="#008b00"><b>const</b></font> zz_pE& a);<br>
<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Input/Output</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
ostream& <font color="#b02f60"><b>operator</b></font><<(ostream& s, <font color="#008b00"><b>const</b></font> zz_pE& a);<br>
<br>
istream& <font color="#b02f60"><b>operator</b></font>>>(istream& s, zz_pE& x);<br>
<font color="#0000ed"><i>// a zz_pX is read and reduced mod p</i></font><br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Modulus Switching </i></font><br>
<br>
<font color="#0000ed"><i>A class zz_pEPush is provided for "backing up" the current modulus</i></font><br>
<font color="#0000ed"><i>and installing a new one.</i></font><br>
<br>
<font color="#0000ed"><i>Here is what you do to save the current modulus, temporarily</i></font><br>
<font color="#0000ed"><i>set it to P, and automatically restore it:</i></font><br>
<br>
<font color="#0000ed"><i> { </i></font><br>
<font color="#0000ed"><i> zz_pEPush push(P); </i></font><br>
<br>
<font color="#0000ed"><i> ...</i></font><br>
<br>
<font color="#0000ed"><i> }</i></font><br>
<br>
<font color="#0000ed"><i>The constructor for push will save the current modulus, and install P as the</i></font><br>
<font color="#0000ed"><i>current modulus. The destructor for push will restore the old modulus when the</i></font><br>
<font color="#0000ed"><i>scope enclosing it exits. This is the so-called RAII (resource acquisition is</i></font><br>
<font color="#0000ed"><i>initialization) paradigm.</i></font><br>
<br>
<font color="#0000ed"><i>You could also do the following:</i></font><br>
<br>
<font color="#0000ed"><i> {</i></font><br>
<font color="#0000ed"><i> zz_pEPush push; // just backup current modulus</i></font><br>
<br>
<font color="#0000ed"><i> ...</i></font><br>
<br>
<font color="#0000ed"><i> zz_pE::init(P1); // install P1 </i></font><br>
<br>
<font color="#0000ed"><i> ...</i></font><br>
<br>
<font color="#0000ed"><i> zz_pE::init(P2); // install P2</i></font><br>
<br>
<font color="#0000ed"><i> // reinstall original modulus as close of scope</i></font><br>
<font color="#0000ed"><i> }</i></font><br>
<br>
<font color="#0000ed"><i> </i></font><br>
<font color="#0000ed"><i>The zz_pEPush interface is good for implementing simple stack-like</i></font><br>
<font color="#0000ed"><i>modulus "context switching". For more general context switching,</i></font><br>
<font color="#0000ed"><i>see zz_pEContext below. There is also an older zz_pEBak class</i></font><br>
<font color="#0000ed"><i>that may also be useful.</i></font><br>
<br>
<font color="#0000ed"><i>..........................................................................</i></font><br>
<br>
<font color="#0000ed"><i>It is critical that zz_pE objects created under one zz_pE modulus are not used in</i></font><br>
<font color="#0000ed"><i>any non-trivial way "out of context", i.e., under a different (or undefined)</i></font><br>
<font color="#0000ed"><i>zz_pE modulus. However, for ease-of-use, some operations may be safely</i></font><br>
<font color="#0000ed"><i>performed out of context. These safe operations include: the default and copy</i></font><br>
<font color="#0000ed"><i>constructor, the destructor, and the assignment operator. In addition is is</i></font><br>
<font color="#0000ed"><i>generally safe to read any zz_pE object out of context (i.e., printing it out, or</i></font><br>
<font color="#0000ed"><i>fetching its underlying representive using the rep() function).</i></font><br>
<br>
<font color="#0000ed"><i>Any unsafe uses out of context are not in general checked, and may </i></font><br>
<font color="#0000ed"><i>lead to unpredictable behavior.</i></font><br>
<br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<font color="#0000ed"><i>// A convenient interface for common cases</i></font><br>
<br>
<font color="#008b00"><b>class</b></font> zz_pEPush {<br>
<br>
<font color="#b02f60"><b>public</b></font>:<br>
zz_pEPush(); <font color="#0000ed"><i>// backup current modulus</i></font><br>
<font color="#008b00"><b>explicit</b></font> zz_pEPush(<font color="#008b00"><b>const</b></font> zz_pX& p); <br>
<font color="#008b00"><b>explicit</b></font> zz_pEPush(<font color="#008b00"><b>const</b></font> zz_pEContext& context); <br>
<font color="#0000ed"><i>// backup current modulus and install the given one</i></font><br>
<br>
<font color="#b02f60"><b>private</b></font>:<br>
zz_pEPush(<font color="#008b00"><b>const</b></font> zz_pEPush&); <font color="#0000ed"><i>// disabled</i></font><br>
<font color="#008b00"><b>void</b></font> <font color="#b02f60"><b>operator</b></font>=(<font color="#008b00"><b>const</b></font> zz_pEPush&); <font color="#0000ed"><i>// disabled</i></font><br>
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};<br>
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<br>
<font color="#0000ed"><i>// more general context switching:</i></font><br>
<font color="#0000ed"><i>// A zz_pEContext object has a modulus Q (possibly "null"),</i></font><br>
<br>
<font color="#008b00"><b>class</b></font> zz_pEContext {<br>
<br>
<br>
<font color="#b02f60"><b>public</b></font>:<br>
<br>
zz_pEContext(); <font color="#0000ed"><i>// Q = "null"</i></font><br>
<font color="#008b00"><b>explicit</b></font> zz_pEContext(<font color="#008b00"><b>const</b></font> zz_pX& P); <font color="#0000ed"><i>// Q = P</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> save(); <font color="#0000ed"><i>// Q = CurrentModulus</i></font><br>
<font color="#008b00"><b>void</b></font> restore() <font color="#008b00"><b>const</b></font>; <font color="#0000ed"><i>// CurrentModulus = Q</i></font><br>
<br>
zz_pEContext(<font color="#008b00"><b>const</b></font> zz_pEContext&); <font color="#0000ed"><i>// copy</i></font><br>
zz_pEContext& <font color="#b02f60"><b>operator</b></font>=(<font color="#008b00"><b>const</b></font> zz_pEContext&); <font color="#0000ed"><i>// assignment</i></font><br>
~zz_pEContext(); <font color="#0000ed"><i>// destructor</i></font><br>
<br>
<br>
};<br>
<br>
<br>
<font color="#0000ed"><i>// An older interface:</i></font><br>
<font color="#0000ed"><i>// To describe this logic, think of a zz_pEBak object</i></font><br>
<font color="#0000ed"><i>// of having two components: a modulus Q (possibly "null") and </i></font><br>
<font color="#0000ed"><i>// an "auto-restore bit" b.</i></font><br>
<br>
<br>
<font color="#008b00"><b>class</b></font> zz_pEBak {<br>
<font color="#b02f60"><b>public</b></font>:<br>
<br>
<br>
zz_pEBak(); <font color="#0000ed"><i>// Q = "null", b = 0</i></font><br>
<br>
~zz_pEBak(); <font color="#0000ed"><i>// if (b) CurrentModulus = Q</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> save(); <font color="#0000ed"><i>// Q = CurrentModulus, b = 1 </i></font><br>
<font color="#008b00"><b>void</b></font> restore(); <font color="#0000ed"><i>// CurrentModulus = Q, b = 0</i></font><br>
<br>
<br>
<font color="#b02f60"><b>private</b></font>:<br>
zz_pEBak(<font color="#008b00"><b>const</b></font> zz_pEBak&); <font color="#0000ed"><i>// copy disabled</i></font><br>
<font color="#008b00"><b>void</b></font> <font color="#b02f60"><b>operator</b></font>=(<font color="#008b00"><b>const</b></font> zz_pEBak&); <font color="#0000ed"><i>// assignment disabled</i></font><br>
};<br>
<br>
<br>
<br>
<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Miscellany</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> clear(zz_pE& x); <font color="#0000ed"><i>// x = 0</i></font><br>
<font color="#008b00"><b>void</b></font> set(zz_pE& x); <font color="#0000ed"><i>// x = 1</i></font><br>
<br>
<font color="#008b00"><b>static</b></font> <font color="#008b00"><b>const</b></font> zz_pE& zz_pE::zero();<br>
<font color="#0000ed"><i>// zz_pE::zero() yields a read-only reference to zero</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> zz_pE::swap(zz_pE& x);<br>
<font color="#008b00"><b>void</b></font> swap(zz_pE& x, zz_pE& y);<br>
<font color="#0000ed"><i>// swap (done by "pointer swapping", if possible).</i></font><br>
<br>
<font color="#008b00"><b>static</b></font> ZZ& zz_pE::cardinality();<br>
<font color="#0000ed"><i>// yields the cardinality, i.e., p^{zz_pE::degree()}</i></font><br>
<br>
zz_pE::zz_pE(INIT_NO_ALLOC_TYPE);<br>
<font color="#0000ed"><i>// special constructor: invoke as zz_pE x(INIT_NO_ALLOC);</i></font><br>
<font color="#0000ed"><i>// initializes x to 0, but allocates no space (this is now the default)</i></font><br>
<br>
zz_pE::zz_pE(INIT_ALLOC_TYPE);<br>
<font color="#0000ed"><i>// special constructor: invoke as zz_pE x(INIT_ALLOC);</i></font><br>
<font color="#0000ed"><i>// initializes x to 0, but allocates space</i></font><br>
<br>
zz_pE::allocate();<br>
<font color="#0000ed"><i>// useful in conjunction with the INIT_NO_ALLLOC constructor:</i></font><br>
<font color="#0000ed"><i>// x.allocate() will pre-allocate space for x, using the</i></font><br>
<font color="#0000ed"><i>// current modulus</i></font><br>
<br>
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