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<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment">MODULE: GF2E</span>
<span class="Comment">SUMMARY:</span>
<span class="Comment">The class GF2E is used to represent polynomials in F_2[X] modulo a</span>
<span class="Comment">polynomial P. The modulus P may be any polynomial with deg(P) > 0,</span>
<span class="Comment">not necessarily irreducible. </span>
<span class="Comment">Objects of the class GF2E are represented as a GF2X of degree < deg(P).</span>
<span class="Comment">An executing program maintains a "current modulus", which is set to P using</span>
<span class="Comment">GF2E::init(P). The current modulus *must* be initialized before any operations</span>
<span class="Comment">on GF2E's are performed. The modulus may be changed, and a mechanism is provided</span>
<span class="Comment">for saving and restoring a modulus (see classes GF2EPush and GF2EContext below).</span>
<span class="Comment">NOTE: if P is a trinomial X^n + X^k + 1, or a pentanomial</span>
<span class="Comment">X^n + X^k3 + X^k2 + X^k1 + 1, or of the form X^n + g, where</span>
<span class="Comment">g has low degree, then performance will be somewhat improved.</span>
<span class="Comment">Such polynomials are constructed by the routines</span>
<span class="Comment">BuildSparseIrred and BuildIrred in GF2XFactoring.</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="PreProc">#include </span><span class="String"><NTL/GF2X.h></span>
<span class="PreProc">#include </span><span class="String"><NTL/SmartPtr.h></span>
<span class="Type">class</span> GF2E {
<span class="Statement">public</span>:
GF2E(); <span class="Comment">// initial value 0</span>
GF2E(<span class="Type">const</span> GF2E& a); <span class="Comment">// copy constructor</span>
<span class="Type">explicit</span> GF2E(GF2 a); <span class="Comment">// promotion constructor</span>
<span class="Type">explicit</span> GF2E(<span class="Type">long</span> a); <span class="Comment">// promotion constructor</span>
GF2E& <span class="Statement">operator</span>=(<span class="Type">const</span> GF2E& a); <span class="Comment">// assignment</span>
GF2E& <span class="Statement">operator</span>=(GF2 a); <span class="Comment">// assignment</span>
GF2E& <span class="Statement">operator</span>=(<span class="Type">long</span> a); <span class="Comment">// assignment</span>
~GF2E(); <span class="Comment">// destructor</span>
<span class="Type">void</span> init(<span class="Type">const</span> GF2X& P);
<span class="Comment">// GF2E::init(P) initializes the current modulus to P;</span>
<span class="Comment">// required: deg(P) >= 1.</span>
<span class="Type">static</span> <span class="Type">const</span> GF2XModulus& modulus();
<span class="Comment">// GF2E::modulus() yields read-only reference to the current modulus </span>
<span class="Type">static</span> <span class="Type">long</span> degree();
<span class="Comment">// GF2E::degree() returns deg(P)</span>
<span class="Comment">// typedefs to aid generic programming</span>
<span class="Type">typedef</span> GF2X rep_type;
<span class="Type">typedef</span> GF2EContext context_type;
<span class="Type">typedef</span> GF2EBak bak_type;
<span class="Type">typedef</span> GF2EPush push_type;
<span class="Type">typedef</span> GF2EX poly_type;
};
<span class="Type">const</span> GF2X& rep(<span class="Type">const</span> GF2E& a); <span class="Comment">// read-only access to representation of a</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Comparison</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">long</span> <span class="Statement">operator</span>==(<span class="Type">const</span> GF2E& a, <span class="Type">const</span> GF2E& b);
<span class="Type">long</span> <span class="Statement">operator</span>!=(<span class="Type">const</span> GF2E& a, <span class="Type">const</span> GF2E& b);
<span class="Type">long</span> IsZero(<span class="Type">const</span> GF2E& a); <span class="Comment">// test for 0</span>
<span class="Type">long</span> IsOne(<span class="Type">const</span> GF2E& a); <span class="Comment">// test for 1</span>
<span class="Comment">// PROMOTIONS: ==, != promote {long, GF2} to GF2E on (a, b).</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Addition </span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Comment">// operator notation:</span>
GF2E <span class="Statement">operator</span>+(<span class="Type">const</span> GF2E& a, <span class="Type">const</span> GF2E& b);
GF2E <span class="Statement">operator</span>-(<span class="Type">const</span> GF2E& a, <span class="Type">const</span> GF2E& b);
GF2E <span class="Statement">operator</span>-(<span class="Type">const</span> GF2E& a);
GF2E& <span class="Statement">operator</span>+=(GF2E& x, <span class="Type">const</span> GF2E& a);
GF2E& <span class="Statement">operator</span>+=(GF2E& x, GF2 a);
GF2E& <span class="Statement">operator</span>+=(GF2E& x, <span class="Type">long</span> a);
GF2E& <span class="Statement">operator</span>++(GF2E& x); <span class="Comment">// prefix</span>
<span class="Type">void</span> <span class="Statement">operator</span>++(GF2E& x, <span class="Type">int</span>); <span class="Comment">// postfix</span>
GF2E& <span class="Statement">operator</span>-=(GF2E& x, <span class="Type">const</span> GF2E& a);
GF2E& <span class="Statement">operator</span>-=(GF2E& x, GF2 a);
GF2E& <span class="Statement">operator</span>-=(GF2E& x, <span class="Type">long</span> a);
GF2E& <span class="Statement">operator</span>--(GF2E& x); <span class="Comment">// prefix</span>
<span class="Type">void</span> <span class="Statement">operator</span>--(GF2E& x, <span class="Type">int</span>); <span class="Comment">// postfix</span>
<span class="Comment">// procedural versions:</span>
<span class="Type">void</span> add(GF2E& x, <span class="Type">const</span> GF2E& a, <span class="Type">const</span> GF2E& b); <span class="Comment">// x = a + b</span>
<span class="Type">void</span> sub(GF2E& x, <span class="Type">const</span> GF2E& a, <span class="Type">const</span> GF2E& b); <span class="Comment">// x = a - b = a + b</span>
<span class="Type">void</span> negate(GF2E& x, <span class="Type">const</span> GF2E& a); <span class="Comment">// x = - a = a</span>
<span class="Comment">// PROMOTIONS: +, -, add, sub promote {long, GF2} to GF2E on (a, b).</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Multiplication </span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Comment">// operator notation:</span>
GF2E <span class="Statement">operator</span>*(<span class="Type">const</span> GF2E& a, <span class="Type">const</span> GF2E& b);
GF2E& <span class="Statement">operator</span>*=(GF2E& x, <span class="Type">const</span> GF2E& a);
GF2E& <span class="Statement">operator</span>*=(GF2E& x, GF2 a);
GF2E& <span class="Statement">operator</span>*=(GF2E& x, <span class="Type">long</span> a);
<span class="Comment">// procedural versions:</span>
<span class="Type">void</span> mul(GF2E& x, <span class="Type">const</span> GF2E& a, <span class="Type">const</span> GF2E& b); <span class="Comment">// x = a * b</span>
<span class="Type">void</span> sqr(GF2E& x, <span class="Type">const</span> GF2E& a); <span class="Comment">// x = a^2</span>
GF2E sqr(<span class="Type">const</span> GF2E& a);
<span class="Comment">// PROMOTIONS: *, mul promote {long, GF2} to GF2E on (a, b).</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Division</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Comment">// operator notation:</span>
GF2E <span class="Statement">operator</span>/(<span class="Type">const</span> GF2E& a, <span class="Type">const</span> GF2E& b);
GF2E& <span class="Statement">operator</span>/=(GF2E& x, <span class="Type">const</span> GF2E& a);
GF2E& <span class="Statement">operator</span>/=(GF2E& x, GF2 a);
GF2E& <span class="Statement">operator</span>/=(GF2E& x, <span class="Type">long</span> a);
<span class="Comment">// procedural versions:</span>
<span class="Type">void</span> div(GF2E& x, <span class="Type">const</span> GF2E& a, <span class="Type">const</span> GF2E& b);
<span class="Comment">// x = a/b. If b is not invertible, an error is raised.</span>
<span class="Type">void</span> inv(GF2E& x, <span class="Type">const</span> GF2E& a);
GF2E inv(<span class="Type">const</span> GF2E& a);
<span class="Comment">// x = 1/a</span>
<span class="Statement">PROMOTIONS</span>: /, div promote {<span class="Type">long</span>, GF2} to GF2E on (a, b).
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Exponentiation</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">void</span> power(GF2E& x, <span class="Type">const</span> GF2E& a, <span class="Type">const</span> ZZ& e);
GF2E power(<span class="Type">const</span> GF2E& a, <span class="Type">const</span> ZZ& e);
<span class="Type">void</span> power(GF2E& x, <span class="Type">const</span> GF2E& a, <span class="Type">long</span> e);
GF2E power(<span class="Type">const</span> GF2E& a, <span class="Type">long</span> e);
<span class="Comment">// x = a^e (e may be negative)</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Random Elements</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">void</span> random(GF2E& x);
GF2E random_GF2E();
<span class="Comment">// x = random element in GF2E.</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Traces</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">void</span> trace(GF2& x, <span class="Type">const</span> GF2E& a); <span class="Comment">// x = trace of a</span>
GF2 trace(<span class="Type">const</span> GF2E& a);
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Input/Output</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
ostream& <span class="Statement">operator</span><<(ostream& s, <span class="Type">const</span> GF2E& a);
istream& <span class="Statement">operator</span>>>(istream& s, GF2E& x);
<span class="Comment">// a GF2X is read and reduced mod p</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Modulus Switching </span>
<span class="Comment">A class GF2EPush is provided for "backing up" the current modulus</span>
<span class="Comment">and installing a new one.</span>
<span class="Comment">Here is what you do to save the current modulus, temporarily</span>
<span class="Comment">set it to P, and automatically restore it:</span>
<span class="Comment"> { </span>
<span class="Comment"> GF2EPush push(P); </span>
<span class="Comment"> ...</span>
<span class="Comment"> }</span>
<span class="Comment">The constructor for push will save the current modulus, and install P as the</span>
<span class="Comment">current modulus. The destructor for push will restore the old modulus when the</span>
<span class="Comment">scope enclosing it exits. This is the so-called RAII (resource acquisition is</span>
<span class="Comment">initialization) paradigm.</span>
<span class="Comment">You could also do the following:</span>
<span class="Comment"> {</span>
<span class="Comment"> GF2EPush push; // just backup current modulus</span>
<span class="Comment"> ...</span>
<span class="Comment"> GF2E::init(P1); // install P1 </span>
<span class="Comment"> ...</span>
<span class="Comment"> GF2E::init(P2); // install P2</span>
<span class="Comment"> // reinstall original modulus as close of scope</span>
<span class="Comment"> }</span>
<span class="Comment"> </span>
<span class="Comment">The GF2EPush interface is good for implementing simple stack-like</span>
<span class="Comment">modulus "context switching". For more general context switching,</span>
<span class="Comment">see GF2EContext below. There is also an older GF2EBak class</span>
<span class="Comment">that may also be useful.</span>
<span class="Comment">..........................................................................</span>
<span class="Comment">It is critical that GF2E objects created under one GF2E modulus are not used in</span>
<span class="Comment">any non-trivial way "out of context", i.e., under a different (or undefined)</span>
<span class="Comment">GF2E modulus. However, for ease-of-use, some operations may be safely</span>
<span class="Comment">performed out of context. These safe operations include: the default and copy</span>
<span class="Comment">constructor, the destructor, and the assignment operator. In addition is is</span>
<span class="Comment">generally safe to read any GF2E object out of context (i.e., printing it out, or</span>
<span class="Comment">fetching its underlying representive using the rep() function).</span>
<span class="Comment">Any unsafe uses out of context are not in general checked, and may </span>
<span class="Comment">lead to unpredictable behavior.</span>
<span class="Comment">NOTE: the implementation of Vec<GF2E> is specialized to manage memory more</span>
<span class="Comment">efficiently than in the default implementation of Vec<T>. Specifically,</span>
<span class="Comment">contiguous elements in a Vec<GF2E> are allocated in a contiguous region of</span>
<span class="Comment">memory. This reduces the number of calls to the memory allocator, and --- more</span>
<span class="Comment">significantly --- leads to greater locality of reference. A consequence of</span>
<span class="Comment">this implementation is that any calls to SetLength on a Vec<GF2E> object will</span>
<span class="Comment">need to use information about the current modulus, and so such calls should</span>
<span class="Comment">only be done "in context". That said, it is still safe to construct a</span>
<span class="Comment">Vec<GF2E> using the default or copy contructor, and to assign or append one</span>
<span class="Comment">Vec<GF2E> to another "out of context".</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Comment">// A convenient interface for common cases</span>
<span class="Type">class</span> GF2EPush {
<span class="Statement">public</span>:
GF2EPush(); <span class="Comment">// backup current modulus</span>
<span class="Type">explicit</span> GF2EPush(<span class="Type">const</span> GF2X& P);
<span class="Type">explicit</span> GF2EPush(<span class="Type">const</span> GF2EContext& context);
<span class="Comment">// backup current modulus and install the given one</span>
<span class="Statement">private</span>:
GF2EPush(<span class="Type">const</span> GF2EPush&); <span class="Comment">// disabled</span>
<span class="Type">void</span> <span class="Statement">operator</span>=(<span class="Type">const</span> GF2EPush&); <span class="Comment">// disabled</span>
};
<span class="Comment">// more general context switching:</span>
<span class="Comment">// A GF2EContext object has a modulus Q (possibly "null"),</span>
<span class="Type">class</span> GF2EContext {
<span class="Statement">public</span>:
GF2EContext(); <span class="Comment">// Q = "null"</span>
<span class="Type">explicit</span> GF2EContext(<span class="Type">const</span> GF2X& P); <span class="Comment">// Q = P</span>
<span class="Type">void</span> save(); <span class="Comment">// Q = CurrentModulus</span>
<span class="Type">void</span> restore() <span class="Type">const</span>; <span class="Comment">// CurrentModulus = Q</span>
GF2EContext(<span class="Type">const</span> GF2EContext&); <span class="Comment">// copy</span>
GF2EContext& <span class="Statement">operator</span>=(<span class="Type">const</span> GF2EContext&); <span class="Comment">// assignment</span>
~GF2EContext(); <span class="Comment">// destructor</span>
};
<span class="Comment">// An older interface:</span>
<span class="Comment">// To describe this logic, think of a GF2EBak object</span>
<span class="Comment">// of having two components: a modulus Q (possibly "null") and </span>
<span class="Comment">// an "auto-restore bit" b.</span>
<span class="Type">class</span> GF2EBak {
<span class="Statement">public</span>:
GF2EBak(); <span class="Comment">// Q = "null", b = 0</span>
~GF2EBak(); <span class="Comment">// if (b) CurrentModulus = Q</span>
<span class="Type">void</span> save(); <span class="Comment">// Q = CurrentModulus, b = 1 </span>
<span class="Type">void</span> restore(); <span class="Comment">// CurrentModulus = Q, b = 0</span>
<span class="Statement">private</span>:
GF2EBak(<span class="Type">const</span> GF2EBak&); <span class="Comment">// copy disabled</span>
<span class="Type">void</span> <span class="Statement">operator</span>=(<span class="Type">const</span> GF2EBak&); <span class="Comment">// assignment disabled</span>
};
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Miscellany</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">void</span> clear(GF2E& x); <span class="Comment">// x = 0</span>
<span class="Type">void</span> set(GF2E& x); <span class="Comment">// x = 1</span>
<span class="Type">static</span> <span class="Type">const</span> GF2E& GF2E::zero();
<span class="Comment">// GF2E::zero() yields a read-only reference to zero</span>
<span class="Type">static</span> <span class="Type">long</span> GF2X::WordLength();
<span class="Comment">// GF2E::size() returns # of words needed to store a polynomial of</span>
<span class="Comment">// degree < GF2E::degree()</span>
<span class="Type">void</span> GF2E::swap(GF2E& x);
<span class="Type">void</span> swap(GF2E& x, GF2E& y);
<span class="Comment">// swap (done by "pointer swapping", if possible).</span>
<span class="Type">static</span> ZZ& GF2E::cardinality();
<span class="Comment">// yields the cardinality, i.e., 2^{GF2E::degree()}</span>
GF2E::GF2E(INIT_NO_ALLOC_TYPE);
<span class="Comment">// special constructor: invoke as GF2E x(INIT_NO_ALLOC);</span>
<span class="Comment">// initializes x to 0, but allocates no space (this is now the default)</span>
GF2E::GF2E(INIT_ALLOC_TYPE);
<span class="Comment">// special constructor: invoke as GF2E x(INIT_ALLOC);</span>
<span class="Comment">// initializes x to 0, but allocates space </span>
GF2E::allocate();
<span class="Comment">// useful in conjunction with the INIT_NO_ALLLOC constructor:</span>
<span class="Comment">// x.allocate() will pre-allocate space for x, using the</span>
<span class="Comment">// current modulus</span>
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