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<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment">MODULE: ZZ_p</span>
<span class="Comment">SUMMARY:</span>
<span class="Comment">The class ZZ_p is used to represent integers mod p. The modulus p may</span>
<span class="Comment">be any positive integer, not necessarily prime. </span>
<span class="Comment">Objects of the class ZZ_p are represented as a ZZ in the range 0..p-1.</span>
<span class="Comment">An executing program maintains a "current modulus", which is set to p using</span>
<span class="Comment">ZZ_p::init(p). The current modulus *must* be initialized before any operations</span>
<span class="Comment">on ZZ_p'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 ZZ_pPush and ZZ_pContext below).</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="PreProc">#include </span><span class="String"><NTL/ZZ.h></span>
<span class="PreProc">#include </span><span class="String"><NTL/ZZVec.h></span>
<span class="PreProc">#include </span><span class="String"><NTL/SmartPtr.h></span>
<span class="Type">class</span> ZZ_p {
<span class="Statement">public</span>:
ZZ_p(); <span class="Comment">// initialize to 0</span>
ZZ_p(<span class="Type">const</span> ZZ_p& a); <span class="Comment">// copy constructor</span>
<span class="Type">explicit</span> ZZ_p(<span class="Type">long</span> a); <span class="Comment">// promotion constructor</span>
~ZZ_p(); <span class="Comment">// destructor</span>
ZZ_p& <span class="Statement">operator</span>=(<span class="Type">const</span> ZZ_p& a); <span class="Comment">// assignment</span>
ZZ_p& <span class="Statement">operator</span>=(<span class="Type">long</span> a); <span class="Comment">// assignment</span>
ZZ_p(ZZ_p&& a);
<span class="Comment">// move constructor (C++11 only)</span>
<span class="Comment">// declared noexcept unless NTL_EXCEPTIONS flag is set</span>
ZZ_p& <span class="Statement">operator</span>=(ZZ_p&& a);
<span class="Comment">// move assignment (C++11 only)</span>
<span class="Comment">// declared noexcept unless NTL_EXCEPTIONS flag is set</span>
<span class="Type">static</span> <span class="Type">void</span> init(<span class="Type">const</span> ZZ& p);
<span class="Comment">// ZZ_p::init(p) sets the modulus to p (p > 1)</span>
<span class="Type">static</span> <span class="Type">const</span> ZZ& modulus();
<span class="Comment">// ZZ_p::modulus() yields read-only reference to the current</span>
<span class="Comment">// modulus</span>
<span class="Comment">// typedefs to aid in generic programming</span>
<span class="Type">typedef</span> ZZ rep_type;
<span class="Type">typedef</span> ZZ_pContext context_type;
<span class="Type">typedef</span> ZZ_pBak bak_type;
<span class="Type">typedef</span> ZZ_pPush push_type;
<span class="Type">typedef</span> ZZ_pX poly_type;
};
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Access to representation</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">const</span> ZZ& rep(<span class="Type">const</span> ZZ_p& a);
<span class="Comment">// read-only access to representation of a</span>
<span class="Comment">/*</span><span class="Comment">***** Example: ******** </span>
<span class="Comment"> ZZ x;</span>
<span class="Comment"> ZZ_p a;</span>
<span class="Comment"> x = rep(a);</span>
<span class="Comment">************************</span><span class="Comment">*/</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> ZZ_p& a, <span class="Type">const</span> ZZ_p& b);
<span class="Type">long</span> <span class="Statement">operator</span>!=(<span class="Type">const</span> ZZ_p& a, <span class="Type">const</span> ZZ_p& b);
<span class="Comment">// PROMOTIONS: the comparison operators provide promotions</span>
<span class="Comment">// from long to ZZ_p on (a, b)</span>
<span class="Type">long</span> IsZero(<span class="Type">const</span> ZZ_p& a); <span class="Comment">// test for 0</span>
<span class="Type">long</span> IsOne(<span class="Type">const</span> ZZ_p& a); <span class="Comment">// test for 1</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>
ZZ_p <span class="Statement">operator</span>+(<span class="Type">const</span> ZZ_p& a, <span class="Type">const</span> ZZ_p& b);
ZZ_p <span class="Statement">operator</span>-(<span class="Type">const</span> ZZ_p& a, <span class="Type">const</span> ZZ_p& b);
ZZ_p <span class="Statement">operator</span>-(<span class="Type">const</span> ZZ_p& a); <span class="Comment">// unary -</span>
ZZ_p& <span class="Statement">operator</span>+=(ZZ_p& x, <span class="Type">const</span> ZZ_p& b);
ZZ_p& <span class="Statement">operator</span>+=(ZZ_p& x, <span class="Type">long</span> b);
ZZ_p& <span class="Statement">operator</span>-=(ZZ_p& x, <span class="Type">const</span> ZZ_p& b);
ZZ_p& <span class="Statement">operator</span>-=(ZZ_p& x, <span class="Type">long</span> b);
ZZ_p& <span class="Statement">operator</span>++(ZZ_p& x); <span class="Comment">// prefix</span>
<span class="Type">void</span> <span class="Statement">operator</span>++(ZZ_p& x, <span class="Type">int</span>); <span class="Comment">// postfix</span>
ZZ_p& <span class="Statement">operator</span>--(ZZ_p& x); <span class="Comment">// prefix</span>
<span class="Type">void</span> <span class="Statement">operator</span>--(ZZ_p& x, <span class="Type">int</span>); <span class="Comment">// postfix</span>
<span class="Comment">// procedural versions:</span>
<span class="Type">void</span> add(ZZ_p& x, <span class="Type">const</span> ZZ_p& a, <span class="Type">const</span> ZZ_p& b); <span class="Comment">// x = a + b</span>
<span class="Type">void</span> sub(ZZ_p& x, <span class="Type">const</span> ZZ_p& a, <span class="Type">const</span> ZZ_p& b); <span class="Comment">// x = a - b </span>
<span class="Type">void</span> negate(ZZ_p& x, <span class="Type">const</span> ZZ_p& a); <span class="Comment">// x = -a</span>
<span class="Comment">// PROMOTIONS: binary +, - and procedures add, sub provide promotions</span>
<span class="Comment">// from long to ZZ_p 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>
ZZ_p <span class="Statement">operator</span>*(<span class="Type">const</span> ZZ_p& a, <span class="Type">const</span> ZZ_p& b);
ZZ_p& <span class="Statement">operator</span>*=(ZZ_p& x, <span class="Type">const</span> ZZ_p& b);
ZZ_p& <span class="Statement">operator</span>*=(ZZ_p& x, <span class="Type">long</span> b);
<span class="Comment">// procedural versions:</span>
<span class="Type">void</span> mul(ZZ_p& x, <span class="Type">const</span> ZZ_p& a, <span class="Type">const</span> ZZ_p& b); <span class="Comment">// x = a * b</span>
<span class="Type">void</span> sqr(ZZ_p& x, <span class="Type">const</span> ZZ_p& a); <span class="Comment">// x = a^2</span>
ZZ_p sqr(<span class="Type">const</span> ZZ_p& a); <span class="Comment">// x = a^2</span>
<span class="Comment">// PROMOTIONS: operator * and procedure mul provide promotions </span>
<span class="Comment">// from long to ZZ_p 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>
ZZ_p <span class="Statement">operator</span>/(<span class="Type">const</span> ZZ_p& a, <span class="Type">const</span> ZZ_p& b);
ZZ_p& <span class="Statement">operator</span>/=(ZZ_p& x, <span class="Type">const</span> ZZ_p& b);
ZZ_p& <span class="Statement">operator</span>/=(ZZ_p& x, <span class="Type">long</span> b);
<span class="Comment">// procedural versions:</span>
<span class="Type">void</span> div(ZZ_p& x, <span class="Type">const</span> ZZ_p& a, <span class="Type">const</span> ZZ_p& b);
<span class="Comment">// x = a/b.</span>
<span class="Comment">// By default, if b is not invertible, an error is raised. </span>
<span class="Comment">// If exceptions are enabled, an InvModErrorObject is thrown</span>
<span class="Comment">// (see documentation in the ZZ module); otherwise, the program</span>
<span class="Comment">// aborts with an error message.</span>
<span class="Comment">// For backward compatibility, one can define an error handler</span>
<span class="Comment">// void H(const ZZ_p& b), and setting ZZ_p::DivHandler = H. Then if b</span>
<span class="Comment">// != 0 and b is not invertible, the function H is invoked with b as</span>
<span class="Comment">// its argument. If the error handler function returns to its caller,</span>
<span class="Comment">// error handling proceeds as described above.</span>
<span class="Type">void</span> inv(ZZ_p& x, <span class="Type">const</span> ZZ_p& a); <span class="Comment">// x = 1/a</span>
ZZ_p inv(<span class="Type">const</span> ZZ_p& a);
<span class="Comment">// Error handling is the same as above.</span>
<span class="Comment">// PROMOTIONS: operator / and procedure div provide promotions</span>
<span class="Comment">// from long to ZZ_p on (a, b)</span>
<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(ZZ_p& x, <span class="Type">const</span> ZZ_p& a, <span class="Type">const</span> ZZ& e); <span class="Comment">// x = a^e (e may be negative)</span>
ZZ_p power(<span class="Type">const</span> ZZ_p& a, <span class="Type">const</span> ZZ& e); <span class="Comment">// functional variants</span>
<span class="Type">void</span> power(ZZ_p& x, <span class="Type">const</span> ZZ_p& a, <span class="Type">long</span> e);
ZZ_p power(ZZ_p& x, <span class="Type">const</span> ZZ_p& a, <span class="Type">long</span> e);
<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(ZZ_p& x);
ZZ_p random_ZZ_p();
<span class="Comment">// x = random element in ZZ_p. </span>
<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> ZZ_p& a);
istream& <span class="Statement">operator</span>>>(istream& s, ZZ_p& x);
<span class="Comment">// a ZZ 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 ZZ_pPush 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"> ZZ_pPush 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"> ZZ_pPush push; // just backup current modulus</span>
<span class="Comment"> ...</span>
<span class="Comment"> ZZ_p::init(p1); // install p1 </span>
<span class="Comment"> ...</span>
<span class="Comment"> ZZ_p::init(p2); // install p2</span>
<span class="Comment"> // reinstall original modulus at close of scope</span>
<span class="Comment"> }</span>
<span class="Comment"> </span>
<span class="Comment">The ZZ_pPush 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 ZZ_pContext below. There is also an older ZZ_pBak class</span>
<span class="Comment">that may also be useful.</span>
<span class="Comment">..........................................................................</span>
<span class="Comment">It is critical that ZZ_p objects created under one ZZ_p 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">ZZ_p 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 ZZ_p 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<ZZ_p> 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<ZZ_p> 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<ZZ_p> 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<ZZ_p> using the default or copy contructor, and to assign or append one</span>
<span class="Comment">Vec<ZZ_p> 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> ZZ_pPush {
<span class="Statement">public</span>:
ZZ_pPush(); <span class="Comment">// backup current modulus</span>
<span class="Type">explicit</span> ZZ_pPush(<span class="Type">const</span> ZZ& p);
<span class="Type">explicit</span> ZZ_pPush(<span class="Type">const</span> ZZ_pContext& context);
<span class="Comment">// backup current modulus and install the given one</span>
<span class="Statement">private</span>:
ZZ_pPush(<span class="Type">const</span> ZZ_pPush&); <span class="Comment">// disabled</span>
<span class="Type">void</span> <span class="Statement">operator</span>=(<span class="Type">const</span> ZZ_pPush&); <span class="Comment">// disabled</span>
};
<span class="Comment">// more general context switching:</span>
<span class="Comment">// A ZZ_pContext object has a modulus q (possibly "null")</span>
<span class="Type">class</span> ZZ_pContext {
<span class="Statement">public</span>:
ZZ_pContext(); <span class="Comment">// q = "null"</span>
<span class="Type">explicit</span> ZZ_pContext(<span class="Type">const</span> ZZ& 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>
ZZ_pContext(<span class="Type">const</span> ZZ_pContext&); <span class="Comment">// copy</span>
ZZ_pContext& <span class="Statement">operator</span>=(<span class="Type">const</span> ZZ_pContext&); <span class="Comment">// assignment</span>
~ZZ_pContext(); <span class="Comment">// destructor</span>
};
<span class="Comment">// An older interface:</span>
<span class="Comment">// To describe this logic, think of a ZZ_pBak 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> ZZ_pBak {
<span class="Statement">public</span>:
ZZ_pBak(); <span class="Comment">// q = "null", b = 0</span>
~ZZ_pBak(); <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>:
ZZ_pBak(<span class="Type">const</span> ZZ_pBak&); <span class="Comment">// copy disabled</span>
<span class="Type">void</span> <span class="Statement">operator</span>=(<span class="Type">const</span> ZZ_pBak&); <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(ZZ_p& x); <span class="Comment">// x = 0</span>
<span class="Type">void</span> set(ZZ_p& x); <span class="Comment">// x = 1</span>
<span class="Type">static</span> <span class="Type">long</span> ZZ_p::ModulusSize();
<span class="Comment">// ZZ_p::ModulusSize() returns ZZ_p::modulus().size()</span>
<span class="Type">static</span> <span class="Type">const</span> ZZ_p& ZZ_p::zero();
<span class="Comment">// ZZ_p::zero() yields a read-only reference to zero</span>
<span class="Type">void</span> swap(ZZ_p& x, ZZ_p& y);
<span class="Comment">// swap x and y (done by "pointer swapping", if possible).</span>
<span class="Type">void</span> ZZ_p::swap(ZZ& x);
<span class="Comment">// swap member function</span>
ZZ_p::ZZ_p(INIT_NO_ALLOC_TYPE);
<span class="Comment">// special constructor: invoke as ZZ_p x(INIT_NO_ALLOC);</span>
<span class="Comment">// initializes x to 0, but allocates no space (this is now the default)</span>
ZZ_p::ZZ_p(INIT_ALLOC_TYPE);
<span class="Comment">// special constructor: invoke as ZZ_p x(INIT_ALLOC);</span>
<span class="Comment">// initializes x to 0, but allocates space</span>
ZZ_p::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>
</pre>
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