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<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i>MODULE: ZZX</i></font><br>
<br>
<font color="#0000ed"><i>SUMMARY:</i></font><br>
<br>
<font color="#0000ed"><i>The class ZZX implements polynomials in ZZ[X], i.e., univariate</i></font><br>
<font color="#0000ed"><i>polynomials with integer coefficients.</i></font><br>
<br>
<font color="#0000ed"><i>Polynomial multiplication is implemented using one of 4 different</i></font><br>
<font color="#0000ed"><i>algorithms:</i></font><br>
<br>
<font color="#0000ed"><i>1) classical </i></font><br>
<br>
<font color="#0000ed"><i>2) Karatsuba</i></font><br>
<br>
<font color="#0000ed"><i>3) Schoenhage & Strassen --- performs an FFT by working</i></font><br>
<font color="#0000ed"><i> modulo a "Fermat number" of appropriate size...</i></font><br>
<font color="#0000ed"><i> good for polynomials with huge coefficients</i></font><br>
<font color="#0000ed"><i> and moderate degree</i></font><br>
<br>
<font color="#0000ed"><i>4) CRT/FFT --- performs an FFT by working modulo several</i></font><br>
<font color="#0000ed"><i> small primes...good for polynomials with moderate coefficients</i></font><br>
<font color="#0000ed"><i> and huge degree.</i></font><br>
<br>
<font color="#0000ed"><i>The choice of algorithm is somewhat heuristic, and may not always be</i></font><br>
<font color="#0000ed"><i>perfect.</i></font><br>
<br>
<font color="#0000ed"><i>Many thanks to Juergen Gerhard <jngerhar@plato.uni-paderborn.de> for</i></font><br>
<font color="#0000ed"><i>pointing out the deficiency in the NTL-1.0 ZZX arithmetic, and for</i></font><br>
<font color="#0000ed"><i>contributing the Schoenhage/Strassen code.</i></font><br>
<br>
<font color="#0000ed"><i>Extensive use is made of modular algorithms to enhance performance</i></font><br>
<font color="#0000ed"><i>(e.g., the GCD algorithm and amny others).</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/vec_ZZ.h></font><br>
<font color="#1773cc">#include </font><font color="#4a6f8b">"zz_pX.h"</font><br>
<font color="#1773cc">#include </font><font color="#4a6f8b"><NTL/ZZ_pX.h></font><br>
<br>
<br>
<font color="#008b00"><b>class</b></font> ZZX {<br>
<font color="#b02f60"><b>public</b></font>:<br>
<br>
<br>
ZZX(); <font color="#0000ed"><i>// initial value 0</i></font><br>
<br>
ZZX(<font color="#008b00"><b>const</b></font> ZZX& a); <font color="#0000ed"><i>// copy</i></font><br>
<font color="#008b00"><b>explicit</b></font> ZZX(<font color="#008b00"><b>const</b></font> ZZ& a); <font color="#0000ed"><i>// promotion</i></font><br>
<font color="#008b00"><b>explicit</b></font> ZZX(<font color="#008b00"><b>long</b></font> a); <font color="#0000ed"><i>// promotion</i></font><br>
<br>
~ZZX();<br>
<br>
ZZX(INIT_MONO_TYPE, <font color="#008b00"><b>long</b></font> i, <font color="#008b00"><b>const</b></font> ZZ& c); <br>
ZZX(INIT_MONO_TYPE, <font color="#008b00"><b>long</b></font> i, <font color="#008b00"><b>long</b></font> c); <br>
<font color="#0000ed"><i>// initial value c*X^i, invoke as ZZX(INIT_MONO, i, c)</i></font><br>
<br>
ZZX(INIT_MONO_TYPE, <font color="#008b00"><b>long</b></font> i); <br>
<font color="#0000ed"><i>// initial value X^i, invoke as ZZX(INIT_MONO, i)</i></font><br>
<br>
ZZX& <font color="#b02f60"><b>operator</b></font>=(<font color="#008b00"><b>const</b></font> ZZX& a); <font color="#0000ed"><i>// assignment</i></font><br>
ZZX& <font color="#b02f60"><b>operator</b></font>=(<font color="#008b00"><b>const</b></font> ZZ& a);<br>
ZZX& <font color="#b02f60"><b>operator</b></font>=(<font color="#008b00"><b>long</b></font> a);<br>
<br>
<font color="#008b00"><b>typedef</b></font> ZZ coeff_type;<br>
<br>
<font color="#0000ed"><i>// ...</i></font><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> Accessing coefficients</i></font><br>
<br>
<font color="#0000ed"><i>The degree of a polynomial f is obtained as deg(f),</i></font><br>
<font color="#0000ed"><i>where the zero polynomial, by definition, has degree -1.</i></font><br>
<br>
<font color="#0000ed"><i>A polynomial f is represented as a coefficient vector.</i></font><br>
<font color="#0000ed"><i>Coefficients may be accesses in one of two ways.</i></font><br>
<br>
<font color="#0000ed"><i>The safe, high-level method is to call the function</i></font><br>
<font color="#0000ed"><i>coeff(f, i) to get the coefficient of X^i in the polynomial f,</i></font><br>
<font color="#0000ed"><i>and to call the function SetCoeff(f, i, a) to set the coefficient</i></font><br>
<font color="#0000ed"><i>of X^i in f to the scalar a.</i></font><br>
<br>
<font color="#0000ed"><i>One can also access the coefficients more directly via a lower level </i></font><br>
<font color="#0000ed"><i>interface. The coefficient of X^i in f may be accessed using </i></font><br>
<font color="#0000ed"><i>subscript notation f[i]. In addition, one may write f.SetLength(n)</i></font><br>
<font color="#0000ed"><i>to set the length of the underlying coefficient vector to n,</i></font><br>
<font color="#0000ed"><i>and f.SetMaxLength(n) to allocate space for n coefficients,</i></font><br>
<font color="#0000ed"><i>without changing the coefficient vector itself.</i></font><br>
<br>
<font color="#0000ed"><i>After setting coefficients using this low-level interface,</i></font><br>
<font color="#0000ed"><i>one must ensure that leading zeros in the coefficient vector</i></font><br>
<font color="#0000ed"><i>are stripped afterwards by calling the function f.normalize().</i></font><br>
<br>
<font color="#0000ed"><i>NOTE: the coefficient vector of f may also be accessed directly</i></font><br>
<font color="#0000ed"><i>as f.rep; however, this is not recommended. Also, for a properly</i></font><br>
<font color="#0000ed"><i>normalized polynomial f, we have f.rep.length() == deg(f)+1,</i></font><br>
<font color="#0000ed"><i>and deg(f) >= 0 => f.rep[deg(f)] != 0.</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<br>
<font color="#008b00"><b>long</b></font> deg(<font color="#008b00"><b>const</b></font> ZZX& a); <font color="#0000ed"><i>// return deg(a); deg(0) == -1.</i></font><br>
<br>
<font color="#008b00"><b>const</b></font> ZZ& coeff(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> i);<br>
<font color="#0000ed"><i>// returns the coefficient of X^i, or zero if i not in range</i></font><br>
<br>
<font color="#008b00"><b>const</b></font> ZZ& LeadCoeff(<font color="#008b00"><b>const</b></font> ZZX& a);<br>
<font color="#0000ed"><i>// returns leading term of a, or zero if a == 0</i></font><br>
<br>
<font color="#008b00"><b>const</b></font> ZZ& ConstTerm(<font color="#008b00"><b>const</b></font> ZZX& a);<br>
<font color="#0000ed"><i>// returns constant term of a, or zero if a == 0</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> SetCoeff(ZZX& x, <font color="#008b00"><b>long</b></font> i, <font color="#008b00"><b>const</b></font> ZZ& a);<br>
<font color="#008b00"><b>void</b></font> SetCoeff(ZZX& x, <font color="#008b00"><b>long</b></font> i, <font color="#008b00"><b>long</b></font> a);<br>
<font color="#0000ed"><i>// makes coefficient of X^i equal to a; error is raised if i < 0</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> SetCoeff(ZZX& x, <font color="#008b00"><b>long</b></font> i);<br>
<font color="#0000ed"><i>// makes coefficient of X^i equal to 1; error is raised if i < 0</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> SetX(ZZX& x); <font color="#0000ed"><i>// x is set to the monomial X</i></font><br>
<br>
<font color="#008b00"><b>long</b></font> IsX(<font color="#008b00"><b>const</b></font> ZZX& a); <font color="#0000ed"><i>// test if x = X</i></font><br>
<br>
<br>
<br>
<br>
ZZ& ZZX::<font color="#b02f60"><b>operator</b></font>[](<font color="#008b00"><b>long</b></font> i); <br>
<font color="#008b00"><b>const</b></font> ZZ& ZZX::<font color="#b02f60"><b>operator</b></font>[](<font color="#008b00"><b>long</b></font> i) <font color="#008b00"><b>const</b></font>;<br>
<font color="#0000ed"><i>// indexing operators: f[i] is the coefficient of X^i ---</i></font><br>
<font color="#0000ed"><i>// i should satsify i >= 0 and i <= deg(f).</i></font><br>
<font color="#0000ed"><i>// No range checking (unless NTL_RANGE_CHECK is defined).</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> ZZX::SetLength(<font color="#008b00"><b>long</b></font> n);<br>
<font color="#0000ed"><i>// f.SetLength(n) sets the length of the inderlying coefficient</i></font><br>
<font color="#0000ed"><i>// vector to n --- after this call, indexing f[i] for i = 0..n-1</i></font><br>
<font color="#0000ed"><i>// is valid.</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> ZZX::normalize(); <br>
<font color="#0000ed"><i>// f.normalize() strips leading zeros from coefficient vector of f</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> ZZX::SetMaxLength(<font color="#008b00"><b>long</b></font> n);<br>
<font color="#0000ed"><i>// f.SetMaxLength(n) pre-allocate spaces for n coefficients. The</i></font><br>
<font color="#0000ed"><i>// polynomial that f represents is unchanged.</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> 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> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
<font color="#008b00"><b>long</b></font> <font color="#b02f60"><b>operator</b></font>!=(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
<br>
<font color="#008b00"><b>long</b></font> IsZero(<font color="#008b00"><b>const</b></font> ZZX& 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> ZZX& a); <font color="#0000ed"><i>// test for 1</i></font><br>
<br>
<font color="#0000ed"><i>// PROMOTIONS: operators ==, != promote {long, ZZ} to ZZX 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>
ZZX <font color="#b02f60"><b>operator</b></font>+(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
ZZX <font color="#b02f60"><b>operator</b></font>-(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
ZZX <font color="#b02f60"><b>operator</b></font>-(<font color="#008b00"><b>const</b></font> ZZX& a); <font color="#0000ed"><i>// unary -</i></font><br>
<br>
ZZX& <font color="#b02f60"><b>operator</b></font>+=(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a);<br>
ZZX& <font color="#b02f60"><b>operator</b></font>-=(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a);<br>
<br>
ZZX& <font color="#b02f60"><b>operator</b></font>++(ZZX& x); <font color="#0000ed"><i>// prefix</i></font><br>
<font color="#008b00"><b>void</b></font> <font color="#b02f60"><b>operator</b></font>++(ZZX& x, <font color="#008b00"><b>int</b></font>); <font color="#0000ed"><i>// postfix</i></font><br>
<br>
ZZX& <font color="#b02f60"><b>operator</b></font>--(ZZX& x); <font color="#0000ed"><i>// prefix</i></font><br>
<font color="#008b00"><b>void</b></font> <font color="#b02f60"><b>operator</b></font>--(ZZX& x, <font color="#008b00"><b>int</b></font>); <font color="#0000ed"><i>// postfix</i></font><br>
<br>
<br>
<font color="#0000ed"><i>// procedural versions:</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> add(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b); <font color="#0000ed"><i>// x = a + b</i></font><br>
<font color="#008b00"><b>void</b></font> sub(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b); <font color="#0000ed"><i>// x = a - b</i></font><br>
<font color="#008b00"><b>void</b></font> negate(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a); <font color="#0000ed"><i>// x = -a</i></font><br>
<br>
<font color="#0000ed"><i>// PROMOTIONS: binary +, - and procedures add, sub promote {long, ZZ} </i></font><br>
<font color="#0000ed"><i>// to ZZX 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>
<font color="#0000ed"><i>// operator notation:</i></font><br>
<br>
ZZX <font color="#b02f60"><b>operator</b></font>*(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
<br>
ZZX& <font color="#b02f60"><b>operator</b></font>*=(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a);<br>
<br>
<br>
<font color="#0000ed"><i>// procedural versions:</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> mul(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b); <font color="#0000ed"><i>// x = a * b</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> sqr(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a); <font color="#0000ed"><i>// x = a^2</i></font><br>
ZZX sqr(<font color="#008b00"><b>const</b></font> ZZX& a);<br>
<br>
<font color="#0000ed"><i>// PROMOTIONS: operator * and procedure mul promote {long, ZZ} to ZZX </i></font><br>
<font color="#0000ed"><i>// 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> Shift Operations</i></font><br>
<br>
<font color="#0000ed"><i>LeftShift by n means multiplication by X^n</i></font><br>
<font color="#0000ed"><i>RightShift by n means division by X^n</i></font><br>
<br>
<font color="#0000ed"><i>A negative shift amount reverses the direction of the shift.</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>
ZZX <font color="#b02f60"><b>operator</b></font><<(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> n);<br>
ZZX <font color="#b02f60"><b>operator</b></font>>>(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> n);<br>
<br>
ZZX& <font color="#b02f60"><b>operator</b></font><<=(ZZX& x, <font color="#008b00"><b>long</b></font> n);<br>
ZZX& <font color="#b02f60"><b>operator</b></font>>>=(ZZX& x, <font color="#008b00"><b>long</b></font> n);<br>
<br>
<font color="#0000ed"><i>// procedural versions:</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> LeftShift(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> n); <br>
ZZX LeftShift(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> n);<br>
<br>
<font color="#008b00"><b>void</b></font> RightShift(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> n); <br>
ZZX RightShift(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> n); <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>// Given polynomials a, b in ZZ[X], there exist polynomials</i></font><br>
<font color="#0000ed"><i>// q, r in QQ[X] such that a = b*q + r, deg(r) < deg(b).</i></font><br>
<font color="#0000ed"><i>// These routines return q and/or r if q and/or r lie(s) in ZZ[X],</i></font><br>
<font color="#0000ed"><i>// and otherwise raise an error. </i></font><br>
<br>
<font color="#0000ed"><i>// Note that if the leading coefficient of b is 1 or -1, </i></font><br>
<font color="#0000ed"><i>// then q and r always lie in ZZ[X], and no error can occur.</i></font><br>
<br>
<font color="#0000ed"><i>// For example, you can write f/2 for a ZZX f. If all coefficients</i></font><br>
<font color="#0000ed"><i>// of f are even, the result is f with a factor of two removed;</i></font><br>
<font color="#0000ed"><i>// otherwise, an error is raised. More generally, f/g will be</i></font><br>
<font color="#0000ed"><i>// evaluate q in ZZ[X] such that f = q*g if such a q exists,</i></font><br>
<font color="#0000ed"><i>// and will otherwise raise an error.</i></font><br>
<br>
<font color="#0000ed"><i>// See also below the routines for pseudo-division and division</i></font><br>
<font color="#0000ed"><i>// predicates for routines that are perhaps more useful in</i></font><br>
<font color="#0000ed"><i>// some situations.</i></font><br>
<br>
<br>
<font color="#0000ed"><i>// operator notation: </i></font><br>
<br>
ZZX <font color="#b02f60"><b>operator</b></font>/(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
ZZX <font color="#b02f60"><b>operator</b></font>/(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZ& b);<br>
ZZX <font color="#b02f60"><b>operator</b></font>/(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> b);<br>
<br>
ZZX <font color="#b02f60"><b>operator</b></font>%(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
<br>
ZZX& <font color="#b02f60"><b>operator</b></font>/=(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
ZZX& <font color="#b02f60"><b>operator</b></font>/=(ZZX& x, <font color="#008b00"><b>const</b></font> ZZ& b);<br>
ZZX& <font color="#b02f60"><b>operator</b></font>/=(ZZX& x, <font color="#008b00"><b>long</b></font> b);<br>
<br>
ZZX& <font color="#b02f60"><b>operator</b></font>%=(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
<br>
<br>
<font color="#0000ed"><i>// procedural versions:</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> DivRem(ZZX& q, ZZX& r, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
<font color="#0000ed"><i>// computes q, r such that a = b q + r and deg(r) < deg(b).</i></font><br>
<font color="#0000ed"><i>// requires LeadCoeff(b) is a unit (+1, -1); otherwise,</i></font><br>
<font color="#0000ed"><i>// an error is raised.</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> div(ZZX& q, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
<font color="#008b00"><b>void</b></font> div(ZZX& q, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZ& b);<br>
<font color="#008b00"><b>void</b></font> div(ZZX& q, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> b);<br>
<font color="#0000ed"><i>// same as DivRem, but only computes q</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> rem(ZZX& r, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
<font color="#0000ed"><i>// same as DivRem, but only computes r</i></font><br>
<br>
<br>
<br>
<font color="#0000ed"><i>// divide predicates:</i></font><br>
<br>
<font color="#008b00"><b>long</b></font> divide(ZZX& q, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
<font color="#008b00"><b>long</b></font> divide(ZZX& q, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZ& b);<br>
<font color="#008b00"><b>long</b></font> divide(ZZX& q, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> b);<br>
<font color="#0000ed"><i>// if b | a, sets q = a/b and returns 1; otherwise returns 0</i></font><br>
<br>
<br>
<font color="#008b00"><b>long</b></font> divide(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
<font color="#008b00"><b>long</b></font> divide(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZ& b);<br>
<font color="#008b00"><b>long</b></font> divide(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> b);<br>
<font color="#0000ed"><i>// if b | a, returns 1; otherwise returns 0</i></font><br>
<br>
<font color="#0000ed"><i>// These algorithms employ a modular approach, performing the division</i></font><br>
<font color="#0000ed"><i>// modulo small primes (reconstructing q via the CRT). It is</i></font><br>
<font color="#0000ed"><i>// usually much faster than the general division routines above</i></font><br>
<font color="#0000ed"><i>// (especially when b does not divide a).</i></font><br>
<br>
<br>
<font color="#008b00"><b>void</b></font> content(ZZ& d, <font color="#008b00"><b>const</b></font> ZZX& f);<br>
ZZ content(<font color="#008b00"><b>const</b></font> ZZX& f);<br>
<font color="#0000ed"><i>// d = content of f, sign(d) == sign(LeadCoeff(f)); content(0) == 0</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> PrimitivePart(ZZX& pp, <font color="#008b00"><b>const</b></font> ZZX& f);<br>
ZZX PrimitivePart(<font color="#008b00"><b>const</b></font> ZZX& f); <br>
<font color="#0000ed"><i>// pp = primitive part of f, LeadCoeff(pp) >= 0; PrimitivePart(0) == 0</i></font><br>
<br>
<br>
<br>
<font color="#0000ed"><i>// pseudo-division:</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> PseudoDivRem(ZZX& q, ZZX& r, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
<font color="#0000ed"><i>// performs pseudo-division: computes q and r with deg(r) < deg(b),</i></font><br>
<font color="#0000ed"><i>// and LeadCoeff(b)^(deg(a)-deg(b)+1) a = b q + r. Only the classical</i></font><br>
<font color="#0000ed"><i>// algorithm is used.</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> PseudoDiv(ZZX& q, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
ZZX PseudoDiv(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
<font color="#0000ed"><i>// same as PseudoDivRem, but only computes q</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> PseudoRem(ZZX& r, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
ZZX PseudoRem(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
<font color="#0000ed"><i>// same as PseudoDivRem, but only computes r</i></font><br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> GCD's</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> GCD(ZZX& d, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b);<br>
ZZX GCD(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b); <br>
<font color="#0000ed"><i>// d = gcd(a, b), LeadCoeff(d) >= 0. Uses a modular algorithm.</i></font><br>
<br>
<br>
<font color="#008b00"><b>void</b></font> XGCD(ZZ& r, ZZX& s, ZZX& t, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b, <br>
<font color="#008b00"><b>long</b></font> deterministic=<font color="#ff8b00">0</font>);<br>
<font color="#0000ed"><i>// r = resultant of a and b; if r != 0, then computes s and t such</i></font><br>
<font color="#0000ed"><i>// that: a*s + b*t = r; otherwise s and t not affected. if</i></font><br>
<font color="#0000ed"><i>// !deterministic, then resultant computation may use a randomized</i></font><br>
<font color="#0000ed"><i>// strategy that errs with probability no more than 2^{-80}.</i></font><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/O format:</i></font><br>
<br>
<font color="#0000ed"><i> [a_0 a_1 ... a_n],</i></font><br>
<br>
<font color="#0000ed"><i>represents the polynomial a_0 + a_1*X + ... + a_n*X^n.</i></font><br>
<br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
istream& <font color="#b02f60"><b>operator</b></font>>>(istream& s, ZZX& x);<br>
ostream& <font color="#b02f60"><b>operator</b></font><<(ostream& s, <font color="#008b00"><b>const</b></font> ZZX& a);<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Some utility routines</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> diff(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a); <font color="#0000ed"><i>// x = derivative of a</i></font><br>
ZZX diff(<font color="#008b00"><b>const</b></font> ZZX& a); <br>
<br>
<font color="#008b00"><b>long</b></font> MaxBits(<font color="#008b00"><b>const</b></font> ZZX& f);<br>
<font color="#0000ed"><i>// returns max NumBits of coefficients of f</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> reverse(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> hi);<br>
ZZX reverse(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> hi);<br>
<br>
<font color="#008b00"><b>void</b></font> reverse(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a);<br>
ZZX reverse(<font color="#008b00"><b>const</b></font> ZZX& a);<br>
<br>
<font color="#0000ed"><i>// x = reverse of a[0]..a[hi] (hi >= -1);</i></font><br>
<font color="#0000ed"><i>// hi defaults to deg(a) in second version</i></font><br>
<br>
<br>
<font color="#008b00"><b>void</b></font> VectorCopy(vec_ZZ& x, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> n);<br>
vec_ZZ VectorCopy(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> n);<br>
<font color="#0000ed"><i>// x = copy of coefficient vector of a of length exactly n.</i></font><br>
<font color="#0000ed"><i>// input is truncated or padded with zeroes as appropriate.</i></font><br>
<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Arithmetic mod X^n</i></font><br>
<br>
<font color="#0000ed"><i>All routines require n >= 0, otherwise an error is raised.</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> trunc(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> m); <font color="#0000ed"><i>// x = a % X^m</i></font><br>
ZZX trunc(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> m);<br>
<br>
<font color="#008b00"><b>void</b></font> MulTrunc(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b, <font color="#008b00"><b>long</b></font> n);<br>
ZZX MulTrunc(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b, <font color="#008b00"><b>long</b></font> n);<br>
<font color="#0000ed"><i>// x = a * b % X^n</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> SqrTrunc(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> n);<br>
ZZX SqrTrunc(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> n);<br>
<font color="#0000ed"><i>// x = a^2 % X^n</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> InvTrunc(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> n);<br>
ZZX InvTrunc(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> n);<br>
<font color="#0000ed"><i>// computes x = a^{-1} % X^m. Must have ConstTerm(a) invertible.</i></font><br>
<br>
<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Modular Arithmetic</i></font><br>
<br>
<font color="#0000ed"><i>The modulus f must be monic with deg(f) > 0, </i></font><br>
<font color="#0000ed"><i>and other arguments must have smaller degree.</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> MulMod(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b, <font color="#008b00"><b>const</b></font> ZZX& f);<br>
ZZX MulMod(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b, <font color="#008b00"><b>const</b></font> ZZX& f);<br>
<font color="#0000ed"><i>// x = a * b mod f</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> SqrMod(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& f);<br>
ZZX SqrMod(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& f);<br>
<font color="#0000ed"><i>// x = a^2 mod f</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> MulByXMod(ZZX& x, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& f);<br>
ZZX MulByXMod(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& f);<br>
<font color="#0000ed"><i>// x = a*X mod f</i></font><br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> traces, norms, resultants, discriminants,</i></font><br>
<font color="#0000ed"><i> minimal and characteristic polynomials</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> TraceMod(ZZ& res, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& f);<br>
ZZ TraceMod(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& f);<br>
<font color="#0000ed"><i>// res = trace of (a mod f). f must be monic, 0 < deg(f), deg(a) <</i></font><br>
<font color="#0000ed"><i>// deg(f)</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> TraceVec(vec_ZZ& S, <font color="#008b00"><b>const</b></font> ZZX& f);<br>
vec_ZZ TraceVec(<font color="#008b00"><b>const</b></font> ZZX& f);<br>
<font color="#0000ed"><i>// S[i] = Trace(X^i mod f), for i = 0..deg(f)-1.</i></font><br>
<font color="#0000ed"><i>// f must be a monic polynomial.</i></font><br>
<br>
<br>
<font color="#0000ed"><i>// The following routines use a modular approach.</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> resultant(ZZ& res, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b, <font color="#008b00"><b>long</b></font> deterministic=<font color="#ff8b00">0</font>);<br>
ZZ resultant(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& b, <font color="#008b00"><b>long</b></font> deterministic=<font color="#ff8b00">0</font>);<br>
<font color="#0000ed"><i>// res = resultant of a and b. If !deterministic, then it may use a</i></font><br>
<font color="#0000ed"><i>// randomized strategy that errs with probability no more than</i></font><br>
<font color="#0000ed"><i>// 2^{-80}.</i></font><br>
<br>
<br>
<br>
<font color="#008b00"><b>void</b></font> NormMod(ZZ& res, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& f, <font color="#008b00"><b>long</b></font> deterministic=<font color="#ff8b00">0</font>);<br>
ZZ NormMod(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& f, <font color="#008b00"><b>long</b></font> deterministic=<font color="#ff8b00">0</font>);<br>
<font color="#0000ed"><i>// res = norm of (a mod f). f must be monic, 0 < deg(f), deg(a) <</i></font><br>
<font color="#0000ed"><i>// deg(f). If !deterministic, then it may use a randomized strategy</i></font><br>
<font color="#0000ed"><i>// that errs with probability no more than 2^{-80}.</i></font><br>
<br>
<br>
<br>
<font color="#008b00"><b>void</b></font> discriminant(ZZ& d, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> deterministic=<font color="#ff8b00">0</font>);<br>
ZZ discriminant(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>long</b></font> deterministic=<font color="#ff8b00">0</font>);<br>
<font color="#0000ed"><i>// d = discriminant of a = (-1)^{m(m-1)/2} resultant(a, a')/lc(a),</i></font><br>
<font color="#0000ed"><i>// where m = deg(a). If !deterministic, then it may use a randomized</i></font><br>
<font color="#0000ed"><i>// strategy that errs with probability no more than 2^{-80}.</i></font><br>
<br>
<br>
<font color="#008b00"><b>void</b></font> CharPolyMod(ZZX& g, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& f, <font color="#008b00"><b>long</b></font> deterministic=<font color="#ff8b00">0</font>);<br>
ZZX CharPolyMod(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& f, <font color="#008b00"><b>long</b></font> deterministic=<font color="#ff8b00">0</font>);<br>
<font color="#0000ed"><i>// g = char poly of (a mod f). f must be monic. If !deterministic,</i></font><br>
<font color="#0000ed"><i>// then it may use a randomized strategy that errs with probability no</i></font><br>
<font color="#0000ed"><i>// more than 2^{-80}.</i></font><br>
<br>
<br>
<font color="#008b00"><b>void</b></font> MinPolyMod(ZZX& g, <font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& f);<br>
ZZX MinPolyMod(<font color="#008b00"><b>const</b></font> ZZX& a, <font color="#008b00"><b>const</b></font> ZZX& f);<br>
<font color="#0000ed"><i>// g = min poly of (a mod f). f must be monic, 0 < deg(f), deg(a) <</i></font><br>
<font color="#0000ed"><i>// deg(f). May use a probabilistic strategy that errs with</i></font><br>
<font color="#0000ed"><i>// probability no more than 2^{-80}.</i></font><br>
<br>
<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Incremental Chinese Remaindering</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> CRT(ZZX& a, ZZ& prod, <font color="#008b00"><b>const</b></font> zz_pX& A);<br>
<font color="#008b00"><b>long</b></font> CRT(ZZX& a, ZZ& prod, <font color="#008b00"><b>const</b></font> ZZ_pX& A);<br>
<font color="#0000ed"><i>// Incremental Chinese Remaindering: If p is the current zz_p/ZZ_p modulus with</i></font><br>
<font color="#0000ed"><i>// (p, prod) = 1; Computes a' such that a' = a mod prod and a' = A mod p,</i></font><br>
<font color="#0000ed"><i>// with coefficients in the interval (-p*prod/2, p*prod/2]; </i></font><br>
<font color="#0000ed"><i>// Sets a := a', prod := p*prod, and returns 1 if a's value changed.</i></font><br>
<br>
<br>
<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> vectors of ZZX's</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<font color="#008b00"><b>typedef</b></font> Vec<ZZX> vec_ZZX; <font color="#0000ed"><i>// backward compatibility</i></font><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>
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<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
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<font color="#008b00"><b>void</b></font> clear(ZZX& x); <font color="#0000ed"><i>// x = 0</i></font><br>
<font color="#008b00"><b>void</b></font> set(ZZX& x); <font color="#0000ed"><i>// x = 1</i></font><br>
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<font color="#008b00"><b>void</b></font> ZZX::kill();<br>
<font color="#0000ed"><i>// f.kill() sets f to 0 and frees all memory held by f. Equivalent to</i></font><br>
<font color="#0000ed"><i>// f.rep.kill().</i></font><br>
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ZZX::ZZX(INIT_SIZE_TYPE, <font color="#008b00"><b>long</b></font> n);<br>
<font color="#0000ed"><i>// ZZX(INIT_SIZE, n) initializes to zero, but space is pre-allocated</i></font><br>
<font color="#0000ed"><i>// for n coefficients</i></font><br>
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<font color="#008b00"><b>static</b></font> <font color="#008b00"><b>const</b></font> ZZX& zero();<br>
<font color="#0000ed"><i>// ZZX::zero() is a read-only reference to 0</i></font><br>
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<font color="#008b00"><b>void</b></font> ZZX::swap(ZZX& x);<br>
<font color="#008b00"><b>void</b></font> swap(ZZX& x, ZZX& y); <br>
<font color="#0000ed"><i>// swap (by swapping pointers)</i></font><br>
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ZZX::ZZX(<font color="#008b00"><b>long</b></font> i, <font color="#008b00"><b>const</b></font> ZZ& c); <br>
ZZX::ZZX(<font color="#008b00"><b>long</b></font> i, <font color="#008b00"><b>long</b></font> c); <br>
<font color="#0000ed"><i>// initial value c*X^i, provided for backward compatibility</i></font><br>
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