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GF2EX.cpp.html
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<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment">MODULE: GF2EX</span>
<span class="Comment">SUMMARY:</span>
<span class="Comment">The class GF2EX represents polynomials over GF2E,</span>
<span class="Comment">and so can be used, for example, for arithmentic in GF(2^n)[X].</span>
<span class="Comment">However, except where mathematically necessary (e.g., GCD computations),</span>
<span class="Comment">GF2E need not be a field.</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="PreProc">#include </span><span class="String"><NTL/GF2E.h></span>
<span class="PreProc">#include </span><span class="String"><NTL/vec_GF2E.h></span>
<span class="Type">class</span> GF2EX {
<span class="Statement">public</span>:
GF2EX(); <span class="Comment">// initial value 0</span>
GF2EX(<span class="Type">const</span> GF2EX& a); <span class="Comment">// copy</span>
<span class="Type">explicit</span> GF2EX(<span class="Type">const</span> GF2E& a); <span class="Comment">// promotion</span>
<span class="Type">explicit</span> GF2EX(GF2 a);
<span class="Type">explicit</span> GF2EX(<span class="Type">long</span> a);
GF2EX& <span class="Statement">operator</span>=(<span class="Type">const</span> GF2EX& a); <span class="Comment">// assignment</span>
GF2EX& <span class="Statement">operator</span>=(<span class="Type">const</span> GF2E& a);
GF2EX& <span class="Statement">operator</span>=(GF2 a);
GF2EX& <span class="Statement">operator</span>=(<span class="Type">long</span> a);
~GF2EX(); <span class="Comment">// destructor</span>
GF2EX(INIT_MONO_TYPE, <span class="Type">long</span> i, <span class="Type">const</span> GF2E& c);
GF2EX(INIT_MONO_TYPE, <span class="Type">long</span> i, GF2 c);
GF2EX(INIT_MONO_TYPE, <span class="Type">long</span> i, <span class="Type">long</span> c);
<span class="Comment">// initialize to c*X^i, invoke as GF2EX(INIT_MONO, i)</span>
GF2EX(INIT_MONO_TYPE, <span class="Type">long</span> i);
<span class="Comment">// initialize to X^i, invoke as GF2EX(INIT_MONO, i)</span>
<span class="Comment">// typedefs to aid in generic programming</span>
<span class="Type">typedef</span> GF2E coeff_type;
<span class="Type">typedef</span> GF2EXModulus modulus_type;
<span class="Comment">// ...</span>
};
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Accessing coefficients</span>
<span class="Comment">The degree of a polynomial f is obtained as deg(f),</span>
<span class="Comment">where the zero polynomial, by definition, has degree -1.</span>
<span class="Comment">A polynomial f is represented as a coefficient vector.</span>
<span class="Comment">Coefficients may be accesses in one of two ways.</span>
<span class="Comment">The safe, high-level method is to call the function</span>
<span class="Comment">coeff(f, i) to get the coefficient of X^i in the polynomial f,</span>
<span class="Comment">and to call the function SetCoeff(f, i, a) to set the coefficient</span>
<span class="Comment">of X^i in f to the scalar a.</span>
<span class="Comment">One can also access the coefficients more directly via a lower level </span>
<span class="Comment">interface. The coefficient of X^i in f may be accessed using </span>
<span class="Comment">subscript notation f[i]. In addition, one may write f.SetLength(n)</span>
<span class="Comment">to set the length of the underlying coefficient vector to n,</span>
<span class="Comment">and f.SetMaxLength(n) to allocate space for n coefficients,</span>
<span class="Comment">without changing the coefficient vector itself.</span>
<span class="Comment">After setting coefficients using this low-level interface,</span>
<span class="Comment">one must ensure that leading zeros in the coefficient vector</span>
<span class="Comment">are stripped afterwards by calling the function f.normalize().</span>
<span class="Comment">NOTE: the coefficient vector of f may also be accessed directly</span>
<span class="Comment">as f.rep; however, this is not recommended. Also, for a properly</span>
<span class="Comment">normalized polynomial f, we have f.rep.length() == deg(f)+1,</span>
<span class="Comment">and deg(f) >= 0 => f.rep[deg(f)] != 0.</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">long</span> deg(<span class="Type">const</span> GF2EX& a); <span class="Comment">// return deg(a); deg(0) == -1.</span>
<span class="Type">const</span> GF2E& coeff(<span class="Type">const</span> GF2EX& a, <span class="Type">long</span> i);
<span class="Comment">// returns the coefficient of X^i, or zero if i not in range</span>
<span class="Type">const</span> GF2E& LeadCoeff(<span class="Type">const</span> GF2EX& a);
<span class="Comment">// returns leading term of a, or zero if a == 0</span>
<span class="Type">const</span> GF2E& ConstTerm(<span class="Type">const</span> GF2EX& a);
<span class="Comment">// returns constant term of a, or zero if a == 0</span>
<span class="Type">void</span> SetCoeff(GF2EX& x, <span class="Type">long</span> i, <span class="Type">const</span> GF2E& a);
<span class="Type">void</span> SetCoeff(GF2EX& x, <span class="Type">long</span> i, GF2 a);
<span class="Type">void</span> SetCoeff(GF2EX& x, <span class="Type">long</span> i, <span class="Type">long</span> a);
<span class="Comment">// makes coefficient of X^i equal to a; error is raised if i < 0</span>
<span class="Type">void</span> SetCoeff(GF2EX& x, <span class="Type">long</span> i);
<span class="Comment">// makes coefficient of X^i equal to 1; error is raised if i < 0</span>
<span class="Type">void</span> SetX(GF2EX& x); <span class="Comment">// x is set to the monomial X</span>
<span class="Type">long</span> IsX(<span class="Type">const</span> GF2EX& a); <span class="Comment">// test if x = X</span>
GF2E& GF2EX::<span class="Statement">operator</span>[](<span class="Type">long</span> i);
<span class="Type">const</span> GF2E& GF2EX::<span class="Statement">operator</span>[](<span class="Type">long</span> i) <span class="Type">const</span>;
<span class="Comment">// indexing operators: f[i] is the coefficient of X^i ---</span>
<span class="Comment">// i should satsify i >= 0 and i <= deg(f).</span>
<span class="Comment">// No range checking (unless NTL_RANGE_CHECK is defined).</span>
<span class="Type">void</span> GF2EX::SetLength(<span class="Type">long</span> n);
<span class="Comment">// f.SetLength(n) sets the length of the inderlying coefficient</span>
<span class="Comment">// vector to n --- after this call, indexing f[i] for i = 0..n-1</span>
<span class="Comment">// is valid.</span>
<span class="Type">void</span> GF2EX::normalize();
<span class="Comment">// f.normalize() strips leading zeros from coefficient vector of f</span>
<span class="Type">void</span> GF2EX::SetMaxLength(<span class="Type">long</span> n);
<span class="Comment">// f.SetMaxLength(n) pre-allocate spaces for n coefficients. The</span>
<span class="Comment">// polynomial that f represents is unchanged.</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> GF2EX& a, <span class="Type">const</span> GF2EX& b);
<span class="Type">long</span> <span class="Statement">operator</span>!=(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b);
<span class="Type">long</span> IsZero(<span class="Type">const</span> GF2EX& a); <span class="Comment">// test for 0</span>
<span class="Type">long</span> IsOne(<span class="Type">const</span> GF2EX& a); <span class="Comment">// test for 1</span>
<span class="Comment">// PROMOTIONS: ==, != promote {long,GF2,GF2E} to GF2EX 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>
GF2EX <span class="Statement">operator</span>+(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b);
GF2EX <span class="Statement">operator</span>-(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b);
GF2EX <span class="Statement">operator</span>-(<span class="Type">const</span> GF2EX& a);
GF2EX& <span class="Statement">operator</span>+=(GF2EX& x, <span class="Type">const</span> GF2EX& a);
GF2EX& <span class="Statement">operator</span>+=(GF2EX& x, <span class="Type">const</span> GF2E& a);
GF2EX& <span class="Statement">operator</span>+=(GF2EX& x, GF2 a);
GF2EX& <span class="Statement">operator</span>+=(GF2EX& x, <span class="Type">long</span> a);
GF2EX& <span class="Statement">operator</span>++(GF2EX& x); <span class="Comment">// prefix</span>
<span class="Type">void</span> <span class="Statement">operator</span>++(GF2EX& x, <span class="Type">int</span>); <span class="Comment">// postfix</span>
GF2EX& <span class="Statement">operator</span>-=(GF2EX& x, <span class="Type">const</span> GF2EX& a);
GF2EX& <span class="Statement">operator</span>-=(GF2EX& x, <span class="Type">const</span> GF2E& a);
GF2EX& <span class="Statement">operator</span>-=(GF2EX& x, GF2 a);
GF2EX& <span class="Statement">operator</span>-=(GF2EX& x, <span class="Type">long</span> a);
GF2EX& <span class="Statement">operator</span>--(GF2EX& x); <span class="Comment">// prefix</span>
<span class="Type">void</span> <span class="Statement">operator</span>--(GF2EX& x, <span class="Type">int</span>); <span class="Comment">// postfix</span>
<span class="Comment">// procedural versions:</span>
<span class="Type">void</span> add(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b); <span class="Comment">// x = a + b</span>
<span class="Type">void</span> sub(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b); <span class="Comment">// x = a - b </span>
<span class="Type">void</span> negate(GF2EX& x, <span class="Type">const</span> GF2EX& a); <span class="Comment">// x = - a </span>
<span class="Comment">// PROMOTIONS: +, -, add, sub promote {long,GF2,GF2E} to GF2EX 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>
GF2EX <span class="Statement">operator</span>*(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b);
GF2EX& <span class="Statement">operator</span>*=(GF2EX& x, <span class="Type">const</span> GF2EX& a);
GF2EX& <span class="Statement">operator</span>*=(GF2EX& x, <span class="Type">const</span> GF2E& a);
GF2EX& <span class="Statement">operator</span>*=(GF2EX& x, GF2 a);
GF2EX& <span class="Statement">operator</span>*=(GF2EX& x, <span class="Type">long</span> a);
<span class="Comment">// procedural versions:</span>
<span class="Type">void</span> mul(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b); <span class="Comment">// x = a * b</span>
<span class="Type">void</span> sqr(GF2EX& x, <span class="Type">const</span> GF2EX& a); <span class="Comment">// x = a^2</span>
GF2EX sqr(<span class="Type">const</span> GF2EX& a);
<span class="Comment">// PROMOTIONS: *, mul promote {long,GF2,GF2E} to GF2EX on (a, b).</span>
<span class="Type">void</span> power(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">long</span> e); <span class="Comment">// x = a^e (e >= 0)</span>
GF2EX power(<span class="Type">const</span> GF2EX& a, <span class="Type">long</span> e);
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Shift Operations</span>
<span class="Comment">LeftShift by n means multiplication by X^n</span>
<span class="Comment">RightShift by n means division by X^n</span>
<span class="Comment">A negative shift amount reverses the direction of the shift.</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Comment">// operator notation:</span>
GF2EX <span class="Statement">operator</span><<(<span class="Type">const</span> GF2EX& a, <span class="Type">long</span> n);
GF2EX <span class="Statement">operator</span>>>(<span class="Type">const</span> GF2EX& a, <span class="Type">long</span> n);
GF2EX& <span class="Statement">operator</span><<=(GF2EX& x, <span class="Type">long</span> n);
GF2EX& <span class="Statement">operator</span>>>=(GF2EX& x, <span class="Type">long</span> n);
<span class="Comment">// procedural versions:</span>
<span class="Type">void</span> LeftShift(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">long</span> n);
GF2EX LeftShift(<span class="Type">const</span> GF2EX& a, <span class="Type">long</span> n);
<span class="Type">void</span> RightShift(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">long</span> n);
GF2EX RightShift(<span class="Type">const</span> GF2EX& a, <span class="Type">long</span> n);
<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>
GF2EX <span class="Statement">operator</span>/(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b);
GF2EX <span class="Statement">operator</span>/(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2E& b);
GF2EX <span class="Statement">operator</span>/(<span class="Type">const</span> GF2EX& a, GF2 b);
GF2EX <span class="Statement">operator</span>/(<span class="Type">const</span> GF2EX& a, <span class="Type">long</span> b);
GF2EX <span class="Statement">operator</span>%(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b);
GF2EX& <span class="Statement">operator</span>/=(GF2EX& x, <span class="Type">const</span> GF2EX& a);
GF2EX& <span class="Statement">operator</span>/=(GF2EX& x, <span class="Type">const</span> GF2E& a);
GF2EX& <span class="Statement">operator</span>/=(GF2EX& x, GF2 a);
GF2EX& <span class="Statement">operator</span>/=(GF2EX& x, <span class="Type">long</span> a);
GF2EX& <span class="Statement">operator</span>%=(GF2EX& x, <span class="Type">const</span> GF2EX& a);
<span class="Comment">// procedural versions:</span>
<span class="Type">void</span> DivRem(GF2EX& q, GF2EX& r, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b);
<span class="Comment">// q = a/b, r = a%b</span>
<span class="Type">void</span> div(GF2EX& q, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b);
<span class="Type">void</span> div(GF2EX& q, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2E& b);
<span class="Type">void</span> div(GF2EX& q, <span class="Type">const</span> GF2EX& a, GF2 b);
<span class="Type">void</span> div(GF2EX& q, <span class="Type">const</span> GF2EX& a, <span class="Type">long</span> b);
<span class="Comment">// q = a/b</span>
<span class="Type">void</span> rem(GF2EX& r, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b);
<span class="Comment">// r = a%b</span>
<span class="Type">long</span> divide(GF2EX& q, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b);
<span class="Comment">// if b | a, sets q = a/b and returns 1; otherwise returns 0</span>
<span class="Type">long</span> divide(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b);
<span class="Comment">// if b | a, sets q = a/b and returns 1; otherwise returns 0</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> GCD's</span>
<span class="Comment">These routines are intended for use when GF2E is a field.</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">void</span> GCD(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b);
GF2EX GCD(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b);
<span class="Comment">// x = GCD(a, b), x is always monic (or zero if a==b==0).</span>
<span class="Type">void</span> XGCD(GF2EX& d, GF2EX& s, GF2EX& t, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b);
<span class="Comment">// d = gcd(a,b), a s + b t = d </span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Input/Output</span>
<span class="Comment">I/O format:</span>
<span class="Comment"> [a_0 a_1 ... a_n],</span>
<span class="Comment">represents the polynomial a_0 + a_1*X + ... + a_n*X^n.</span>
<span class="Comment">On output, all coefficients will be polynomials of degree < GF2E::degree() and</span>
<span class="Comment">a_n not zero (the zero polynomial is [ ]). On input, the coefficients</span>
<span class="Comment">are arbitrary polynomials which are reduced modulo GF2E::modulus(), and leading</span>
<span class="Comment">zeros stripped.</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
istream& <span class="Statement">operator</span>>>(istream& s, GF2EX& x);
ostream& <span class="Statement">operator</span><<(ostream& s, <span class="Type">const</span> GF2EX& a);
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Some utility routines</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">void</span> diff(GF2EX& x, <span class="Type">const</span> GF2EX& a); <span class="Comment">// x = derivative of a</span>
GF2EX diff(<span class="Type">const</span> GF2EX& a);
<span class="Type">void</span> MakeMonic(GF2EX& x);
<span class="Comment">// if x != 0 makes x into its monic associate; LeadCoeff(x) must be</span>
<span class="Comment">// invertible in this case</span>
<span class="Type">void</span> reverse(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">long</span> hi);
GF2EX reverse(<span class="Type">const</span> GF2EX& a, <span class="Type">long</span> hi);
<span class="Type">void</span> reverse(GF2EX& x, <span class="Type">const</span> GF2EX& a);
GF2EX reverse(<span class="Type">const</span> GF2EX& a);
<span class="Comment">// x = reverse of a[0]..a[hi] (hi >= -1);</span>
<span class="Comment">// hi defaults to deg(a) in second version</span>
<span class="Type">void</span> VectorCopy(vec_GF2E& x, <span class="Type">const</span> GF2EX& a, <span class="Type">long</span> n);
vec_GF2E VectorCopy(<span class="Type">const</span> GF2EX& a, <span class="Type">long</span> n);
<span class="Comment">// x = copy of coefficient vector of a of length exactly n.</span>
<span class="Comment">// input is truncated or padded with zeroes as appropriate.</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Random Polynomials</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">void</span> random(GF2EX& x, <span class="Type">long</span> n);
GF2EX random_GF2EX(<span class="Type">long</span> n);
<span class="Comment">// x = random polynomial of degree < n </span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Polynomial Evaluation and related problems</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">void</span> BuildFromRoots(GF2EX& x, <span class="Type">const</span> vec_GF2E& a);
GF2EX BuildFromRoots(<span class="Type">const</span> vec_GF2E& a);
<span class="Comment">// computes the polynomial (X-a[0]) ... (X-a[n-1]), where n = a.length()</span>
<span class="Type">void</span> eval(GF2E& b, <span class="Type">const</span> GF2EX& f, <span class="Type">const</span> GF2E& a);
GF2E eval(<span class="Type">const</span> GF2EX& f, <span class="Type">const</span> GF2E& a);
<span class="Comment">// b = f(a)</span>
<span class="Type">void</span> eval(GF2E& b, <span class="Type">const</span> GF2X& f, <span class="Type">const</span> GF2E& a);
GF2E eval(<span class="Type">const</span> GF2EX& f, <span class="Type">const</span> GF2E& a);
<span class="Comment">// b = f(a); uses ModComp algorithm for GF2X</span>
<span class="Type">void</span> eval(vec_GF2E& b, <span class="Type">const</span> GF2EX& f, <span class="Type">const</span> vec_GF2E& a);
vec_GF2E eval(<span class="Type">const</span> GF2EX& f, <span class="Type">const</span> vec_GF2E& a);
<span class="Comment">// b.SetLength(a.length()); b[i] = f(a[i]) for 0 <= i < a.length()</span>
<span class="Type">void</span> interpolate(GF2EX& f, <span class="Type">const</span> vec_GF2E& a, <span class="Type">const</span> vec_GF2E& b);
GF2EX interpolate(<span class="Type">const</span> vec_GF2E& a, <span class="Type">const</span> vec_GF2E& b);
<span class="Comment">// interpolates the polynomial f satisfying f(a[i]) = b[i]. </span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Arithmetic mod X^n</span>
<span class="Comment">Required: n >= 0; otherwise, an error is raised.</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">void</span> trunc(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">long</span> n); <span class="Comment">// x = a % X^n</span>
GF2EX trunc(<span class="Type">const</span> GF2EX& a, <span class="Type">long</span> n);
<span class="Type">void</span> MulTrunc(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b, <span class="Type">long</span> n);
GF2EX MulTrunc(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b, <span class="Type">long</span> n);
<span class="Comment">// x = a * b % X^n</span>
<span class="Type">void</span> SqrTrunc(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">long</span> n);
GF2EX SqrTrunc(<span class="Type">const</span> GF2EX& a, <span class="Type">long</span> n);
<span class="Comment">// x = a^2 % X^n</span>
<span class="Type">void</span> InvTrunc(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">long</span> n);
GF2EX InvTrunc(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">long</span> n);
<span class="Comment">// computes x = a^{-1} % X^m. Must have ConstTerm(a) invertible.</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Modular Arithmetic (without pre-conditioning)</span>
<span class="Comment">Arithmetic mod f.</span>
<span class="Comment">All inputs and outputs are polynomials of degree less than deg(f), and</span>
<span class="Comment">deg(f) > 0.</span>
<span class="Comment">NOTE: if you want to do many computations with a fixed f, use the</span>
<span class="Comment">GF2EXModulus data structure and associated routines below for better</span>
<span class="Comment">performance.</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">void</span> MulMod(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b, <span class="Type">const</span> GF2EX& f);
GF2EX MulMod(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b, <span class="Type">const</span> GF2EX& f);
<span class="Comment">// x = (a * b) % f</span>
<span class="Type">void</span> SqrMod(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& f);
GF2EX SqrMod(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& f);
<span class="Comment">// x = a^2 % f</span>
<span class="Type">void</span> MulByXMod(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& f);
GF2EX MulByXMod(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& f);
<span class="Comment">// x = (a * X) mod f</span>
<span class="Type">void</span> InvMod(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& f);
GF2EX InvMod(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& f);
<span class="Comment">// x = a^{-1} % f, error is a is not invertible</span>
<span class="Type">long</span> InvModStatus(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& f);
<span class="Comment">// if (a, f) = 1, returns 0 and sets x = a^{-1} % f; otherwise,</span>
<span class="Comment">// returns 1 and sets x = (a, f)</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Modular Arithmetic with Pre-Conditioning</span>
<span class="Comment">If you need to do a lot of arithmetic modulo a fixed f, build</span>
<span class="Comment">GF2EXModulus F for f. This pre-computes information about f that</span>
<span class="Comment">speeds up subsequent computations.</span>
<span class="Comment">As an example, the following routine the product modulo f of a vector</span>
<span class="Comment">of polynomials.</span>
<span class="Comment">#include <NTL/GF2EX.h></span>
<span class="Comment">void product(GF2EX& x, const vec_GF2EX& v, const GF2EX& f)</span>
<span class="Comment">{</span>
<span class="Comment"> GF2EXModulus F(f);</span>
<span class="Comment"> GF2EX res;</span>
<span class="Comment"> res = 1;</span>
<span class="Comment"> long i;</span>
<span class="Comment"> for (i = 0; i < v.length(); i++)</span>
<span class="Comment"> MulMod(res, res, v[i], F); </span>
<span class="Comment"> x = res;</span>
<span class="Comment">}</span>
<span class="Comment">NOTE: A GF2EX may be used wherever a GF2EXModulus is required,</span>
<span class="Comment">and a GF2EXModulus may be used wherever a GF2EX is required.</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">class</span> GF2EXModulus {
<span class="Statement">public</span>:
GF2EXModulus(); <span class="Comment">// initially in an unusable state</span>
GF2EXModulus(<span class="Type">const</span> GF2EX& f); <span class="Comment">// initialize with f, deg(f) > 0</span>
GF2EXModulus(<span class="Type">const</span> GF2EXModulus&); <span class="Comment">// copy</span>
GF2EXModulus& <span class="Statement">operator</span>=(<span class="Type">const</span> GF2EXModulus&); <span class="Comment">// assignment</span>
~GF2EXModulus(); <span class="Comment">// destructor</span>
<span class="Statement">operator</span> <span class="Type">const</span> GF2EX& () <span class="Type">const</span>; <span class="Comment">// implicit read-only access to f</span>
<span class="Type">const</span> GF2EX& val() <span class="Type">const</span>; <span class="Comment">// explicit read-only access to f</span>
};
<span class="Type">void</span> build(GF2EXModulus& F, <span class="Type">const</span> GF2EX& f);
<span class="Comment">// pre-computes information about f and stores it in F. Must have</span>
<span class="Comment">// deg(f) > 0. Note that the declaration GF2EXModulus F(f) is</span>
<span class="Comment">// equivalent to GF2EXModulus F; build(F, f).</span>
<span class="Comment">// In the following, f refers to the polynomial f supplied to the</span>
<span class="Comment">// build routine, and n = deg(f).</span>
<span class="Type">long</span> deg(<span class="Type">const</span> GF2EXModulus& F); <span class="Comment">// return n=deg(f)</span>
<span class="Type">void</span> MulMod(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b, <span class="Type">const</span> GF2EXModulus& F);
GF2EX MulMod(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// x = (a * b) % f; deg(a), deg(b) < n</span>
<span class="Type">void</span> SqrMod(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EXModulus& F);
GF2EX SqrMod(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// x = a^2 % f; deg(a) < n</span>
<span class="Type">void</span> PowerMod(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> ZZ& e, <span class="Type">const</span> GF2EXModulus& F);
GF2EX PowerMod(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> ZZ& e, <span class="Type">const</span> GF2EXModulus& F);
<span class="Type">void</span> PowerMod(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">long</span> e, <span class="Type">const</span> GF2EXModulus& F);
GF2EX PowerMod(<span class="Type">const</span> GF2EX& a, <span class="Type">long</span> e, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// x = a^e % f; e >= 0, deg(a) < n. Uses a sliding window algorithm.</span>
<span class="Comment">// (e may be negative)</span>
<span class="Type">void</span> PowerXMod(GF2EX& x, <span class="Type">const</span> ZZ& e, <span class="Type">const</span> GF2EXModulus& F);
GF2EX PowerXMod(<span class="Type">const</span> ZZ& e, <span class="Type">const</span> GF2EXModulus& F);
<span class="Type">void</span> PowerXMod(GF2EX& x, <span class="Type">long</span> e, <span class="Type">const</span> GF2EXModulus& F);
GF2EX PowerXMod(<span class="Type">long</span> e, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// x = X^e % f (e may be negative)</span>
<span class="Type">void</span> rem(GF2EX& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// x = a % f</span>
<span class="Type">void</span> DivRem(GF2EX& q, GF2EX& r, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// q = a/f, r = a%f</span>
<span class="Type">void</span> div(GF2EX& q, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// q = a/f</span>
<span class="Comment">// operator notation:</span>
GF2EX <span class="Statement">operator</span>/(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EXModulus& F);
GF2EX <span class="Statement">operator</span>%(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EXModulus& F);
GF2EX& <span class="Statement">operator</span>/=(GF2EX& x, <span class="Type">const</span> GF2EXModulus& F);
GF2EX& <span class="Statement">operator</span>%=(GF2EX& x, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> vectors of GF2EX's</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">typedef</span> Vec<GF2EX> vec_GF2EX; <span class="Comment">// backward compatibility</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Modular Composition</span>
<span class="Comment">Modular composition is the problem of computing g(h) mod f for</span>
<span class="Comment">polynomials f, g, and h.</span>
<span class="Comment">The algorithm employed is that of Brent & Kung (Fast algorithms for</span>
<span class="Comment">manipulating formal power series, JACM 25:581-595, 1978), which uses</span>
<span class="Comment">O(n^{1/2}) modular polynomial multiplications, and O(n^2) scalar</span>
<span class="Comment">operations.</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">void</span> CompMod(GF2EX& x, <span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EX& h, <span class="Type">const</span> GF2EXModulus& F);
GF2EX CompMod(<span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EX& h,
<span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// x = g(h) mod f; deg(h) < n</span>
<span class="Type">void</span> Comp2Mod(GF2EX& x1, GF2EX& x2, <span class="Type">const</span> GF2EX& g1, <span class="Type">const</span> GF2EX& g2,
<span class="Type">const</span> GF2EX& h, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// xi = gi(h) mod f (i=1,2); deg(h) < n.</span>
<span class="Type">void</span> Comp3Mod(GF2EX& x1, GF2EX& x2, GF2EX& x3,
<span class="Type">const</span> GF2EX& g1, <span class="Type">const</span> GF2EX& g2, <span class="Type">const</span> GF2EX& g3,
<span class="Type">const</span> GF2EX& h, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// xi = gi(h) mod f (i=1..3); deg(h) < n.</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Composition with Pre-Conditioning</span>
<span class="Comment">If a single h is going to be used with many g's then you should build</span>
<span class="Comment">a GF2EXArgument for h, and then use the compose routine below. The</span>
<span class="Comment">routine build computes and stores h, h^2, ..., h^m mod f. After this</span>
<span class="Comment">pre-computation, composing a polynomial of degree roughly n with h</span>
<span class="Comment">takes n/m multiplies mod f, plus n^2 scalar multiplies. Thus,</span>
<span class="Comment">increasing m increases the space requirement and the pre-computation</span>
<span class="Comment">time, but reduces the composition time.</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">struct</span> GF2EXArgument {
vec_GF2EX H;
};
<span class="Type">void</span> build(GF2EXArgument& H, <span class="Type">const</span> GF2EX& h, <span class="Type">const</span> GF2EXModulus& F, <span class="Type">long</span> m);
<span class="Comment">// Pre-Computes information about h. m > 0, deg(h) < n.</span>
<span class="Type">void</span> CompMod(GF2EX& x, <span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXArgument& H,
<span class="Type">const</span> GF2EXModulus& F);
GF2EX CompMod(<span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXArgument& H,
<span class="Type">const</span> GF2EXModulus& F);
<span class="Type">extern</span> <span class="Type">long</span> GF2EXArgBound;
<span class="Comment">// Initially 0. If this is set to a value greater than zero, then</span>
<span class="Comment">// composition routines will allocate a table of no than about</span>
<span class="Comment">// GF2EXArgBound KB. Setting this value affects all compose routines</span>
<span class="Comment">// and the power projection and minimal polynomial routines below, </span>
<span class="Comment">// and indirectly affects many routines in GF2EXFactoring.</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> power projection routines</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">void</span> project(GF2E& x, <span class="Type">const</span> GF2EVector& a, <span class="Type">const</span> GF2EX& b);
GF2E project(<span class="Type">const</span> GF2EVector& a, <span class="Type">const</span> GF2EX& b);
<span class="Comment">// x = inner product of a with coefficient vector of b</span>
<span class="Type">void</span> ProjectPowers(vec_GF2E& x, <span class="Type">const</span> vec_GF2E& a, <span class="Type">long</span> k,
<span class="Type">const</span> GF2EX& h, <span class="Type">const</span> GF2EXModulus& F);
vec_GF2E ProjectPowers(<span class="Type">const</span> vec_GF2E& a, <span class="Type">long</span> k,
<span class="Type">const</span> GF2EX& h, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// Computes the vector</span>
<span class="Comment">// project(a, 1), project(a, h), ..., project(a, h^{k-1} % f). </span>
<span class="Comment">// This operation is the "transpose" of the modular composition operation.</span>
<span class="Type">void</span> ProjectPowers(vec_GF2E& x, <span class="Type">const</span> vec_GF2E& a, <span class="Type">long</span> k,
<span class="Type">const</span> GF2EXArgument& H, <span class="Type">const</span> GF2EXModulus& F);
vec_GF2E ProjectPowers(<span class="Type">const</span> vec_GF2E& a, <span class="Type">long</span> k,
<span class="Type">const</span> GF2EXArgument& H, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// same as above, but uses a pre-computed GF2EXArgument</span>
<span class="Type">class</span> GF2EXTransMultiplier { <span class="Comment">/*</span><span class="Comment"> ... </span><span class="Comment">*/</span> };
<span class="Type">void</span> build(GF2EXTransMultiplier& B, <span class="Type">const</span> GF2EX& b, <span class="Type">const</span> GF2EXModulus& F);
<span class="Type">void</span> UpdateMap(vec_GF2E& x, <span class="Type">const</span> vec_GF2E& a,
<span class="Type">const</span> GF2EXMultiplier& B, <span class="Type">const</span> GF2EXModulus& F);
vec_GF2E UpdateMap(<span class="Type">const</span> vec_GF2E& a,
<span class="Type">const</span> GF2EXMultiplier& B, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// Computes the vector</span>
<span class="Comment">// project(a, b), project(a, (b*X)%f), ..., project(a, (b*X^{n-1})%f)</span>
<span class="Comment">// Restriction: a.length() <= deg(F), deg(b) < deg(F).</span>
<span class="Comment">// This is "transposed" MulMod by B.</span>
<span class="Comment">// Input may have "high order" zeroes stripped.</span>
<span class="Comment">// Output always has high order zeroes stripped.</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Minimum Polynomials</span>
<span class="Comment">These routines should be used only when GF2E is a field.</span>
<span class="Comment">All of these routines implement the algorithm from [Shoup, J. Symbolic</span>
<span class="Comment">Comp. 17:371-391, 1994] and [Shoup, J. Symbolic Comp. 20:363-397,</span>
<span class="Comment">1995], based on transposed modular composition and the</span>
<span class="Comment">Berlekamp/Massey algorithm.</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">void</span> MinPolySeq(GF2EX& h, <span class="Type">const</span> vec_GF2E& a, <span class="Type">long</span> m);
GF2EX MinPolySeq(<span class="Type">const</span> vec_GF2E& a, <span class="Type">long</span> m);
<span class="Comment">// computes the minimum polynomial of a linealy generated sequence; m</span>
<span class="Comment">// is a bound on the degree of the polynomial; required: a.length() >=</span>
<span class="Comment">// 2*m</span>
<span class="Type">void</span> ProbMinPolyMod(GF2EX& h, <span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F, <span class="Type">long</span> m);
GF2EX ProbMinPolyMod(<span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F, <span class="Type">long</span> m);
<span class="Type">void</span> ProbMinPolyMod(GF2EX& h, <span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F);
GF2EX ProbMinPolyMod(<span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// computes the monic minimal polynomial if (g mod f). m = a bound on</span>
<span class="Comment">// the degree of the minimal polynomial; in the second version, this</span>
<span class="Comment">// argument defaults to n. The algorithm is probabilistic, always</span>
<span class="Comment">// returns a divisor of the minimal polynomial, and returns a proper</span>
<span class="Comment">// divisor with probability at most m/2^{GF2E::degree()}.</span>
<span class="Type">void</span> MinPolyMod(GF2EX& h, <span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F, <span class="Type">long</span> m);
GF2EX MinPolyMod(<span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F, <span class="Type">long</span> m);
<span class="Type">void</span> MinPolyMod(GF2EX& h, <span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F);
GF2EX MinPolyMod(<span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// same as above, but guarantees that result is correct</span>
<span class="Type">void</span> IrredPolyMod(GF2EX& h, <span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F, <span class="Type">long</span> m);
GF2EX IrredPolyMod(<span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F, <span class="Type">long</span> m);
<span class="Type">void</span> IrredPolyMod(GF2EX& h, <span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F);
GF2EX IrredPolyMod(<span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// same as above, but assumes that f is irreducible, or at least that</span>
<span class="Comment">// the minimal poly of g is itself irreducible. The algorithm is</span>
<span class="Comment">// deterministic (and is always correct).</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Composition and Minimal Polynomials in towers</span>
<span class="Comment">These are implementations of algorithms that will be described</span>
<span class="Comment">and analyzed in a forthcoming paper.</span>
<span class="Comment">GF2E need not be a field.</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">void</span> CompTower(GF2EX& x, <span class="Type">const</span> GF2X& g, <span class="Type">const</span> GF2EXArgument& h,
<span class="Type">const</span> GF2EXModulus& F);
GF2EX CompTower(<span class="Type">const</span> GF2X& g, <span class="Type">const</span> GF2EXArgument& h,
<span class="Type">const</span> GF2EXModulus& F);
<span class="Type">void</span> CompTower(GF2EX& x, <span class="Type">const</span> GF2X& g, <span class="Type">const</span> GF2EX& h,
<span class="Type">const</span> GF2EXModulus& F);
GF2EX CompTower(<span class="Type">const</span> GF2X& g, <span class="Type">const</span> GF2EX& h,
<span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// x = g(h) mod f</span>
<span class="Type">void</span> ProbMinPolyTower(GF2X& h, <span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F,
<span class="Type">long</span> m);
GF2X ProbMinPolyTower(<span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F, <span class="Type">long</span> m);
<span class="Type">void</span> ProbMinPolyTower(GF2X& h, <span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F);
GF2X ProbMinPolyTower(<span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// Uses a probabilistic algorithm to compute the minimal</span>
<span class="Comment">// polynomial of (g mod f) over GF2.</span>
<span class="Comment">// The parameter m is a bound on the degree of the minimal polynomial</span>
<span class="Comment">// (default = deg(f)*GF2E::degree()).</span>
<span class="Comment">// In general, the result will be a divisor of the true minimimal</span>
<span class="Comment">// polynomial. For correct results, use the MinPoly routines below.</span>
<span class="Type">void</span> MinPolyTower(GF2X& h, <span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F, <span class="Type">long</span> m);
GF2X MinPolyTower(<span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F, <span class="Type">long</span> m);
<span class="Type">void</span> MinPolyTower(GF2X& h, <span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F);
GF2X MinPolyTower(<span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// Same as above, but result is always correct.</span>
<span class="Type">void</span> IrredPolyTower(GF2X& h, <span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F, <span class="Type">long</span> m);
GF2X IrredPolyTower(<span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F, <span class="Type">long</span> m);
<span class="Type">void</span> IrredPolyTower(GF2X& h, <span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F);
GF2X IrredPolyTower(<span class="Type">const</span> GF2EX& g, <span class="Type">const</span> GF2EXModulus& F);
<span class="Comment">// Same as above, but assumes the minimal polynomial is</span>
<span class="Comment">// irreducible, and uses a slightly faster, deterministic algorithm.</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Traces, norms, resultants</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">void</span> TraceMod(GF2E& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EXModulus& F);
GF2E TraceMod(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EXModulus& F);
<span class="Type">void</span> TraceMod(GF2E& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& f);
GF2E TraceMod(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EXModulus& f);
<span class="Comment">// x = Trace(a mod f); deg(a) < deg(f)</span>
<span class="Type">void</span> TraceVec(vec_GF2E& S, <span class="Type">const</span> GF2EX& f);
vec_GF2E TraceVec(<span class="Type">const</span> GF2EX& f);
<span class="Comment">// S[i] = Trace(X^i mod f), i = 0..deg(f)-1; 0 < deg(f)</span>
<span class="Comment">// The above trace routines implement the asymptotically fast trace</span>
<span class="Comment">// algorithm from [von zur Gathen and Shoup, Computational Complexity,</span>
<span class="Comment">// 1992].</span>
<span class="Type">void</span> NormMod(GF2E& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& f);
GF2E NormMod(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& f);
<span class="Comment">// x = Norm(a mod f); 0 < deg(f), deg(a) < deg(f)</span>
<span class="Type">void</span> resultant(GF2E& x, <span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b);
GF2E resultant(<span class="Type">const</span> GF2EX& a, <span class="Type">const</span> GF2EX& b);
<span class="Comment">// x = resultant(a, b)</span>
<span class="Comment">// NormMod and resultant require that GF2E is a field.</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(GF2EX& x) <span class="Comment">// x = 0</span>
<span class="Type">void</span> set(GF2EX& x); <span class="Comment">// x = 1</span>
<span class="Type">void</span> GF2EX::kill();
<span class="Comment">// f.kill() sets f to 0 and frees all memory held by f. Equivalent to</span>
<span class="Comment">// f.rep.kill().</span>
GF2EX::GF2EX(INIT_SIZE_TYPE, <span class="Type">long</span> n);
<span class="Comment">// GF2EX(INIT_SIZE, n) initializes to zero, but space is pre-allocated</span>
<span class="Comment">// for n coefficients</span>
<span class="Type">static</span> <span class="Type">const</span> GF2EX& zero();
<span class="Comment">// GF2EX::zero() is a read-only reference to 0</span>
<span class="Type">void</span> GF2EX::swap(GF2EX& x);
<span class="Type">void</span> swap(GF2EX& x, GF2EX& y);
<span class="Comment">// swap (via "pointer swapping")</span>
GF2EX::GF2EX(<span class="Type">long</span> i, <span class="Type">const</span> GF2E& c);
GF2EX::GF2EX(<span class="Type">long</span> i, GF2 c);
GF2EX::GF2EX(<span class="Type">long</span> i, <span class="Type">long</span> c);
<span class="Comment">// initialize to X^i*c, provided for backward compatibility</span>
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