/**************************************************************************\
MODULE: zz_pX
SUMMARY:
The class zz_pX implements polynomial arithmetic modulo p.
Polynomial arithmetic is implemented using a combination of classical
routines, Karatsuba, and FFT.
\**************************************************************************/
#include "zz_p.h"
#include "vec_zz_p.h"
class zz_pX {
public:
zz_pX(); // initial value 0
zz_pX(const zz_pX& a); // copy
explicit zz_pX(zz_p a); // promotion
explicit zz_pX(long a); // promotion
zz_pX& operator=(const zz_pX& a); // assignment
zz_pX& operator=(zz_p a);
zz_pX& operator=(long a);
~zz_pX(); // destructor
zz_pX(INIT_MONO_TYPE, long i, zz_p c);
zz_pX(INIT_MONO_TYPE, long i, long c);
// initialize to c*X^i, invoke as zz_pX(INIT_MONO, i, c)
zz_pX(INIT_MONO_TYPE, long i);
// initialize to X^i, invoke as zz_pX(INIT_MONO, i)
typedef zz_p coeff_type;
// ...
};
/**************************************************************************\
Accessing coefficients
The degree of a polynomial f is obtained as deg(f),
where the zero polynomial, by definition, has degree -1.
A polynomial f is represented as a coefficient vector.
Coefficients may be accesses in one of two ways.
The safe, high-level method is to call the function
coeff(f, i) to get the coefficient of X^i in the polynomial f,
and to call the function SetCoeff(f, i, a) to set the coefficient
of X^i in f to the scalar a.
One can also access the coefficients more directly via a lower level
interface. The coefficient of X^i in f may be accessed using
subscript notation f[i]. In addition, one may write f.SetLength(n)
to set the length of the underlying coefficient vector to n,
and f.SetMaxLength(n) to allocate space for n coefficients,
without changing the coefficient vector itself.
After setting coefficients using this low-level interface,
one must ensure that leading zeros in the coefficient vector
are stripped afterwards by calling the function f.normalize().
NOTE: the coefficient vector of f may also be accessed directly
as f.rep; however, this is not recommended. Also, for a properly
normalized polynomial f, we have f.rep.length() == deg(f)+1,
and deg(f) >= 0 => f.rep[deg(f)] != 0.
\**************************************************************************/
long deg(const zz_pX& a); // return deg(a); deg(0) == -1.
const zz_p coeff(const zz_pX& a, long i);
// returns the coefficient of X^i, or zero if i not in range
const zz_p LeadCoeff(const zz_pX& a);
// returns leading term of a, or zero if a == 0
const zz_p ConstTerm(const zz_pX& a);
// returns constant term of a, or zero if a == 0
void SetCoeff(zz_pX& x, long i, zz_p a);
void SetCoeff(zz_pX& x, long i, long a);
// makes coefficient of X^i equal to a; error is raised if i < 0
void SetCoeff(zz_pX& x, long i);
// makes coefficient of X^i equal to 1; error is raised if i < 0
void SetX(zz_pX& x); // x is set to the monomial X
long IsX(const zz_pX& a); // test if x = X
zz_p& zz_pX::operator[](long i);
const zz_p& zz_pX::operator[](long i) const;
// indexing operators: f[i] is the coefficient of X^i ---
// i should satsify i >= 0 and i <= deg(f).
// No range checking (unless NTL_RANGE_CHECK is defined).
void zz_pX::SetLength(long n);
// f.SetLength(n) sets the length of the inderlying coefficient
// vector to n --- after this call, indexing f[i] for i = 0..n-1
// is valid.
void zz_pX::normalize();
// f.normalize() strips leading zeros from coefficient vector of f
void zz_pX::SetMaxLength(long n);
// f.SetMaxLength(n) pre-allocate spaces for n coefficients. The
// polynomial that f represents is unchanged.
/**************************************************************************\
Comparison
\**************************************************************************/
long operator==(const zz_pX& a, const zz_pX& b);
long operator!=(const zz_pX& a, const zz_pX& b);
long IsZero(const zz_pX& a); // test for 0
long IsOne(const zz_pX& a); // test for 1
// PROMOTIONS: operators ==, != promote {long, zz_p} to zz_pX on (a, b)
/**************************************************************************\
Addition
\**************************************************************************/
// operator notation:
zz_pX operator+(const zz_pX& a, const zz_pX& b);
zz_pX operator-(const zz_pX& a, const zz_pX& b);
zz_pX operator-(const zz_pX& a); // unary -
zz_pX& operator+=(zz_pX& x, const zz_pX& a);
zz_pX& operator+=(zz_pX& x, zz_p a);
zz_pX& operator+=(zz_pX& x, long a);
zz_pX& operator-=(zz_pX& x, const zz_pX& a);
zz_pX& operator-=(zz_pX& x, zz_p a);
zz_pX& operator-=(zz_pX& x, long a);
zz_pX& operator++(zz_pX& x); // prefix
void operator++(zz_pX& x, int); // postfix
zz_pX& operator--(zz_pX& x); // prefix
void operator--(zz_pX& x, int); // postfix
// procedural versions:
void add(zz_pX& x, const zz_pX& a, const zz_pX& b); // x = a + b
void sub(zz_pX& x, const zz_pX& a, const zz_pX& b); // x = a - b
void negate(zz_pX& x, const zz_pX& a); // x = -a
// PROMOTIONS: binary +, - and procedures add, sub promote {long, zz_p}
// to zz_pX on (a, b).
/**************************************************************************\
Multiplication
\**************************************************************************/
// operator notation:
zz_pX operator*(const zz_pX& a, const zz_pX& b);
zz_pX& operator*=(zz_pX& x, const zz_pX& a);
zz_pX& operator*=(zz_pX& x, zz_p a);
zz_pX& operator*=(zz_pX& x, long a);
// procedural versions:
void mul(zz_pX& x, const zz_pX& a, const zz_pX& b); // x = a * b
void sqr(zz_pX& x, const zz_pX& a); // x = a^2
zz_pX sqr(const zz_pX& a);
// PROMOTIONS: operator * and procedure mul promote {long, zz_p} to zz_pX
// on (a, b).
void power(zz_pX& x, const zz_pX& a, long e); // x = a^e (e >= 0)
zz_pX power(const zz_pX& a, long e);
/**************************************************************************\
Shift Operations
LeftShift by n means multiplication by X^n
RightShift by n means division by X^n
A negative shift amount reverses the direction of the shift.
\**************************************************************************/
// operator notation:
zz_pX operator<<(const zz_pX& a, long n);
zz_pX operator>>(const zz_pX& a, long n);
zz_pX& operator<<=(zz_pX& x, long n);
zz_pX& operator>>=(zz_pX& x, long n);
// procedural versions:
void LeftShift(zz_pX& x, const zz_pX& a, long n);
zz_pX LeftShift(const zz_pX& a, long n);
void RightShift(zz_pX& x, const zz_pX& a, long n);
zz_pX RightShift(const zz_pX& a, long n);
/**************************************************************************\
Division
\**************************************************************************/
// operator notation:
zz_pX operator/(const zz_pX& a, const zz_pX& b);
zz_pX operator%(const zz_pX& a, const zz_pX& b);
zz_pX& operator/=(zz_pX& x, const zz_pX& a);
zz_pX& operator/=(zz_pX& x, zz_p a);
zz_pX& operator/=(zz_pX& x, long a);
zz_pX& operator%=(zz_pX& x, const zz_pX& b);
// procedural versions:
void DivRem(zz_pX& q, zz_pX& r, const zz_pX& a, const zz_pX& b);
// q = a/b, r = a%b
void div(zz_pX& q, const zz_pX& a, const zz_pX& b);
// q = a/b
void rem(zz_pX& r, const zz_pX& a, const zz_pX& b);
// r = a%b
long divide(zz_pX& q, const zz_pX& a, const zz_pX& b);
// if b | a, sets q = a/b and returns 1; otherwise returns 0
long divide(const zz_pX& a, const zz_pX& b);
// if b | a, sets q = a/b and returns 1; otherwise returns 0
// PROMOTIONS: operator / and procedure div promote {long, zz_p} to zz_pX
// on (a, b).
/**************************************************************************\
GCD's
These routines are intended for use when p is prime.
\**************************************************************************/
void GCD(zz_pX& x, const zz_pX& a, const zz_pX& b);
zz_pX GCD(const zz_pX& a, const zz_pX& b);
// x = GCD(a, b), x is always monic (or zero if a==b==0).
void XGCD(zz_pX& d, zz_pX& s, zz_pX& t, const zz_pX& a, const zz_pX& b);
// d = gcd(a,b), a s + b t = d
// NOTE: A classical algorithm is used, switching over to a
// "half-GCD" algorithm for large degree
/**************************************************************************\
Input/Output
I/O format:
[a_0 a_1 ... a_n],
represents the polynomial a_0 + a_1*X + ... + a_n*X^n.
On output, all coefficients will be integers between 0 and p-1, amd
a_n not zero (the zero polynomial is [ ]). On input, the coefficients
are arbitrary integers which are reduced modulo p, and leading zeros
stripped.
\**************************************************************************/
istream& operator>>(istream& s, zz_pX& x);
ostream& operator<<(ostream& s, const zz_pX& a);
/**************************************************************************\
Some utility routines
\**************************************************************************/
void diff(zz_pX& x, const zz_pX& a);
zz_pX diff(const zz_pX& a);
// x = derivative of a
void MakeMonic(zz_pX& x);
// if x != 0 makes x into its monic associate; LeadCoeff(x) must be
// invertible in this case.
void reverse(zz_pX& x, const zz_pX& a, long hi);
zz_pX reverse(const zz_pX& a, long hi);
void reverse(zz_pX& x, const zz_pX& a);
zz_pX reverse(const zz_pX& a);
// x = reverse of a[0]..a[hi] (hi >= -1);
// hi defaults to deg(a) in second version
void VectorCopy(vec_zz_p& x, const zz_pX& a, long n);
vec_zz_p VectorCopy(const zz_pX& a, long n);
// x = copy of coefficient vector of a of length exactly n.
// input is truncated or padded with zeroes as appropriate.
/**************************************************************************\
Random Polynomials
\**************************************************************************/
void random(zz_pX& x, long n);
zz_pX random_zz_pX(long n);
// x = random polynomial of degree < n
/**************************************************************************\
Polynomial Evaluation and related problems
\**************************************************************************/
void BuildFromRoots(zz_pX& x, const vec_zz_p& a);
zz_pX BuildFromRoots(const vec_zz_p& a);
// computes the polynomial (X-a[0]) ... (X-a[n-1]), where n =
// a.length()
void eval(zz_p& b, const zz_pX& f, zz_p a);
zz_p eval(const zz_pX& f, zz_p a);
// b = f(a)
void eval(vec_zz_p& b, const zz_pX& f, const vec_zz_p& a);
vec_zz_p eval(const zz_pX& f, const vec_zz_p& a);
// b.SetLength(a.length()); b[i] = f(a[i]) for 0 <= i < a.length()
void interpolate(zz_pX& f, const vec_zz_p& a, const vec_zz_p& b);
zz_pX interpolate(const vec_zz_p& a, const vec_zz_p& b);
// interpolates the polynomial f satisfying f(a[i]) = b[i]. p should
// be prime.
/**************************************************************************\
Arithmetic mod X^n
It is required that n >= 0, otherwise an error is raised.
\**************************************************************************/
void trunc(zz_pX& x, const zz_pX& a, long n); // x = a % X^n
zz_pX trunc(const zz_pX& a, long n);
void MulTrunc(zz_pX& x, const zz_pX& a, const zz_pX& b, long n);
zz_pX MulTrunc(const zz_pX& a, const zz_pX& b, long n);
// x = a * b % X^n
void SqrTrunc(zz_pX& x, const zz_pX& a, long n);
zz_pX SqrTrunc(const zz_pX& a, long n);
// x = a^2 % X^n
void InvTrunc(zz_pX& x, const zz_pX& a, long n);
zz_pX InvTrunc(const zz_pX& a, long n);
// computes x = a^{-1} % X^n. Must have ConstTerm(a) invertible.
/**************************************************************************\
Modular Arithmetic (without pre-conditioning)
Arithmetic mod f.
All inputs and outputs are polynomials of degree less than deg(f), and
deg(f) > 0.
NOTE: if you want to do many computations with a fixed f, use the
zz_pXModulus data structure and associated routines below for better
performance.
\**************************************************************************/
void MulMod(zz_pX& x, const zz_pX& a, const zz_pX& b, const zz_pX& f);
zz_pX MulMod(const zz_pX& a, const zz_pX& b, const zz_pX& f);
// x = (a * b) % f
void SqrMod(zz_pX& x, const zz_pX& a, const zz_pX& f);
zz_pX SqrMod(const zz_pX& a, const zz_pX& f);
// x = a^2 % f
void MulByXMod(zz_pX& x, const zz_pX& a, const zz_pX& f);
zz_pX MulByXMod(const zz_pX& a, const zz_pX& f);
// x = (a * X) mod f
void InvMod(zz_pX& x, const zz_pX& a, const zz_pX& f);
zz_pX InvMod(const zz_pX& a, const zz_pX& f);
// x = a^{-1} % f, error is a is not invertible
long InvModStatus(zz_pX& x, const zz_pX& a, const zz_pX& f);
// if (a, f) = 1, returns 0 and sets x = a^{-1} % f; otherwise,
// returns 1 and sets x = (a, f)
// for modular exponentiation, see below
/**************************************************************************\
Modular Arithmetic with Pre-Conditioning
If you need to do a lot of arithmetic modulo a fixed f, build
zz_pXModulus F for f. This pre-computes information about f that
speeds up subsequent computations. Required: deg(f) > 0 and LeadCoeff(f)
invertible.
As an example, the following routine computes the product modulo f of a vector
of polynomials.
#include "zz_pX.h"
void product(zz_pX& x, const vec_zz_pX& v, const zz_pX& f)
{
zz_pXModulus F(f);
zz_pX res;
res = 1;
long i;
for (i = 0; i < v.length(); i++)
MulMod(res, res, v[i], F);
x = res;
}
Note that automatic conversions are provided so that a zz_pX can
be used wherever a zz_pXModulus is required, and a zz_pXModulus
can be used wherever a zz_pX is required.
\**************************************************************************/
class zz_pXModulus {
public:
zz_pXModulus(); // initially in an unusable state
~zz_pXModulus();
zz_pXModulus(const zz_pXModulus&); // copy
zz_pXModulus& operator=(const zz_pXModulus&); // assignment
zz_pXModulus(const zz_pX& f); // initialize with f, deg(f) > 0
operator const zz_pX& () const;
// read-only access to f, implicit conversion operator
const zz_pX& val() const;
// read-only access to f, explicit notation
};
void build(zz_pXModulus& F, const zz_pX& f);
// pre-computes information about f and stores it in F.
// Note that the declaration zz_pXModulus F(f) is equivalent to
// zz_pXModulus F; build(F, f).
// In the following, f refers to the polynomial f supplied to the
// build routine, and n = deg(f).
long deg(const zz_pXModulus& F); // return deg(f)
void MulMod(zz_pX& x, const zz_pX& a, const zz_pX& b, const zz_pXModulus& F);
zz_pX MulMod(const zz_pX& a, const zz_pX& b, const zz_pXModulus& F);
// x = (a * b) % f; deg(a), deg(b) < n
void SqrMod(zz_pX& x, const zz_pX& a, const zz_pXModulus& F);
zz_pX SqrMod(const zz_pX& a, const zz_pXModulus& F);
// x = a^2 % f; deg(a) < n
void PowerMod(zz_pX& x, const zz_pX& a, const ZZ& e, const zz_pXModulus& F);
zz_pX PowerMod(const zz_pX& a, const ZZ& e, const zz_pXModulus& F);
void PowerMod(zz_pX& x, const zz_pX& a, long e, const zz_pXModulus& F);
zz_pX PowerMod(const zz_pX& a, long e, const zz_pXModulus& F);
// x = a^e % f; deg(a) < n (e may be negative)
void PowerXMod(zz_pX& x, const ZZ& e, const zz_pXModulus& F);
zz_pX PowerXMod(const ZZ& e, const zz_pXModulus& F);
void PowerXMod(zz_pX& x, long e, const zz_pXModulus& F);
zz_pX PowerXMod(long e, const zz_pXModulus& F);
// x = X^e % f (e may be negative)
void PowerXPlusAMod(zz_pX& x, const zz_p& a, const ZZ& e,
const zz_pXModulus& F);
zz_pX PowerXPlusAMod(const zz_p& a, const ZZ& e,
const zz_pXModulus& F);
void PowerXPlusAMod(zz_pX& x, const zz_p& a, long e,
const zz_pXModulus& F);
zz_pX PowerXPlusAMod(const zz_p& a, long e,
const zz_pXModulus& F);
// x = (X + a)^e % f (e may be negative)
void rem(zz_pX& x, const zz_pX& a, const zz_pXModulus& F);
// x = a % f
void DivRem(zz_pX& q, zz_pX& r, const zz_pX& a, const zz_pXModulus& F);
// q = a/f, r = a%f
void div(zz_pX& q, const zz_pX& a, const zz_pXModulus& F);
// q = a/f
// operator notation:
zz_pX operator/(const zz_pX& a, const zz_pXModulus& F);
zz_pX operator%(const zz_pX& a, const zz_pXModulus& F);
zz_pX& operator/=(zz_pX& x, const zz_pXModulus& F);
zz_pX& operator%=(zz_pX& x, const zz_pXModulus& F);
/**************************************************************************\
More Pre-Conditioning
If you need to compute a * b % f for a fixed b, but for many a's, it
is much more efficient to first build a zz_pXMultiplier B for b, and
then use the MulMod routine below.
Here is an example that multiplies each element of a vector by a fixed
polynomial modulo f.
#include "zz_pX.h"
void mul(vec_zz_pX& v, const zz_pX& b, const zz_pX& f)
{
zz_pXModulus F(f);
zz_pXMultiplier B(b, F);
long i;
for (i = 0; i < v.length(); i++)
MulMod(v[i], v[i], B, F);
}
Note that a (trivial) conversion operator from zz_pXMultiplier to zz_pX
is provided, so that a zz_pXMultiplier can be used in a context
where a zz_pX is required.
\**************************************************************************/
class zz_pXMultiplier {
public:
zz_pXMultiplier(); // initially zero
zz_pXMultiplier(const zz_pX& b, const zz_pXModulus& F);
// initializes with b mod F, where deg(b) < deg(F)
zz_pXMultiplier(const zz_pXMultiplier&);
zz_pXMultiplier& operator=(const zz_pXMultiplier&);
~zz_pXMultiplier();
const zz_pX& val() const; // read-only access to b
};
void build(zz_pXMultiplier& B, const zz_pX& b, const zz_pXModulus& F);
// pre-computes information about b and stores it in B; deg(b) <
// deg(F)
void MulMod(zz_pX& x, const zz_pX& a, const zz_pXMultiplier& B,
const zz_pXModulus& F);
zz_pX MulMod(const zz_pX& a, const zz_pXMultiplier& B,
const zz_pXModulus& F);
// x = (a * b) % F; deg(a) < deg(F)
/**************************************************************************\
vectors of zz_pX's
\**************************************************************************/
typedef Vec<zz_pX> vec_zz_pX; // backward compatibility
/**************************************************************************\
Modular Composition
Modular composition is the problem of computing g(h) mod f for
polynomials f, g, and h.
The algorithm employed is that of Brent & Kung (Fast algorithms for
manipulating formal power series, JACM 25:581-595, 1978), which uses
O(n^{1/2}) modular polynomial multiplications, and O(n^2) scalar
operations.
\**************************************************************************/
void CompMod(zz_pX& x, const zz_pX& g, const zz_pX& h, const zz_pXModulus& F);
zz_pX CompMod(const zz_pX& g, const zz_pX& h, const zz_pXModulus& F);
// x = g(h) mod f; deg(h) < n
void Comp2Mod(zz_pX& x1, zz_pX& x2, const zz_pX& g1, const zz_pX& g2,
const zz_pX& h, const zz_pXModulus& F);
// xi = gi(h) mod f (i=1,2), deg(h) < n.
void CompMod3(zz_pX& x1, zz_pX& x2, zz_pX& x3,
const zz_pX& g1, const zz_pX& g2, const zz_pX& g3,
const zz_pX& h, const zz_pXModulus& F);
// xi = gi(h) mod f (i=1..3), deg(h) < n
/**************************************************************************\
Composition with Pre-Conditioning
If a single h is going to be used with many g's then you should build
a zz_pXArgument for h, and then use the compose routine below. The
routine build computes and stores h, h^2, ..., h^m mod f. After this
pre-computation, composing a polynomial of degree roughly n with h
takes n/m multiplies mod f, plus n^2 scalar multiplies. Thus,
increasing m increases the space requirement and the pre-computation
time, but reduces the composition time.
\**************************************************************************/
struct zz_pXArgument {
vec_zz_pX H;
};
void build(zz_pXArgument& H, const zz_pX& h, const zz_pXModulus& F, long m);
// Pre-Computes information about h. m > 0, deg(h) < n
void CompMod(zz_pX& x, const zz_pX& g, const zz_pXArgument& H,
const zz_pXModulus& F);
zz_pX CompMod(const zz_pX& g, const zz_pXArgument& H,
const zz_pXModulus& F);
extern long zz_pXArgBound;
// Initially 0. If this is set to a value greater than zero, then
// composition routines will allocate a table of no than about
// zz_pXArgBound KB. Setting this value affects all compose routines
// and the power projection and minimal polynomial routines below,
// and indirectly affects many routines in zz_pXFactoring.
/**************************************************************************\
power projection routines
\**************************************************************************/
void project(zz_p& x, const zz_pVector& a, const zz_pX& b);
zz_p project(const zz_pVector& a, const zz_pX& b);
// x = inner product of a with coefficient vector of b
void ProjectPowers(vec_zz_p& x, const vec_zz_p& a, long k,
const zz_pX& h, const zz_pXModulus& F);
vec_zz_p ProjectPowers(const vec_zz_p& a, long k,
const zz_pX& h, const zz_pXModulus& F);
// Computes the vector
// project(a, 1), project(a, h), ..., project(a, h^{k-1} % f).
// This operation is the "transpose" of the modular composition operation.
// Input and output may have "high order" zeroes stripped.
void ProjectPowers(vec_zz_p& x, const vec_zz_p& a, long k,
const zz_pXArgument& H, const zz_pXModulus& F);
vec_zz_p ProjectPowers(const vec_zz_p& a, long k,
const zz_pXArgument& H, const zz_pXModulus& F);
// same as above, but uses a pre-computed zz_pXArgument
void UpdateMap(vec_zz_p& x, const vec_zz_p& a,
const zz_pXMultiplier& B, const zz_pXModulus& F);
vec_zz_p UpdateMap(const vec_zz_p& a,
const zz_pXMultiplier& B, const zz_pXModulus& F);
// Computes the vector
// project(a, b), project(a, (b*X)%f), ..., project(a, (b*X^{n-1})%f)
// Restriction: a.length() <= deg(F).
// This is "transposed" MulMod by B.
// Input vector may have "high order" zeroes striped.
// The output will always have high order zeroes stripped.
/**************************************************************************\
Minimum Polynomials
These routines should be used with prime p.
All of these routines implement the algorithm from [Shoup, J. Symbolic
Comp. 17:371-391, 1994] and [Shoup, J. Symbolic Comp. 20:363-397,
1995], based on transposed modular composition and the
Berlekamp/Massey algorithm.
\**************************************************************************/
void MinPolySeq(zz_pX& h, const vec_zz_p& a, long m);
// computes the minimum polynomial of a linealy generated sequence; m
// is a bound on the degree of the polynomial; required: a.length() >=
// 2*m
void ProbMinPolyMod(zz_pX& h, const zz_pX& g, const zz_pXModulus& F, long m);
zz_pX ProbMinPolyMod(const zz_pX& g, const zz_pXModulus& F, long m);
void ProbMinPolyMod(zz_pX& h, const zz_pX& g, const zz_pXModulus& F);
zz_pX ProbMinPolyMod(const zz_pX& g, const zz_pXModulus& F);
// computes the monic minimal polynomial if (g mod f). m = a bound on
// the degree of the minimal polynomial; in the second version, this
// argument defaults to n. The algorithm is probabilistic, always
// returns a divisor of the minimal polynomial, and returns a proper
// divisor with probability at most m/p.
void MinPolyMod(zz_pX& h, const zz_pX& g, const zz_pXModulus& F, long m);
zz_pX MinPolyMod(const zz_pX& g, const zz_pXModulus& F, long m);
void MinPolyMod(zz_pX& h, const zz_pX& g, const zz_pXModulus& F);
zz_pX MinPolyMod(const zz_pX& g, const zz_pXModulus& F);
// same as above, but guarantees that result is correct
void IrredPoly(zz_pX& h, const zz_pX& g, const zz_pXModulus& F, long m);
zz_pX IrredPoly(const zz_pX& g, const zz_pXModulus& F, long m);
void IrredPoly(zz_pX& h, const zz_pX& g, const zz_pXModulus& F);
zz_pX IrredPoly(const zz_pX& g, const zz_pXModulus& F);
// same as above, but assumes that f is irreducible, or at least that
// the minimal poly of g is itself irreducible. The algorithm is
// deterministic (and is always correct).
/**************************************************************************\
Traces, norms, resultants
These routines should be used with prime p.
\**************************************************************************/
void TraceMod(zz_p& x, const zz_pX& a, const zz_pXModulus& F);
zz_p TraceMod(const zz_pX& a, const zz_pXModulus& F);
void TraceMod(zz_p& x, const zz_pX& a, const zz_pX& f);
zz_p TraceMod(const zz_pX& a, const zz_pXModulus& f);
// x = Trace(a mod f); deg(a) < deg(f)
void TraceVec(vec_zz_p& S, const zz_pX& f);
vec_zz_p TraceVec(const zz_pX& f);
// S[i] = Trace(X^i mod f), i = 0..deg(f)-1; 0 < deg(f)
// The above routines implement the asymptotically fast trace
// algorithm from [von zur Gathen and Shoup, Computational Complexity,
// 1992].
void NormMod(zz_p& x, const zz_pX& a, const zz_pX& f);
zz_p NormMod(const zz_pX& a, const zz_pX& f);
// x = Norm(a mod f); 0 < deg(f), deg(a) < deg(f)
void resultant(zz_p& x, const zz_pX& a, const zz_pX& b);
zz_pX resultant(zz_p& x, const zz_pX& a, const zz_pX& b);
// x = resultant(a, b)
void CharPolyMod(zz_pX& g, const zz_pX& a, const zz_pX& f);
zz_pX CharPolyMod(const zz_pX& a, const zz_pX& f);
// g = charcteristic polynomial of (a mod f); 0 < deg(f), deg(g) <
// deg(f). This routine works for arbitrary f. For irreducible f,
// is it faster to use IrredPolyMod, and then exponentiate as
// necessary, since in this case the characterstic polynomial
// is a power of the minimal polynomial.
/**************************************************************************\
Miscellany
\**************************************************************************/
void clear(zz_pX& x) // x = 0
void set(zz_pX& x); // x = 1
void zz_pX::kill();
// f.kill() sets f to 0 and frees all memory held by f. Equivalent to
// f.rep.kill().
zz_pX::zz_pX(INIT_SIZE_TYPE, long n);
// zz_pX(INIT_SIZE, n) initializes to zero, but space is pre-allocated
// for n coefficients
static const zz_pX& zero();
// zz_pX::zero() is a read-only reference to 0
void swap(zz_pX& x, zz_pX& y);
// swap x and y (via "pointer swapping")
zz_pX::zz_pX(long i, zz_p c);
zz_pX::zz_pX(long i, long c);
// initialize to c*X^i, provided for backward compatibility