/**************************************************************************\
MODULE: zz_pEX
SUMMARY:
The class zz_pEX represents polynomials over zz_pE,
and so can be used, for example, for arithmentic in GF(p^n)[X].
However, except where mathematically necessary (e.g., GCD computations),
zz_pE need not be a field.
\**************************************************************************/
#include <NTL/lzz_pE.h>
#include <NTL/vec_lzz_pE.h>
class zz_pEX {
public:
zz_pEX(); // initial value 0
zz_pEX(const zz_pEX& a); // copy
zz_pEX(const zz_pE& a); // promotion
zz_pEX(const zz_p& a);
zz_pEX(long a);
zz_pEX& operator=(const zz_pEX& a); // assignment
zz_pEX& operator=(const zz_pE& a);
zz_pEX& operator=(const zz_p& a);
zz_pEX& operator=(long a);
~zz_pEX(); // destructor
zz_pEX(INIT_MONO_TYPE, long i, const zz_pE& c);
zz_pEX(INIT_MONO_TYPE, long i, const zz_p& c);
zz_pEX(INIT_MONO_TYPE, long i, long c);
// initilaize to c*X^i; invoke as zz_pEX(INIT_MONO, i, c)
zz_pEX(INIT_MONO_TYPE, long i);
// initilaize to X^i; invoke as zz_pEX(INIT_MONO, i)
// typedefs to aid in generic programming
typedef zz_pE coeff_type;
typedef zz_pEXModulus modulus_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_pEX& a); // return deg(a); deg(0) == -1.
const zz_pE& coeff(const zz_pEX& a, long i);
// returns the coefficient of X^i, or zero if i not in range
const zz_pE& LeadCoeff(const zz_pEX& a);
// returns leading term of a, or zero if a == 0
const zz_pE& ConstTerm(const zz_pEX& a);
// returns constant term of a, or zero if a == 0
void SetCoeff(zz_pEX& x, long i, const zz_pE& a);
void SetCoeff(zz_pEX& x, long i, const zz_p& a);
void SetCoeff(zz_pEX& x, long i, long a);
// makes coefficient of X^i equal to a; error is raised if i < 0
void SetCoeff(zz_pEX& x, long i);
// makes coefficient of X^i equal to 1; error is raised if i < 0
void SetX(zz_pEX& x); // x is set to the monomial X
long IsX(const zz_pEX& a); // test if x = X
zz_pE& zz_pEX::operator[](long i);
const zz_pE& zz_pEX::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_pEX::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_pEX::normalize();
// f.normalize() strips leading zeros from coefficient vector of f
void zz_pEX::SetMaxLength(long n);
// f.SetMaxLength(n) pre-allocate spaces for n coefficients. The
// polynomial that f represents is unchanged.
/**************************************************************************\
Comparison
\**************************************************************************/
long operator==(const zz_pEX& a, const zz_pEX& b);
long operator!=(const zz_pEX& a, const zz_pEX& b);
long IsZero(const zz_pEX& a); // test for 0
long IsOne(const zz_pEX& a); // test for 1
// PROMOTIONS: ==, != promote {long,zz_p,zz_pE} to zz_pEX on (a, b).
/**************************************************************************\
Addition
\**************************************************************************/
// operator notation:
zz_pEX operator+(const zz_pEX& a, const zz_pEX& b);
zz_pEX operator-(const zz_pEX& a, const zz_pEX& b);
zz_pEX operator-(const zz_pEX& a);
zz_pEX& operator+=(zz_pEX& x, const zz_pEX& a);
zz_pEX& operator+=(zz_pEX& x, const zz_pE& a);
zz_pEX& operator+=(zz_pEX& x, const zz_p& a);
zz_pEX& operator+=(zz_pEX& x, long a);
zz_pEX& operator++(zz_pEX& x); // prefix
void operator++(zz_pEX& x, int); // postfix
zz_pEX& operator-=(zz_pEX& x, const zz_pEX& a);
zz_pEX& operator-=(zz_pEX& x, const zz_pE& a);
zz_pEX& operator-=(zz_pEX& x, const zz_p& a);
zz_pEX& operator-=(zz_pEX& x, long a);
zz_pEX& operator--(zz_pEX& x); // prefix
void operator--(zz_pEX& x, int); // postfix
// procedural versions:
void add(zz_pEX& x, const zz_pEX& a, const zz_pEX& b); // x = a + b
void sub(zz_pEX& x, const zz_pEX& a, const zz_pEX& b); // x = a - b
void negate(zz_pEX& x, const zz_pEX& a); // x = - a
// PROMOTIONS: +, -, add, sub promote {long,zz_p,zz_pE} to zz_pEX on (a, b).
/**************************************************************************\
Multiplication
\**************************************************************************/
// operator notation:
zz_pEX operator*(const zz_pEX& a, const zz_pEX& b);
zz_pEX& operator*=(zz_pEX& x, const zz_pEX& a);
zz_pEX& operator*=(zz_pEX& x, const zz_pE& a);
zz_pEX& operator*=(zz_pEX& x, const zz_p& a);
zz_pEX& operator*=(zz_pEX& x, long a);
// procedural versions:
void mul(zz_pEX& x, const zz_pEX& a, const zz_pEX& b); // x = a * b
void sqr(zz_pEX& x, const zz_pEX& a); // x = a^2
zz_pEX sqr(const zz_pEX& a);
// PROMOTIONS: *, mul promote {long,zz_p,zz_pE} to zz_pEX on (a, b).
void power(zz_pEX& x, const zz_pEX& a, long e); // x = a^e (e >= 0)
zz_pEX power(const zz_pEX& 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_pEX operator<<(const zz_pEX& a, long n);
zz_pEX operator>>(const zz_pEX& a, long n);
zz_pEX& operator<<=(zz_pEX& x, long n);
zz_pEX& operator>>=(zz_pEX& x, long n);
// procedural versions:
void LeftShift(zz_pEX& x, const zz_pEX& a, long n);
zz_pEX LeftShift(const zz_pEX& a, long n);
void RightShift(zz_pEX& x, const zz_pEX& a, long n);
zz_pEX RightShift(const zz_pEX& a, long n);
/**************************************************************************\
Division
\**************************************************************************/
// operator notation:
zz_pEX operator/(const zz_pEX& a, const zz_pEX& b);
zz_pEX operator/(const zz_pEX& a, const zz_pE& b);
zz_pEX operator/(const zz_pEX& a, const zz_p& b);
zz_pEX operator/(const zz_pEX& a, long b);
zz_pEX operator%(const zz_pEX& a, const zz_pEX& b);
zz_pEX& operator/=(zz_pEX& x, const zz_pEX& a);
zz_pEX& operator/=(zz_pEX& x, const zz_pE& a);
zz_pEX& operator/=(zz_pEX& x, const zz_p& a);
zz_pEX& operator/=(zz_pEX& x, long a);
zz_pEX& operator%=(zz_pEX& x, const zz_pEX& a);
// procedural versions:
void DivRem(zz_pEX& q, zz_pEX& r, const zz_pEX& a, const zz_pEX& b);
// q = a/b, r = a%b
void div(zz_pEX& q, const zz_pEX& a, const zz_pEX& b);
void div(zz_pEX& q, const zz_pEX& a, const zz_pE& b);
void div(zz_pEX& q, const zz_pEX& a, const zz_p& b);
void div(zz_pEX& q, const zz_pEX& a, long b);
// q = a/b
void rem(zz_pEX& r, const zz_pEX& a, const zz_pEX& b);
// r = a%b
long divide(zz_pEX& q, const zz_pEX& a, const zz_pEX& b);
// if b | a, sets q = a/b and returns 1; otherwise returns 0
long divide(const zz_pEX& a, const zz_pEX& b);
// if b | a, sets q = a/b and returns 1; otherwise returns 0
/**************************************************************************\
GCD's
These routines are intended for use when zz_pE is a field.
\**************************************************************************/
void GCD(zz_pEX& x, const zz_pEX& a, const zz_pEX& b);
zz_pEX GCD(const zz_pEX& a, const zz_pEX& b);
// x = GCD(a, b), x is always monic (or zero if a==b==0).
void XGCD(zz_pEX& d, zz_pEX& s, zz_pEX& t, const zz_pEX& a, const zz_pEX& b);
// d = gcd(a,b), a s + b t = d
/**************************************************************************\
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 polynomials of degree < zz_pE::degree() and
a_n not zero (the zero polynomial is [ ]). On input, the coefficients
are arbitrary polynomials which are reduced modulo zz_pE::modulus(),
and leading zeros stripped.
\**************************************************************************/
istream& operator>>(istream& s, zz_pEX& x);
ostream& operator<<(ostream& s, const zz_pEX& a);
/**************************************************************************\
Some utility routines
\**************************************************************************/
void diff(zz_pEX& x, const zz_pEX& a); // x = derivative of a
zz_pEX diff(const zz_pEX& a);
void MakeMonic(zz_pEX& x);
// if x != 0 makes x into its monic associate; LeadCoeff(x) must be
// invertible in this case
void reverse(zz_pEX& x, const zz_pEX& a, long hi);
zz_pEX reverse(const zz_pEX& a, long hi);
void reverse(zz_pEX& x, const zz_pEX& a);
zz_pEX reverse(const zz_pEX& a);
// x = reverse of a[0]..a[hi] (hi >= -1);
// hi defaults to deg(a) in second version
void VectorCopy(vec_zz_pE& x, const zz_pEX& a, long n);
vec_zz_pE VectorCopy(const zz_pEX& 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_pEX& x, long n);
zz_pEX random_zz_pEX(long n);
// x = random polynomial of degree < n
/**************************************************************************\
Polynomial Evaluation and related problems
\**************************************************************************/
void BuildFromRoots(zz_pEX& x, const vec_zz_pE& a);
zz_pEX BuildFromRoots(const vec_zz_pE& a);
// computes the polynomial (X-a[0]) ... (X-a[n-1]), where n = a.length()
void eval(zz_pE& b, const zz_pEX& f, const zz_pE& a);
zz_pE eval(const zz_pEX& f, const zz_pE& a);
// b = f(a)
void eval(zz_pE& b, const zz_pX& f, const zz_pE& a);
zz_pE eval(const zz_pEX& f, const zz_pE& a);
// b = f(a); uses ModComp algorithm for zz_pX
void eval(vec_zz_pE& b, const zz_pEX& f, const vec_zz_pE& a);
vec_zz_pE eval(const zz_pEX& f, const vec_zz_pE& a);
// b.SetLength(a.length()); b[i] = f(a[i]) for 0 <= i < a.length()
void interpolate(zz_pEX& f, const vec_zz_pE& a, const vec_zz_pE& b);
zz_pEX interpolate(const vec_zz_pE& a, const vec_zz_pE& b);
// interpolates the polynomial f satisfying f(a[i]) = b[i].
/**************************************************************************\
Arithmetic mod X^n
Required: n >= 0; otherwise, an error is raised.
\**************************************************************************/
void trunc(zz_pEX& x, const zz_pEX& a, long n); // x = a % X^n
zz_pEX trunc(const zz_pEX& a, long n);
void MulTrunc(zz_pEX& x, const zz_pEX& a, const zz_pEX& b, long n);
zz_pEX MulTrunc(const zz_pEX& a, const zz_pEX& b, long n);
// x = a * b % X^n
void SqrTrunc(zz_pEX& x, const zz_pEX& a, long n);
zz_pEX SqrTrunc(const zz_pEX& a, long n);
// x = a^2 % X^n
void InvTrunc(zz_pEX& x, const zz_pEX& a, long n);
zz_pEX InvTrunc(zz_pEX& x, const zz_pEX& a, long n);
// computes x = a^{-1} % X^m. 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_pEXModulus data structure and associated routines below for better
performance.
\**************************************************************************/
void MulMod(zz_pEX& x, const zz_pEX& a, const zz_pEX& b, const zz_pEX& f);
zz_pEX MulMod(const zz_pEX& a, const zz_pEX& b, const zz_pEX& f);
// x = (a * b) % f
void SqrMod(zz_pEX& x, const zz_pEX& a, const zz_pEX& f);
zz_pEX SqrMod(const zz_pEX& a, const zz_pEX& f);
// x = a^2 % f
void MulByXMod(zz_pEX& x, const zz_pEX& a, const zz_pEX& f);
zz_pEX MulByXMod(const zz_pEX& a, const zz_pEX& f);
// x = (a * X) mod f
void InvMod(zz_pEX& x, const zz_pEX& a, const zz_pEX& f);
zz_pEX InvMod(const zz_pEX& a, const zz_pEX& f);
// x = a^{-1} % f, error is a is not invertible
long InvModStatus(zz_pEX& x, const zz_pEX& a, const zz_pEX& f);
// if (a, f) = 1, returns 0 and sets x = a^{-1} % f; otherwise,
// returns 1 and sets x = (a, f)
/**************************************************************************\
Modular Arithmetic with Pre-Conditioning
If you need to do a lot of arithmetic modulo a fixed f, build
zz_pEXModulus F for f. This pre-computes information about f that
speeds up subsequent computations.
As an example, the following routine the product modulo f of a vector
of polynomials.
#include <NTL/lzz_pEX.h>
void product(zz_pEX& x, const vec_zz_pEX& v, const zz_pEX& f)
{
zz_pEXModulus F(f);
zz_pEX res;
res = 1;
long i;
for (i = 0; i < v.length(); i++)
MulMod(res, res, v[i], F);
x = res;
}
NOTE: A zz_pEX may be used wherever a zz_pEXModulus is required,
and a zz_pEXModulus may be used wherever a zz_pEX is required.
\**************************************************************************/
class zz_pEXModulus {
public:
zz_pEXModulus(); // initially in an unusable state
zz_pEXModulus(const zz_pEX& f); // initialize with f, deg(f) > 0
zz_pEXModulus(const zz_pEXModulus&); // copy
zz_pEXModulus& operator=(const zz_pEXModulus&); // assignment
~zz_pEXModulus(); // destructor
operator const zz_pEX& () const; // implicit read-only access to f
const zz_pEX& val() const; // explicit read-only access to f
};
void build(zz_pEXModulus& F, const zz_pEX& f);
// pre-computes information about f and stores it in F. Must have
// deg(f) > 0. Note that the declaration zz_pEXModulus F(f) is
// equivalent to zz_pEXModulus 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_pEXModulus& F); // return n=deg(f)
void MulMod(zz_pEX& x, const zz_pEX& a, const zz_pEX& b,
const zz_pEXModulus& F);
zz_pEX MulMod(const zz_pEX& a, const zz_pEX& b, const zz_pEXModulus& F);
// x = (a * b) % f; deg(a), deg(b) < n
void SqrMod(zz_pEX& x, const zz_pEX& a, const zz_pEXModulus& F);
zz_pEX SqrMod(const zz_pEX& a, const zz_pEXModulus& F);
// x = a^2 % f; deg(a) < n
void PowerMod(zz_pEX& x, const zz_pEX& a, const ZZ& e, const zz_pEXModulus& F);
zz_pEX PowerMod(const zz_pEX& a, const ZZ& e, const zz_pEXModulus& F);
void PowerMod(zz_pEX& x, const zz_pEX& a, long e, const zz_pEXModulus& F);
zz_pEX PowerMod(const zz_pEX& a, long e, const zz_pEXModulus& F);
// x = a^e % f; e >= 0, deg(a) < n. Uses a sliding window algorithm.
// (e may be negative)
void PowerXMod(zz_pEX& x, const ZZ& e, const zz_pEXModulus& F);
zz_pEX PowerXMod(const ZZ& e, const zz_pEXModulus& F);
void PowerXMod(zz_pEX& x, long e, const zz_pEXModulus& F);
zz_pEX PowerXMod(long e, const zz_pEXModulus& F);
// x = X^e % f (e may be negative)
void rem(zz_pEX& x, const zz_pEX& a, const zz_pEXModulus& F);
// x = a % f
void DivRem(zz_pEX& q, zz_pEX& r, const zz_pEX& a, const zz_pEXModulus& F);
// q = a/f, r = a%f
void div(zz_pEX& q, const zz_pEX& a, const zz_pEXModulus& F);
// q = a/f
// operator notation:
zz_pEX operator/(const zz_pEX& a, const zz_pEXModulus& F);
zz_pEX operator%(const zz_pEX& a, const zz_pEXModulus& F);
zz_pEX& operator/=(zz_pEX& x, const zz_pEXModulus& F);
zz_pEX& operator%=(zz_pEX& x, const zz_pEXModulus& F);
/**************************************************************************\
vectors of zz_pEX's
\**************************************************************************/
typedef Vec<zz_pEX> vec_zz_pEX; // 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_pEX& x, const zz_pEX& g, const zz_pEX& h,
const zz_pEXModulus& F);
zz_pEX CompMod(const zz_pEX& g, const zz_pEX& h,
const zz_pEXModulus& F);
// x = g(h) mod f; deg(h) < n
void Comp2Mod(zz_pEX& x1, zz_pEX& x2, const zz_pEX& g1, const zz_pEX& g2,
const zz_pEX& h, const zz_pEXModulus& F);
// xi = gi(h) mod f (i=1,2); deg(h) < n.
void Comp3Mod(zz_pEX& x1, zz_pEX& x2, zz_pEX& x3,
const zz_pEX& g1, const zz_pEX& g2, const zz_pEX& g3,
const zz_pEX& h, const zz_pEXModulus& 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_pEXArgument 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_pEXArgument {
vec_zz_pEX H;
};
void build(zz_pEXArgument& H, const zz_pEX& h, const zz_pEXModulus& F, long m);
// Pre-Computes information about h. m > 0, deg(h) < n.
void CompMod(zz_pEX& x, const zz_pEX& g, const zz_pEXArgument& H,
const zz_pEXModulus& F);
zz_pEX CompMod(const zz_pEX& g, const zz_pEXArgument& H,
const zz_pEXModulus& F);
extern long zz_pEXArgBound;
// Initially 0. If this is set to a value greater than zero, then
// composition routines will allocate a table of no than about
// zz_pEXArgBound KB. Setting this value affects all compose routines
// and the power projection and minimal polynomial routines below,
// and indirectly affects many routines in zz_pEXFactoring.
/**************************************************************************\
power projection routines
\**************************************************************************/
void project(zz_pE& x, const zz_pEVector& a, const zz_pEX& b);
zz_pE project(const zz_pEVector& a, const zz_pEX& b);
// x = inner product of a with coefficient vector of b
void ProjectPowers(vec_zz_pE& x, const vec_zz_pE& a, long k,
const zz_pEX& h, const zz_pEXModulus& F);
vec_zz_pE ProjectPowers(const vec_zz_pE& a, long k,
const zz_pEX& h, const zz_pEXModulus& 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.
void ProjectPowers(vec_zz_pE& x, const vec_zz_pE& a, long k,
const zz_pEXArgument& H, const zz_pEXModulus& F);
vec_zz_pE ProjectPowers(const vec_zz_pE& a, long k,
const zz_pEXArgument& H, const zz_pEXModulus& F);
// same as above, but uses a pre-computed zz_pEXArgument
class zz_pEXTransMultiplier { /* ... */ };
void build(zz_pEXTransMultiplier& B, const zz_pEX& b, const zz_pEXModulus& F);
void UpdateMap(vec_zz_pE& x, const vec_zz_pE& a,
const zz_pEXMultiplier& B, const zz_pEXModulus& F);
vec_zz_pE UpdateMap(const vec_zz_pE& a,
const zz_pEXMultiplier& B, const zz_pEXModulus& F);
// Computes the vector
// project(a, b), project(a, (b*X)%f), ..., project(a, (b*X^{n-1})%f)
// Required: a.length() <= deg(F), deg(b) < deg(F).
// This is "transposed" MulMod by B.
// Input may have "high order" zeroes stripped.
// Output always has high order zeroes stripped.
/**************************************************************************\
Minimum Polynomials
These routines should be used only when zz_pE is a field.
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_pEX& h, const vec_zz_pE& a, long m);
zz_pEX MinPolySeq(const vec_zz_pE& 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_pEX& h, const zz_pEX& g, const zz_pEXModulus& F, long m);
zz_pEX ProbMinPolyMod(const zz_pEX& g, const zz_pEXModulus& F, long m);
void ProbMinPolyMod(zz_pEX& h, const zz_pEX& g, const zz_pEXModulus& F);
zz_pEX ProbMinPolyMod(const zz_pEX& g, const zz_pEXModulus& 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/2^{zz_pE::degree()}.
void MinPolyMod(zz_pEX& h, const zz_pEX& g, const zz_pEXModulus& F, long m);
zz_pEX MinPolyMod(const zz_pEX& g, const zz_pEXModulus& F, long m);
void MinPolyMod(zz_pEX& h, const zz_pEX& g, const zz_pEXModulus& F);
zz_pEX MinPolyMod(const zz_pEX& g, const zz_pEXModulus& F);
// same as above, but guarantees that result is correct
void IrredPolyMod(zz_pEX& h, const zz_pEX& g, const zz_pEXModulus& F, long m);
zz_pEX IrredPolyMod(const zz_pEX& g, const zz_pEXModulus& F, long m);
void IrredPolyMod(zz_pEX& h, const zz_pEX& g, const zz_pEXModulus& F);
zz_pEX IrredPolyMod(const zz_pEX& g, const zz_pEXModulus& 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).
/**************************************************************************\
Composition and Minimal Polynomials in towers
These are implementations of algorithms that will be described
and analyzed in a forthcoming paper.
The routines require that p is prime, but zz_pE need not be a field.
\**************************************************************************/
void CompTower(zz_pEX& x, const zz_pX& g, const zz_pEXArgument& h,
const zz_pEXModulus& F);
zz_pEX CompTower(const zz_pX& g, const zz_pEXArgument& h,
const zz_pEXModulus& F);
void CompTower(zz_pEX& x, const zz_pX& g, const zz_pEX& h,
const zz_pEXModulus& F);
zz_pEX CompTower(const zz_pX& g, const zz_pEX& h,
const zz_pEXModulus& F);
// x = g(h) mod f
void ProbMinPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F,
long m);
zz_pX ProbMinPolyTower(const zz_pEX& g, const zz_pEXModulus& F, long m);
void ProbMinPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F);
zz_pX ProbMinPolyTower(const zz_pEX& g, const zz_pEXModulus& F);
// Uses a probabilistic algorithm to compute the minimal
// polynomial of (g mod f) over zz_p.
// The parameter m is a bound on the degree of the minimal polynomial
// (default = deg(f)*zz_pE::degree()).
// In general, the result will be a divisor of the true minimimal
// polynomial. For correct results, use the MinPoly routines below.
void MinPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F, long m);
zz_pX MinPolyTower(const zz_pEX& g, const zz_pEXModulus& F, long m);
void MinPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F);
zz_pX MinPolyTower(const zz_pEX& g, const zz_pEXModulus& F);
// Same as above, but result is always correct.
void IrredPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F, long m);
zz_pX IrredPolyTower(const zz_pEX& g, const zz_pEXModulus& F, long m);
void IrredPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F);
zz_pX IrredPolyTower(const zz_pEX& g, const zz_pEXModulus& F);
// Same as above, but assumes the minimal polynomial is
// irreducible, and uses a slightly faster, deterministic algorithm.
/**************************************************************************\
Traces, norms, resultants
\**************************************************************************/
void TraceMod(zz_pE& x, const zz_pEX& a, const zz_pEXModulus& F);
zz_pE TraceMod(const zz_pEX& a, const zz_pEXModulus& F);
void TraceMod(zz_pE& x, const zz_pEX& a, const zz_pEX& f);
zz_pE TraceMod(const zz_pEX& a, const zz_pEXModulus& f);
// x = Trace(a mod f); deg(a) < deg(f)
void TraceVec(vec_zz_pE& S, const zz_pEX& f);
vec_zz_pE TraceVec(const zz_pEX& f);
// S[i] = Trace(X^i mod f), i = 0..deg(f)-1; 0 < deg(f)
// The above trace routines implement the asymptotically fast trace
// algorithm from [von zur Gathen and Shoup, Computational Complexity,
// 1992].
void NormMod(zz_pE& x, const zz_pEX& a, const zz_pEX& f);
zz_pE NormMod(const zz_pEX& a, const zz_pEX& f);
// x = Norm(a mod f); 0 < deg(f), deg(a) < deg(f)
void resultant(zz_pE& x, const zz_pEX& a, const zz_pEX& b);
zz_pE resultant(const zz_pEX& a, const zz_pEX& b);
// x = resultant(a, b)
// NormMod and resultant require that zz_pE is a field.
/**************************************************************************\
Miscellany
\**************************************************************************/
void clear(zz_pEX& x) // x = 0
void set(zz_pEX& x); // x = 1
void zz_pEX::kill();
// f.kill() sets f to 0 and frees all memory held by f. Equivalent to
// f.rep.kill().
zz_pEX::zz_pEX(INIT_SIZE_TYPE, long n);
// zz_pEX(INIT_SIZE, n) initializes to zero, but space is pre-allocated
// for n coefficients
static const zz_pEX& zero();
// zz_pEX::zero() is a read-only reference to 0
void zz_pEX::swap(zz_pEX& x);
void swap(zz_pEX& x, zz_pEX& y);
// swap (via "pointer swapping")
zz_pEX::zz_pEX(long i, const zz_pE& c);
zz_pEX::zz_pEX(long i, const zz_p& c);
zz_pEX::zz_pEX(long i, long c);
// initilaize to c*X^i; provided for backward compatibility