MODULE: zz_p


The class zz_p is used to represent integers mod p, where 1 <= p <
NTL_SP_BOUND.  Note that NTL_SP_BOUND is usually 2^30 on 32-bit machines and
2^50 on 64-bit machines.

The modulus p may be any positive integer, not necessarily prime.

Objects of the class zz_p are represented as a long in the range 0..p-1.

An executing program maintains a "current modulus", which is set to p using
zz_p::init(p).  The current modulus *must* be initialized before any operations
on zz_p's are performed.  The modulus may be changed, and a mechanism is provided
for saving and restoring a modulus (see classes zz_pPush and zz_pContext below).


#include <NTL/ZZ.h>
#include <NTL/FFT.h>
#include <NTL/SmartPtr.h>

class zz_p {

   zz_p(); // initial value 0

   zz_p(const zz_p& a); // copy constructor
   explicit zz_p(long a); // promotion constructor

   zz_p& operator=(const zz_p& a); // assignment
   zz_p& operator=(long a); // assignment

   static void init(long p);
   // set the modulus to p, where p > 1.  This must be called before any
   // zz_p constructors are invoked.
   // The number p must have at most NTL_SP_NBITS bits.

   static long modulus();
   // zz_p::modulus() yields read-only reference to the current
   // modulus

   // typedefs to aid in generic programming
   typedef long rep_type;
   typedef zz_pContext context_type;
   typedef zz_pBak bak_type;
   typedef zz_pPush push_type;
   typedef zz_pX poly_type;


long rep(zz_p a); // read-only access to representation of a




long operator==(zz_p a, zz_p b);
long operator!=(zz_p a, zz_p b);

long IsZero(zz_p a);  // test for 0
long IsOne(zz_p a);  // test for 1

// PROMOTIONS: operators ==, != promote long to zz_p on (a, b).




// operator notation:

zz_p operator+(zz_p a, zz_p b);
zz_p operator-(zz_p a, zz_p b);

zz_p operator-(zz_p a); // unary -

zz_p& operator+=(zz_p& x, zz_p a);
zz_p& operator+=(zz_p& x, long a);

zz_p& operator-=(zz_p& x, zz_p a);
zz_p& operator-=(zz_p& x, long a);

zz_p& operator++(zz_p& x);  // prefix
void operator++(zz_p& x, int);  // postfix

zz_p& operator--(zz_p& x);  // prefix
void operator--(zz_p& x, int);  // postfix

// procedural versions:

void add(zz_p& x, zz_p a, zz_p b); // x = a + b
void sub(zz_p& x, zz_p a, zz_p b); // x = a - b 
void negate(zz_p& x, zz_p a); // x = -a

// PROMOTIONS: binary +, -, and procedures add, sub promote
// from long to zz_p on (a, b).




// operator notation:

zz_p operator*(zz_p a, zz_p b);

zz_p& operator*=(zz_p& x, zz_p a);
zz_p& operator*=(zz_p& x, long a);

// procedural versions:

void mul(zz_p& x, zz_p a, zz_p b); // x = a * b

void sqr(zz_p& x, zz_p a); // x = a^2
zz_p sqr(zz_p a);

// PROMOTIONS: operator * and procedure mul promote from long to zz_p
// on (a, b).




operator notation:

zz_p operator/(z_p a, zz_p b);

zz_p& operator/=(zz_p& x, zz_p a);
zz_p& operator/=(zz_p& x, long a);

procedural versions:

void div(zz_p& x, zz_p a, zz_p b);
// x = a/b

void inv(zz_p& x, zz_p a);
zz_p inv(zz_p a);
// x = 1/a

// PROMOTIONS: operator / and procedure div promote from long to zz_p
// on (a, b).




void power(zz_p& x, zz_p a, long e); // x = a^e (e may be negative)
zz_p power(zz_p a, long e);


                               Random Elements


void random(zz_p& x);
zz_p random_zz_p();
// x = random element in zz_p.  Uses RandomBnd from ZZ.

void VectorRandom(long k, zz_p *x);
// equivalent to random(x[i]) for i in [0..k), but fatser




ostream& operator<<(ostream& s, zz_p a);

istream& operator>>(istream& s, zz_p& x);
// a ZZ is read and reduced mod p


                       Modulus Switching 

A class zz_pPush is provided for "backing up" the current modulus
and installing a new one.

Here is what you do to save the current modulus, temporarily
set it to p, and automatically restore it:

      zz_pPush push(p); 



The constructor for push will save the current modulus, and install p as the
current modulus.  The destructor for push will restore the old modulus when the
scope enclosing it exits.  This is the so-called RAII (resource acquisition is
initialization) paradigm.

You could also do the following:

      zz_pPush push; // just backup current modulus


      zz_p::init(p1); // install p1 


      zz_p::init(p2); // install p2

      // reinstall original modulus as close of scope

The zz_pPush interface is good for implementing simple stack-like
modulus "context switching".  For more general context switching,
see zz_pContext below.  There is also an older zz_pBak class
that may also be useful.


It is critical that zz_p objects created under one zz_p modulus are not used in
any non-trivial way "out of context", i.e., under a different (or undefined)
zz_p modulus.  However, for ease-of-use, some operations may be safely
performed out of context.  These safe operations include: the default and copy
constructor, the destructor, and the assignment operator.  In addition is is
generally safe to read any zz_p object out of context (i.e., printing it out, or
fetching its underlying representive using the rep() function).

Any unsafe uses out of context are not in general checked, and may 
lead to unpredictable behavior.


// A convenient interface for common cases:

class zz_pPush {

zz_pPush();  // just backup current modulus

explicit zz_pPush(long p, long maxroot=NTL_FFTMaxRoot);
zz_pPush(INIT_FFT_TYPE, long index);
zz_pPush(INIT_USER_FFT_TYPE, long p);
explicit zz_pPush(const zz_pContext& context);
  // backup current modulus and install the given one
  // see documentation for zz_p::init for more details

zz_pPush(const zz_pPush&); // disabled
void operator=(const zz_pPush&); // disabled


// more general context switching:
// A zz_pContext object has a modulus q (possibly "null")

class zz_pContext {


zz_pContext();  // q = "null"

explicit zz_pContext(long p);
zz_pContext(INIT_FFT_TYPE, long index);
zz_pContext(INIT_USER_FFT_TYPE, long p);
  // q = the given modulus
  // see documentation for zz_p::init for more details

void save(); // q = CurrentModulus
void restore() const; // CurrentModulus = q

zz_pContext(const zz_pContext&);  // copy
zz_pContext& operator=(const zz_pContext&); // assignment
~zz_pContext(); // destructor


/ An older interface:
// To describe this logic, think of a zz_pBak object
// of having two components: a modulus q (possibly "null") and 
// an "auto-restore bit" b.

class zz_pBak {

   zz_pBak();  // q = "null", b = 0

   ~zz_pBak();  // if (b) CurrentModulus = q

   void save();  // q = CurrentModulus, b = 1 
   void restore();  // CurrentModulus = q, b = 0

   zz_pBak(const zz_pBak&);  // copy disabled
   void operator=(const zz_pBak&);  // assignment disabled




void clear(zz_p& x); // x = 0
void set(zz_p& x); // x = 1

static mulmod_t zz_p::ModulusInverse();
// zz_p::ModulusInverse() returns PrepMulMod(zz_p::modulus()) 

static zz_p zz_p::zero();
// zz_p::zero() yields a read-only reference to zero

void swap(zz_p& x, zz_p& y);
// swap x and y 

static void zz_p::init(long p, long maxroot);
// Same as ordinary zz_p::init(p), but somewhat more efficient.  If you are
// going to perform arithmetic modulo a degree n polynomial, in which
// case set maxroot to NextPowerOfTwo(n)+1.  This is useful, for
// example, if you are going to factor a polynomial of degree n modulo
// p, and you know n in advance.
// If maxroot is set too low, the program will abort with an
// appropriate error message.

static void zz_p::FFTInit(long i);
// sets modulus to the i-th FFT prime (counting from 0).  FFT primes
// are NTL_SP_NBITS-bit primes p, where p-1 is divisible by a high power
// of two.  Thus, polynomial arithmetic mod p can be implemented
// particularly efficiently using the FFT.  As i increases, the power
// of 2 that divides p-1 gets smaller, thus placing a more severe
// restriction on the degrees of the polynomials to be multiplied.

static void zz_p::UserFFTInit(long p);
// set the modulus to a user-provided FFT prime p. To be useful,
// p-1 should be divisibly by a high power of 2. 
// The function is a utility routine that may be used to 
// calculate this value (see below). 
// If you are going to perform arithmetic modulo a degree n polynomial, 
// you will want CalcMaxRoot(p) >= NextPowerOfTwo(n)+1. 

zz_pContext::zz_pContext(long p, long maxroot);
// constructor for a zz_pContext with same semantics
// as zz_p::init(p, maxroot) above.

zz_pContext::zz_pContext(INIT_FFT_TYPE, long i);
// constructor for a zz_pContext with same semantics
// as zz_p::FFTInit(i) above; invoke as zz_pContext(INIT_FFT, i).

zz_pContext::zz_pContext(INIT_USER_FFT_TYPE, long p);
// constructor for a zz_pContext with same semantics
// as zz_p::UserFFTInit(p) above; invoke as zz_pContext(INIT_USER_FFT, p).

// provided for consistency with other classes, initialize to zero

// provided for consistency with other classes, initialize to zero

// provided for consistency with other classes, no action

long CalcMaxRoot(long p);
// p is assumed to be an odd prime.
// Returns the largest k such that 2^k divides p-1
// and such that k does not exceed an implementation defined
// constant.  This represents the max power of two for which
// an FFT mod p is supported.

void VectorConv(long k, zz_p *x, const ZZ *a);
void VectorConv(long k, zz_p *x, const long *a);
// equivalent to conv(x[i], a[i]) for i in [0..k), but fatser