SmallPrimeField.hpp

#ifndef _SMALL_PRIME_FIELD_H
#define _SMALL_PRIME_FIELD_H

#include "BPASFiniteField.hpp"
#include <iostream>
#include <sstream>
#include <math.h>

using std::cout;
using std::endl;

//forward declarations
class Integer;
class RationalNumber;
class ComplexRationalNumber;
class BigPrimeField;
class GeneralizedFermatPrimeField;
class DenseUnivariateIntegerPolynomial;
class DenseUnivariateRationalPolynomial;
template <class Ring>
class SparseUnivariatePolynomial;

//NOTE the below code is not properly formatted with the corresponding 
//implementation file. Only the montegomery version is setup properly.
#if ( 0 )
// simple small prime field using long long int to represent the number
class SmallPrimeField : public BPASField{

private:

    static long int prime;
    long long int a;
    
public:

    static mpz_class characteristic;
    static bool isPrimeField;
    static bool isSmallPrimeField;
    static bool isComplexField;
    

    SmallPrimeField () : a(0) {}
    SmallPrimeField (long long int _a) {
        while (_a < 0){
            _a = _a + prime;
        }
        a = _a%prime;
    }
    SmallPrimeField (const SmallPrimeField& c) {
        a = c.a;
    }
    //SmallPrimeField (Integer c);
    SmallPrimeField (RationalNumber c);
    SmallPrimeField (BigPrimeField c);
    SmallPrimeField (GeneralizedFermatPrimeField c);
    //SmallPrimeField (DenseUnivariateIntegerPolynomial c);
    //SmallPrimeField (DenseUnivariateRationalPolynomial c);
    //SmallPrimeField (SparseUnivariatePolynomial<Integer> c);
    //SmallPrimeField (SparseUnivariatePolynomial<RationalNumber> c);
    //SmallPrimeField (SparseUnivariatePolynomial<ComplexRationalNumber> c);
    //template <class Ring>
    //SmallPrimeField (SparseUnivariatePolynomial<Ring> c);
    
    SmallPrimeField* SPFpointer(SmallPrimeField* b) {
        return b;
    }
    SmallPrimeField* SPFpointer(RationalNumber* a) {
        std::cout << "BPAS error, try to cast pointer to Rational Number to pointer to SmallPrimeField" << std::endl;
        exit(1);
    }
    SmallPrimeField* SPFpointer(BigPrimeField* a) {
        std::cout << "BPAS error, try to cast pointer to BigPrimeField to pointer to SmallPrimeField" << std::endl;
        exit(1);
    }
    SmallPrimeField* SPFpointer(GeneralizedFermatPrimeField* a) {
        std::cout << "BPAS error, try to cast pointer to GeneralizedFermatPrimeField to pointer to SmallPrimeField" << std::endl;
        exit(1);
    }
    
    static void setPrime(long int p){
        
        prime = p;
        //using string for mpz_class assignment
        std::ostringstream ss;
        ss << p;
        std::string sp = ss.str();
        
        //mpz_t z;
        //mpz_init(z
        
        mpz_class w(sp);
    //  mpz_class w(p);
        characteristic = w;
    }

    long int Prime(){
        return prime;
    }

    void whichprimefield(){
        cout << "SmallPrimeField" << endl;
    }

    long long int number(){
        return a;
    }

//fing the nth primitive root of unity
    static SmallPrimeField findPrimitiveRootofUnity(long int n){
        if ( ((prime - 1) % n != 0)){
            cout << "ERROR: n does not divide prime - 1." << endl;
            return -1;
        }
        bool flag = false;
        long int  q = (prime - 1) / n;
        SmallPrimeField p1 (prime - 1);
        SmallPrimeField c;
        int i = 0;
        long int test = q * n / 2;
        srand (time(NULL));
        while(i < 20){
            c =  rand();
            if ((c^test) == p1) {
                flag = true;
                return (c^q);
            }
            i ++;
        }
        if (!flag ){
            cout << "No primitive root found!"<< endl;
            return 0;
        }
    // If no primitive root found
    }

    SmallPrimeField& operator= (SmallPrimeField c) {
        if (this != &c) {
        a = c.a;
        }
        return *this;
    }
    SmallPrimeField& operator= (long long int k) {
        while (k < 0){
        k = k + prime;
        }
        a = k%prime;
        return *this;
    }

    inline bool isZero() {
        return (a == 0);
    }
    inline void zero() {
        a = 0;
    }
    inline bool isOne() {
        return (a == 1);
    }
    inline void one() {
        a = 1;
    }

    inline bool isNegativeOne() {
        return (a == (prime - 1));
    }
    inline void negativeOne() {
        a = prime - 1;
    }

    inline int isConstant() {
        if (a >= 0)
        return 1;
        else { return -1; }
    }


    inline SmallPrimeField gcd (SmallPrimeField c) {
        SmallPrimeField q (0);
        SmallPrimeField r (0);
        SmallPrimeField b (*this);
        if(b.a < c.a){
            return c.gcd(b);
        }
        while (c.a > 0) {
            q.a = b.a / c.a;
            r.a = b.a % c.a;
            b = c;
            c = r;
        }
        return b;
    }
    inline SmallPrimeField gcd (long long int c){
        SmallPrimeField b (c);
        SmallPrimeField r (*this);
        return r.gcd(b);
    }

    inline bool operator== (SmallPrimeField& c) {
        if (a == c.a)
            return 1;
        else { return 0; }
    }

    inline bool operator== (long long int k){
        SmallPrimeField r (*this);
        SmallPrimeField b (k);
        if (b == r){
        return 1;
        }
        else {
        return 0;
        }
    }
    inline bool operator!= (SmallPrimeField& c) {
        if (a == c.a)
            return 0;
        else { return 1; }
    }

    inline bool operator!= (long long int k){
        SmallPrimeField r (*this);
        SmallPrimeField b (k);
        if (b == r){
        return 0;
        }
        else {
        return 1;
        }
    }


    inline SmallPrimeField operator+ (SmallPrimeField& c) {
        SmallPrimeField r (*this);
        return (r += c);
    }
    inline SmallPrimeField operator+ (long long int& c) {
        SmallPrimeField r (*this);
        SmallPrimeField b (c);
        return (r += b);
    }
    inline SmallPrimeField operator+ (int& c) {
        SmallPrimeField r (*this);
        SmallPrimeField b (c);
        return (r += b);
    }
    inline SmallPrimeField operator+= (long long int& c) {
        SmallPrimeField r (*this);
        SmallPrimeField b (c);
        return (r += b);
    }
    inline SmallPrimeField& operator+= (SmallPrimeField c) {
        a = (a + c.a)%prime;
        return *this;
    }
 
    inline SmallPrimeField operator- (SmallPrimeField& c) {
        SmallPrimeField r (*this);
        return (r -= c);
    }
    inline SmallPrimeField operator- (long long int& c) {
        SmallPrimeField r (*this);
        SmallPrimeField b (c);
        return (r -= b);
    }
    inline SmallPrimeField operator-= (long long int& c) {
        SmallPrimeField r (*this);
        SmallPrimeField b (c);
        return (r -= b);
    }
    inline SmallPrimeField& operator-= (SmallPrimeField c) {
        if ((a - c.a)<0){
        a = prime+(a - c.a);
        }
        else{
        a = a - c.a;
        }
        return *this;
    }
    inline SmallPrimeField operator- () {
        a = prime - a;
        return *this;
    }
    inline SmallPrimeField operator* (SmallPrimeField& c) {
        SmallPrimeField r (*this);
        return (r *= c);
    }
    inline SmallPrimeField operator* (long long int c) {
        SmallPrimeField r (*this);
        return (r *= c);
    }

    inline SmallPrimeField& operator*= (SmallPrimeField c) {
        a = (a * c.a)%prime;
        return *this;
    }
    inline SmallPrimeField operator*= (long long int& c) {
        a = a * c;
        SmallPrimeField r (a);
        return r;
    }
  void egcd (long long int x, long long int y, long long int *ao, long long int *bo, long long int *vo, long long int P){
        long long int t, A, B, C, D, u, v, q;

        u = y;
        v = x;
        A = 1;
        B = 0;
        C = 0;
        D = 1;

    do
    {
        q = u / v;
        t = u;
        u = v;
        v = t - q * v;
        t = A;
        A = B;
        B = t - q * B;
        t = C;
        C = D;
        D = t - q * D;
    }
    while (v != 0);

    *ao = A;
    *bo = C;
    *vo = u;
  }

  inline SmallPrimeField inverse2(){
    long long int n, b, v;
    egcd (a, prime, &n, &b, &v, prime);
    if (b < 0)
        b += prime;
    SmallPrimeField r (b);
    return r;
  }


//Binary inverse

  SmallPrimeField inverse(){
    long long int u = a;
    long long int v = prime;
    long long int x1 = 1;
    long long int x2 = 0;
    SmallPrimeField r(0);
    while(u!=1 && v!=1){
        while (u%2 == 0){
            u = u >> 1;
            if(x1%2==0)
                x1 = x1 >> 1;
            else
                x1 = (x1 + prime) >> 1;
        }
        while (v%2 ==0){
            v = v >> 1;
            if(x2%2 == 0)
                x2 = x2 >> 1;
            else
                x2 = (x2 + prime) >> 1;
        }
        if(u>=v){
            u = u - v;
            x1 = x1 - x2;
        }
        else{
            v = v - u;
            x2 = x2 - x1;
        }
    }
    if(u == 1)
        r = x1;
    else
        r = x2;
    return r;
  }

  inline SmallPrimeField operator/ (SmallPrimeField& c) {
    SmallPrimeField r (*this);
    return (r /= c);
  }
  inline SmallPrimeField operator/ (long long int c) {
    SmallPrimeField r (*this);
    SmallPrimeField b (c);
    return (r /= b);
  }
  inline SmallPrimeField operator/= (long long int c) {
    SmallPrimeField r (*this);
    SmallPrimeField b (c);
    return (r /= b);
  }
  inline SmallPrimeField operator/= (SmallPrimeField c) {
    if (c.isZero()) {
      std::cout << "BPAS: error, dividend is zero from SmallPrimeField."<< std::endl;
            exit(1);
    }
    c=c.inverse();
    SmallPrimeField r (*this);
    r*=c;
    a = r.a;
    return *this;
  }

  inline SmallPrimeField operator^ (long long int e){
    SmallPrimeField r;
    if (isZero() || isOne() || e == 1)
      r = *this;
    else if (e == 2) {
      r = *this * *this;
    }
    else if (e > 2) {
      SmallPrimeField x (*this);
      r.one();

      while (e != 0) {
        if (e % 2)
        r *= x;
        x = x * x;
        e >>= 1;
      }
    }
    else if (e == 0) {
      r.one();
    }
    else {
      r = *this ^ (-e);
      r=r.inverse();
    }
    return r;
  }
  

  inline friend std::ostream& operator<< (std::ostream &out, SmallPrimeField c){            
    out << c.a;
    return out;
  }
};


#else

/**
 * A prime field whose prime is 32 bits or less. Elements of this field
 * are encoded using montgomery trick.
 */
class SmallPrimeField : public BPASFiniteField<SmallPrimeField> {

private:

    long long int a;
    static long long int prime;     
    //static long long int R;
    static unsigned long long int Pp;

//R = 2^Wt;
//  static long long int Rp;

//Pp = -p^-1 mod R;
//  static long long int N;

public:

    static RingProperties properties;
    static mpz_class characteristic;
// static bool isPrimeField;
// static bool isSmallPrimeField;
// static bool isComplexField;

    SmallPrimeField ();

// using aR mod p to represent the number
    SmallPrimeField (long long int _a);

    SmallPrimeField (const SmallPrimeField& c);

    explicit SmallPrimeField (const Integer& c);

    explicit SmallPrimeField (const RationalNumber& c);

    explicit SmallPrimeField (const ComplexRationalNumber& c);

    explicit SmallPrimeField (const BigPrimeField& c);

    explicit SmallPrimeField (const GeneralizedFermatPrimeField& c);

    explicit SmallPrimeField (const DenseUnivariateIntegerPolynomial& c);

    explicit SmallPrimeField (const DenseUnivariateRationalPolynomial& c);

    explicit SmallPrimeField (const SparseUnivariatePolynomial<Integer>& c);

    explicit SmallPrimeField (const SparseUnivariatePolynomial<RationalNumber>& c);

    explicit SmallPrimeField (const SparseUnivariatePolynomial<ComplexRationalNumber>& c);

    template <class Ring>
    explicit SmallPrimeField (const SparseUnivariatePolynomial<Ring>& c);

    SmallPrimeField* SPFpointer(SmallPrimeField* b);

    SmallPrimeField* SPFpointer(RationalNumber* a);

    SmallPrimeField* SPFpointer(BigPrimeField* a);

    SmallPrimeField* SPFpointer(GeneralizedFermatPrimeField* a);

// return the actual number
    long long int number() const;

    void whichprimefield();

    static void setPrime(long long int p){

        prime = p;
//using string for mpz_class assignment
        std::ostringstream ss;
        ss << p;
        std::string sp = ss.str();

//mpz_t z;
//mpz_init(z

        mpz_class w(sp);
//  mpz_class w(p);
        SmallPrimeField::characteristic = w;

        // long long int p1 = R - prime;

        // for(int i = 0; i < R; i ++){
        //  if ((i * p1)%R == 1){
        //      Pp = i;
        //      break;
        //  }
        // }
//cout << Pp << endl;
      long long int t, A, B, C, D, u, v,q;
    __asm__(                          //compute the first step of egcd where R = q*prime + v
       // "movq %%rax,%%rsi\n\t"
        "xor %%rax,%%rax\n\t"
        "movq $1,%%rdx\n\t"
        "divq %2\n\t"
       // "movq %%rsi,%%rax\n\t"
       // "movq %%rdx,%0\n\t"
       // "movq %%rax,%1\n\t"
        : "=&d" (v),"=&a" (q) 
        : "b"(prime)
        :"rsi","rdi");
    A = 1;
    B = 0;
    C = 0;
    D = 1;
   // q = R/prime;
    u = prime;
    //v = R - R mod prime;
    t = A;
    A = B;
    B = t - q * B;
    t = C;
    C = D;
    D = t - q * D;

    while (v != 0){
      q = u / v;
      t = u;
      u = v;
      v = t - q * v;
      t = A;
      A = B;
      B = t - q * B;
      t = C;
      C = D;
      D = t - q * D;
   //   printf("%llu,%llu\n",A,C);
    }
     //unsigned long long int res;
    if(C < 0){
      C = 0-C;
      Pp = (unsigned long long int)C;
    }
    else{
      __asm__(                          //compute -C mod R;
       // "movq %%rax,%%rsi\n\t"
        "xor %%rax,%%rax\n\t"
        "movq $1,%%rdx\n\t"
        "sub %1,%%rax\n\t"
        "sbb $0,%%rdx\n\t"
       // "movq %%rsi,%%rax\n\t"
       // "movq %%rdx,%0\n\t"
        "movq %%rax,%%rdx\n\t"
        : "=&d" (Pp)
        : "b"((unsigned long long int)C)
        :"rsi","rdi");
    }

    }

    long long int Prime();

    SmallPrimeField& operator= (const SmallPrimeField& c);

    SmallPrimeField& operator= (long long int k);

    SmallPrimeField findPrimitiveRootOfUnity(long int n) const {
        return SmallPrimeField::findPrimitiveRootofUnity(n);
    }


    static SmallPrimeField findPrimitiveRootofUnity(long long int n){
        if ( ((prime - 1) % n != 0)){
            cout << "ERROR: n does not divide prime - 1." << endl;
            return -1;
        }
        bool flag = false;
        long long int  q = (prime - 1) / n;
        SmallPrimeField p1 (prime - 1);
        SmallPrimeField c;
        int i = 0;
        long long int test = q * n / 2;
        srand (time(NULL));
        while(i < 20){
            c =  rand();
            if ((c^test) == p1) {
                flag = true;
                return (c^q);
            }
            i ++;
        }
        if (!flag ){
            cout << "No primitive root found!"<< endl;
            return 0;
        }
// If no primitive root found
    }

    inline bool isZero() const {
        return (a == 0);
    }

    inline void zero() {
        a = 0;
    }

    inline bool isOne() const {
    SmallPrimeField b(1);
        return  (a == b.a);
    }

    inline void one() {
    SmallPrimeField b(1);
        a = b.a;// R % prime;
    }

    inline bool isNegativeOne() {
    SmallPrimeField b(-1);
    return (a == b.a);
    }
    inline void negativeOne() {
    SmallPrimeField b(-1);
      a = b.a;
    }

    inline int isConstant() {
        if (a >= 0)
            return 1;
        else { return -1; }
    }

  SmallPrimeField unitCanonical(SmallPrimeField* u = NULL, SmallPrimeField* v = NULL) const ;

    inline SmallPrimeField operator+ (const SmallPrimeField& c) const {
        SmallPrimeField r (*this);
        return (r += c);
    }

    inline SmallPrimeField operator+ (const long long int& c) const {
        SmallPrimeField r (*this);
        SmallPrimeField b (c);
        return (r += b);
    }
    inline SmallPrimeField operator+ (const int& c) const {
        SmallPrimeField r (*this);
        SmallPrimeField b (c);
        return (r += b);
    }
    inline SmallPrimeField operator+= (const long long int& c) {
        SmallPrimeField r (*this);
        SmallPrimeField b (c);
        return (r += b);
    }

    inline SmallPrimeField& operator+= (const SmallPrimeField& c) {
        //a = (a + c.a)%prime;
        //return *this;
    a = a + c.a;
    a -= prime;
    a += (a >> 63) & prime;
    return *this;
    }

    inline SmallPrimeField operator- (const SmallPrimeField& c) const {
        SmallPrimeField r (*this);
        return (r -= c);
    }

    inline SmallPrimeField operator- (const long long int& c) const {
        SmallPrimeField r (*this);
        SmallPrimeField b (c);
        return (r -= b);
    }
    inline SmallPrimeField operator-= (const long long int& c) {
        SmallPrimeField r (*this);
        SmallPrimeField b (c);
        return (r -= b);
    }

    inline SmallPrimeField& operator-= (const SmallPrimeField& c) {
        // if ((a - c.a)<0){
        //  a = prime+(a - c.a);
        // }
        // else{
        //  a = a - c.a;
        // }
        // return *this;
    a = a - c.a;
    a += (a >> 63) & prime;
    return *this;
    }
    inline SmallPrimeField operator- () const {
        SmallPrimeField ret (*this);
        ret.a = prime - a;
        return ret;
    }

    inline SmallPrimeField operator* (const SmallPrimeField& c) const {
        SmallPrimeField r (*this);

        r*=c;
//cout << r << endl;
        return r;
    }

    inline SmallPrimeField operator* (long long int c) const {
        SmallPrimeField r (*this);
        SmallPrimeField b(c);
        return (r *= b);
    }
// Mont(a,b) = ab/R mod p
// Mont(aR,bR) = abR mod p
    inline SmallPrimeField& operator*= (const SmallPrimeField& c) {
        // long long int x = (c.a * a);
        // long long int w = x*Pp;
        // long long int y = x + prime*(w&(R-1));
        // long long int z =  y>>32 ;
        // if(z >= prime){
        //  z = z - prime;
        // }
        // a = z;
        // return *this;
    long long int _a = a;
  __asm__(
      "mulq %2\n\t"
      "movq %%rax,%%rsi\n\t"
      "movq %%rdx,%%rdi\n\t"
      "imulq %3,%%rax\n\t"
      "mulq %4\n\t"
      "add %%rsi,%%rax\n\t"
      "adc %%rdi,%%rdx\n\t"
      "subq %4,%%rdx\n\t"
      "mov %%rdx,%%rax\n\t"
      "sar $63,%%rax\n\t"
      "andq %4,%%rax\n\t"
      "addq %%rax,%%rdx\n\t"
      : "=&d" (_a)
      : "a"(_a),"rm"(c.a),"b"((unsigned long long int)Pp),"c"(prime)
      :"rsi","rdi");
  a = _a;
    return *this;
    }

//partial Montgomery inversion
    long long int* pinverse();
//  long long int MontMul(long long int a, long long int b);

//return bc/R mod p;
    static long long int Mont(long long int b, long long int c);
  static long long int getRsquare();
//Montgomery inversion
    inline SmallPrimeField inverse() const {
         SmallPrimeField r(*this);
    //   SmallPrimeField rsquare(R);
     long long int rsquare = getRsquare();
        // long long int* result = r.pinverse();
        // if(result[0] < 32){
        //  result[0] = Mont(result[0], rsquare.a);
        //  result[1] += 32;
        // }
        // result[0] = Mont(result[0], rsquare.a);
        // result[0] = Mont(result[0], pow(2,64-result[1]));
        // r.a = result[0];
        // return r;

      if (r.a == 0)
      {
        printf("Not invertible!\n");
        return 0;
      }
   // long long int rsquare =  getRsquare(prime);
    long long int* result = r.pinverse();
    if(result[1] < 64){
      result[0] = Mont(result[0], rsquare);
      result[1] += 64;
    }
    result[0] = Mont(result[0], rsquare);
    long long int tmp;
    if(result[1] != 64){
      r.a = 1L << (128 - result[1]);
      result[0] = Mont(result[0], r.a);
    }
    r.a = result[0];
    free(result);
    return r;
    }

//Binary inverse
    SmallPrimeField inverse2();

    inline SmallPrimeField operator^ (long long int e) const {
        SmallPrimeField r;
        if (isZero() || isOne() || e == 1)
            r = *this;
        else if (e == 2) {
            r = *this * *this;
        }
        else if (e > 2) {
            SmallPrimeField x (*this);
            r.one();

            while (e != 0) {
                if (e % 2)
                    r *= x;
                x = x * x;
                e >>= 1;
            }
        }
        else if (e == 0) {
            r.one();
        }
        else {
            r = *this ^ (-e);
            r=r.inverse();
        }
        return r;

    }

    inline SmallPrimeField& operator^= (long long int e) {
        *this = *this ^ e;
        return *this;
    }

    inline bool operator== (const SmallPrimeField& c) const {
        if ((*this).number() == c.number())
            return 1;
        else { return 0; }
    }

    inline bool operator== (long long int k) const {
        SmallPrimeField r (*this);
        SmallPrimeField b (k);
        if (b.number() == r.number()){
            return 1;
        }
        else {
            return 0;
        }
    }

    inline bool operator!= (const SmallPrimeField& c) const {
        if ((*this).number() == c.number())
            return 0;
        else { return 1; }
    }


    inline bool operator!= (long long int k) const {
        SmallPrimeField r (*this);
        SmallPrimeField b (k);
        if (b.number() == r.number()){
            return 0;
        }
        else {
            return 1;
        }
    }

    inline ExpressionTree convertToExpressionTree() const {
        return ExpressionTree(new ExprTreeNode(this->number()));
    }

    inline SmallPrimeField operator/ (const SmallPrimeField& c) const {
        SmallPrimeField r (*this);
        r /= c;
        return r;
    }

    inline SmallPrimeField operator/ (long long int c) const {
        SmallPrimeField r (*this);
        SmallPrimeField b(c);
        return (r /= b);
    }

    inline SmallPrimeField& operator/= (const SmallPrimeField& c) {
        if (c.isZero()) {
            std::cout << "BPAS: error, dividend is zero from SmallPrimeField."<< std::endl;
            exit(1);
        }
    //cout << "division" << endl;
        SmallPrimeField inv = c.inverse();
        SmallPrimeField r (*this);
        r *= inv;
    (*this).a = r.a;
        return *this;
    }

  inline SmallPrimeField operator% (const SmallPrimeField& c) const {
    return *this % c;
  }

  inline SmallPrimeField& operator%= (const SmallPrimeField& c) {
    (*this).a  = (*this).a % c.a;
    return *this;
  }

    inline SmallPrimeField gcd (const SmallPrimeField& other) const {
        SmallPrimeField q (0);
        SmallPrimeField r (0);
        SmallPrimeField b (*this);
        SmallPrimeField c (other);
        if(b.a < c.a){
            return c.gcd(b);
        }
        while (c.a > 0) {
            q.a = b.a / c.a;
            r.a = b.a % c.a;
            b = c;
            c = r;
        }
        return b;
    }
    
    /**
     * Compute squarefree factorization of *this
     */
    inline Factors<SmallPrimeField> squareFree() const {
        std::vector<SmallPrimeField> ret;
        ret.push_back(*this);
        return ret;
    }

    /** 
     * Get the euclidean size of *this.
     */
    inline SmallPrimeField euclideanSize() const {
        return (*this).number();
    }
    
    /**
     * Perform the eucldiean division of *this and b. Returns the 
     * remainder. If q is not NULL, then returns the quotient in q. 
     */ 
    SmallPrimeField euclideanDivision(const SmallPrimeField& b, SmallPrimeField* q = NULL) const;

    /**
     * Perform the extended euclidean division on *this and b. 
     * Returns the GCD. If s and t are not NULL, returns the bezout coefficients in them.
     */
    SmallPrimeField extendedEuclidean(const SmallPrimeField& b, SmallPrimeField* s = NULL, SmallPrimeField* t = NULL) const;
    
    /**
     * Get the quotient of *this and b.
     */
    SmallPrimeField quotient(const SmallPrimeField& b) const;

    /** 
     * Get the remainder of *this and b.
     */
    SmallPrimeField remainder(const SmallPrimeField& b) const;

};

#endif //montgomery switch

#endif //include guard