大数类的简单实现

本文仅做参考，主要是体现大数实现原理。

 

大数操作是使用计算机计算经常遇到的问题。

使用VS2005，在32位的机器上

 

各个类型的大小：

sizeof(int): 4

sizeof(unsigned): 4

sizeof(long): 4

sizeof(long long): 8

sizeof(char): 1

 

因此，unsigned的二进制位数为：4*8=32位，能够表示的最大数为：4294967295 即

11111111 11111111 11111111 11111111

而 long long 最大能够表示的无符号数为2^64 -1=18446744073709551615

因此，实现一个大数类是很有必要的。

网上有很多以实现不错的大数类，我这里只是练习练习。

 

使用数组记录大数据：

 

 * 这里列举vector<char> dats表示大数

 *100000的二进制为:
 * 1111 01000010 01000000
 *将之存放到数组中为:
 * -----------------     -----------------        -----------------
 * |   第2个元素   |   |   第1个元素   |       |   第0个元素   |
 * -----------------     -----------------        -----------------
 * | | | | |1|1|1|1| | |1|0|0|0|0|1|0| | |1|0|0|0|0|0|0|
 * ----------------- ----------------- -----------------
 * 即:dats[0] = 64,dats[1] =66,dats[2]=15
 * 还原方法:((dats[2]*2^8)+dats[1])*2^8+dats[0] = 100000

 

 

定义一个大数类的结构：

class bigint 

{ 

    char sign;      //符号0:+, 1:-                   

    vector<unsigned> dats;

}

 

1.将string转到dats中,可以通过用2除来转为二进制。

例如：string numStr = ‘76543210’;

首先由于末位为0，为偶数，所以记录0，结果为0

7&1==1 有补位给6, 7>>1 = 3

16&1==0 无补位， 16>>1 = 8

5&1 ==1 有补位给4， 5>>1 = 2

14&1 ==0 无补位…

 

最后得到38271605

由于末位为5，为奇数，所以再记录1，结果为10

在次除2得到19135802

由于末位为2，为偶数，所以再记录0，结果为010

….

最后当末位为1，为奇数，再记录1，结果为100100011111111010011101010

 

当然也可以采用分块计算，即把numStr分成多个整数，如：

numStr = “755432109876543210”;

可以每次取8位放入到vector<unsigned> arr中，得到：

Arr = {75,54321098,76543210};

同样，也可以按照上面的除法，只是数组元素之间的补位值相当于10^8。

如，第一个数75&1==1，所以补位为1，也就说，

下一个数是154321098>>1.

 

 

2.加操作：

对应dats各块，进行加，数组的每个位置和用long long carry记录，则该为的值 = carry & (1<<32 -1)

而进位值为carry>>32.具体需要注意位对齐细节，以及最后的进位处理，可能会进位多次。还要注意数的符号。

 

3.减操作：

减法是通过反码来实现的，其实就是利用越界来实现。

如：正数A – 正数B，首先求正数B的反码

若当相减之后为正数，那就是（正数A + 正数B的反码）再加1.

若相减之后为负数，那么就是（正数A + 正数B的反码）再取反码。当然别忘了修改符号sign。

 

对于乘除，则需要对数组各个部分进行分开来看，考虑进位即可。

跟加法很相似，只是原来各位置相加变为一个位置对应多个位置的相加。

 

4.除法操作：

按照位运算来操纵。

 

贴上代码，代码只是半成品，未经过测试，不能直接使用。

/* bigInt.h made by g_idea 20110301**/ #include <string> #include <vector> #include <algorithm> #define JUDGECARRY 4294967296 class bigInt { public: bigInt() {} bigInt(const std::string src) { if( !StrToBigInt(src) ) throw "Input num Error!"; } bigInt( int num ) { if( num < 0 ) { sign = 1; dats.push_back(-num); } else { sign = 0; dats.push_back(num); } } bigInt( const bigInt& otherNum ):sign(otherNum.sign),dats(otherNum.dats) {} ~bigInt() {} public: //重置 void reset() { this->sign = 0; this->dats.clear(); this->dats.push_back(0); } //左移 bigInt operator<<(unsigned length) { bigInt result; result.sign = this->sign; result.dats = std::vector<unsigned>(length/32,0); length %= 32; long long carry = 0; for( size_t i = 0; i < dats.size(); ++i ) { long long temp = dats[i]; temp <<= length; result.dats.push_back( unsigned((temp^carry)&(JUDGECARRY-1)) ); carry = temp>>32; } if( carry > 0 ) result.dats.push_back(unsigned(carry)); result.bigIntReserve(); return result; } //右移 bigInt operator>>(unsigned length) { bigInt result; result.sign = this->sign; int pos(length/32); length %= 32; unsigned selectBits = ((1<<length) - 1); long long carry = 0; result.dats = std::vector<unsigned>(dats.size() - pos); for( int i = int(dats.size()-1); i >= pos; --i ) { result.dats[i-pos] = unsigned((((carry<<32)+dats[i])>>length)&(JUDGECARRY-1)); carry = (dats[i]&selectBits); } VecReserve<unsigned>(result.dats); return result; } //逻辑与 bigInt operator&( bigInt& right ) { bigInt result; size_t length = this->dats.size()<right.dats.size()?dats.size():right.dats.size(); for(size_t i=0; i< length; ++i) { result.dats.push_back(dats[i] & right.dats[i] ); } VecReserve<unsigned>(result.dats); return result; } //逻辑或 bigInt operator|( bigInt& right ) { bigInt result; result.sign = 0; std::vector<unsigned> *minPos = &dats;//指向位数较小 std::vector<unsigned> *maxPos = &right.dats;//指向位数较大 if( dats.size() > right.dats.size() ) { maxPos = &dats; minPos = &right.dats; } for(int i=0; i< int(minPos->size()); ++i) { result.dats.push_back(minPos->at(i)|maxPos->at(i) ); } //将剩余高位直接拷贝 for(int i = int(minPos->size()); i < int(maxPos->size()); ++i) { result.dats.push_back(maxPos->at(i)); } return result; } //逻辑异或 bigInt operator^( bigInt& right ) { bigInt result; result.sign = 0; std::vector<unsigned> *minPos = &dats;//指向位数较小 std::vector<unsigned> *maxPos = &right.dats;//指向位数较大 if( dats.size() > right.dats.size() ) { maxPos = &dats; minPos = &right.dats; } for(int i=0; i< int(minPos->size()); ++i) { result.dats.push_back(minPos->at(i)^maxPos->at(i) ); } //将剩余高位直接拷贝 for(int i = int(minPos->size()); i < int(maxPos->size()); ++i) { result.dats.push_back(maxPos->at(i)); } VecReserve<unsigned>(result.dats); return result; } bigInt bigInt::operator|=(bigInt right) { *this=*this|right; return *this; } bigInt bigInt::operator&=(bigInt right) { *this=*this&right; return *this; } bigInt bigInt::operator^=(bigInt right) { *this=*this^right; return *this; } bigInt bigInt::operator<<=(unsigned right) { *this=*this<<right; return *this; } bigInt bigInt::operator>>=(unsigned right) { *this = *this>>right; return *this; } std::string GetDecimalStr() { if( dats.size() == 1 && dats.front() == 0 ) return "0"; int length = int(dats.size()); std::string result = "0"; //2^32 = 4294967296 std::string two_32 = "6927694924"; for( int i = length-1; i >= 0; --i ) { unsigned long num = dats[i]; std::string temp; while(num>0) { temp.push_back( char(num%10 + '0')); num /= 10; } if( i == length-1 ) { result = temp; continue; } result = strNumAdd(strNumMul(result,two_32),temp); } if( sign == 1 ) result.push_back('-'); std::reverse(result.begin(),result.end()); return result; } // 计算n的p次方,如：3^7 = 3*3*3*3*3*3*3 = ((3*3)*(3*3)) * (3*3) * 3 bigInt Power(bigInt n, bigInt p) { bigInt ONE(1); bigInt odd(1); //oddk用来计算“剩下的”乘积 while (p > ONE)// 一直计算到指数小于或等于1 { if ( (p&ONE) == ONE )// 判断p是否奇数 { odd = odd*n; // 若odd为奇数，则把“剩下的”乘起来 } n = n*n; // 主体乘方 p >>=1; // 指数除以2 } return n * odd; // 最后把主体和“剩下的”乘起来作为结果 } ////快速计算(n^p)%m的值，用到定理(X*Y)%Z=((X%Z)*(Y%Z))%Z ////算法实现原理同Power一样 //unsigned Montgomery(unsigned n,unsigned p,unsigned m) //{ // unsigned k=1; // n%=m; // while(p!=1) // { // if( (p&1) == 1 ) // k=(k*n)%m; // n=(n*n)%m; // p>>=1;// 指数除以2 // } // return(n*k)%m; //} friend bigInt operator+(bigInt& left, bigInt& right); friend bigInt operator-(bigInt& left, bigInt& right); friend bigInt operator*(bigInt& left, bigInt& right); friend bigInt operator/(bigInt& left, bigInt& right); friend bool operator>(bigInt& left, bigInt& right); friend bool operator<(bigInt& left, bigInt& right); friend bool operator==(bigInt& left, bigInt& right); friend bool operator>=(bigInt& left, bigInt& right); friend bool operator<=(bigInt& left, bigInt& right); friend bool operator!=(bigInt& left, bigInt& right); friend bigInt operator+(bigInt& left, int& right); friend bigInt operator-(bigInt& left, int& right); friend bigInt operator*(bigInt& left, int& right); friend bigInt operator/(bigInt& left, int& right); friend bool operator>(bigInt& left, int& right); friend bool operator<(bigInt& left, int& right); friend bool operator==(bigInt& left, int& right); friend bool operator>=(bigInt& left, int& right); friend bool operator<=(bigInt& left, int& right); friend bool operator!=(bigInt& left, int& right); //friend bigInt operator%(bigInt& left, bigInt& right); //friend bigInt operator<<(bigInt& left, bigInt& right); //friend bigInt operator>>(bigInt& left, bigInt& right); //friend bigInt operator+=(bigInt& left, bigInt& right); //friend bigInt operator-=(bigInt& left, bigInt& right); //friend bigInt operator*=(bigInt& left, bigInt& right); //friend bigInt operator/=(bigInt& left, bigInt& right); //friend bigInt operator-(bigInt& right); private: char sign; //+:0, -:1 std::vector<unsigned> dats;//低位到高位存放 private: //清除多余位 void bigIntReserve() { VecReserve(this->dats); if( this->dats.empty() ) { this->dats.push_back(0); this->sign = 0; } } //清除高位上的0 template<typename _T> void VecReserve(std::vector<_T>& arr) { while(!arr.empty()) { if( arr.back() == 0 ) arr.pop_back(); else break; } } //获取二进制总位数 unsigned getBitsNum() { unsigned length = unsigned(this->dats.size()-1)*32; long long carry = 1; while( carry <= this->dats.back() ) { carry<<=1; ++length; } return length; } //正数比较大小 int compPlus( bigInt&left, bigInt& right ) { size_t lengthL = left.dats.size(); size_t lengthR = right.dats.size(); if( lengthL > lengthR ) return 1; else if( lengthL < lengthR ) return -1; else { for( int i = int(lengthL-1); i >=0; --i ) { if( left.dats[i] > right.dats[i] ) return 1; if( left.dats[i] < right.dats[i] ) return -1; } } return 0; } //比较大小:1:>, 0:==, -1:< int compVal( bigInt& right ) { if( sign == 1 && right.sign == 0 ) return -1; if( sign == 0 && right.sign == 1 ) return 1; int comp = compPlus(*this,right); if( sign == 1 ) comp = -comp; return comp; } //读取字符型输入数据 bool StrToBigInt(const std::string src) { this->sign = 0;//默认为正 this->dats = std::vector<unsigned>(src.size()/9 +1,0);//分配空间,只多不少 //将字符串转为数字 std::vector<int> NumTemp; for( int i = int(src.size() - 1); i > 0; --i ) { if( '0'<=src[0] && '9'>=src[0] ) NumTemp.push_back(src[i]-'0'); else return false; } if( '0'<src[0] && '9'>=src[0] ) { NumTemp.push_back(src[0]-'0'); }else if( '-' == src[0]) this->sign = 1; else if( '+'!= src[0] ) return false; long long tempCarry = 1;//标记位置 size_t Pos = 0;//对应数组元素位置 while( !NumTemp.empty() ) { if( tempCarry >= JUDGECARRY ) { ++Pos; tempCarry>>=32; } if( 1 == (1&NumTemp[0]) )//为奇数 { this->dats[Pos] ^= tempCarry; } //将NumTemp表示的数除2 for( int i = int(NumTemp.size()-1); i >=0; --i ) { if( 1==(1&NumTemp[i]) && 0<i ) NumTemp[i-1] += 10; NumTemp[i]>>=1; } tempCarry<<=1; VecReserve<int>(NumTemp);//清理为0元素 } this->bigIntReserve(); return true; } //取反 void Negate( size_t length = 0 ) { size_t i=0; for( ; i < dats.size(); ++i) dats[i] = ~dats[i]; for( ; i < length; ++i ) dats.push_back(~unsigned(0)); } //加,取this的符号为结果符号 bigInt Add(bigInt &right) { std::vector<unsigned> *minPos = &dats;//指向位数较小 std::vector<unsigned> *maxPos = &right.dats;//指向位数较大 if( dats.size() > right.dats.size() ) { maxPos = &dats; minPos = &right.dats; } bigInt result; long long carry = 0; for(int i=0; i< int(minPos->size()); ++i) { carry = (long long)minPos->at(i) + (long long)maxPos->at(i)+ carry; result.dats.push_back( unsigned( carry & (JUDGECARRY-1) ) ); carry>>=32; } //将进位carry再与本数相加,可能会多次进位. for(int i = int(minPos->size()); i < int(maxPos->size()); ++i) { carry += (long long)maxPos->at(i); result.dats.push_back(unsigned( carry&(JUDGECARRY-1) )); carry>>=32; } if( carry != 0 ) result.dats.push_back(unsigned(carry)); result.sign = this->sign; return result; } //减操作,其中:this和right均为正数 bigInt Sub(bigInt &right) { size_t maxLength = dats.size()>right.dats.size()?dats.size():right.dats.size(); bigInt result(*this); bigInt subtrahend(right); //减数 subtrahend.Negate(maxLength); result = result.Add(subtrahend); if(result.dats.size() > maxLength)//数组长度增加则为正,得到结果需再加1. { result.dats.pop_back(); result = result.Add(bigInt(1));//再加1. } else//为负则再取反码 { result.Negate(); result.sign = 1; } result.bigIntReserve(); return result; } //乘操作 bigInt Mul(bigInt &right) { bigInt result; result.dats = std::vector<unsigned>(dats.size()+right.dats.size()+1,0); size_t posl, posr; unsigned long long carry; for(posl=0; posl<dats.size(); ++posl) { bigInt temp; temp.dats = std::vector<unsigned>(dats.size()+right.dats.size()+1,0); for(carry=posr=0; posr<right.dats.size(); ++posr) { carry = (long long)dats[posl] * (long long)right.dats[posr] + carry; temp.dats[posr+posl] = (unsigned)carry&(JUDGECARRY-1);//posr+posl确定位置 carry>>=32; } temp.dats[posr+posl]=(unsigned)carry; result = result.Add(temp); } VecReserve<unsigned>(result.dats); if( this->sign == right.sign ) result.sign = 0; else result.sign = 1; return result; } //除操作,this为被除数,right为除数,qoutient为商,remainder为余数 bool Div(bigInt &right,bigInt"ient, bigInt& remainder) { if( right == bigInt(0) ) return false; quotient.reset(); remainder.reset(); int test = compPlus(*this,right); if( test == -1 ) { quotient = 0; remainder = *this; return true; } if( test == 0 ) { quotient = 1; remainder = 0; return true; } bigInt dividend = *this; int dis = int(this->getBitsNum() - right.getBitsNum()); while(dis>=0) { bigInt temp = right<<unsigned(dis); int comp = compPlus(dividend,temp); if( comp >= 0 ) { dividend = dividend.Sub(temp); quotient = quotient.Add(bigInt(1)<<unsigned(dis)); } --dis; } remainder = dividend; VecReserve<unsigned>(remainder.dats); if( this->sign == right.sign ) quotient.sign = 0; else quotient.sign = 1; return true; } std::string strNumAdd( std::string left, std::string right ) { size_t maxLength = (left.length() > right.length() ? left.length():right.length())+1; std::string result = left; result.resize(maxLength,'0'); int carry = 0; size_t pos = 0; for( ; pos < right.length(); ++pos ) { carry = result[pos]-'0' + right[pos] - '0' + carry; result[pos] = carry%10 + '0'; carry /= 10; } for( ; carry > 0; ++pos ) { carry = result[pos] - '0' + carry; result[pos] = carry%10 + '0'; carry /= 10; } if( result[result.length()-1] == '0' ) result.resize(result.length()-1); return result; } std::string strNumMul( std::string left, std::string right ) { std::string result, tmp; size_t i,j; size_t carry; for( i = 0; i < right.length(); ++i ) { tmp.resize( (left.length() + 1 + i),'0' ); for( carry = j = 0; j< left.length(); j++ ) { carry = (left[j] - '0') * (right[i] - '0' ) + carry; tmp[j+i] = char(carry%10 + '0');//j+i确定位置 carry /= 10; } tmp[j+i] = char(carry + '0'); result = strNumAdd(result,tmp); tmp.clear(); } while( result[result.length()-1] == '0' ) result.resize(result.length()-1); return result; } }; //友元函数 //加法 bigInt operator+(bigInt& left, bigInt& right) { if( left.sign == right.sign ) return left.Add(right); if( left.sign == 0 && right.sign == 1 ) return left.Sub(right); else return right.Sub(left); } //减法 bigInt operator-(bigInt& left, bigInt& right) { if( left.sign == right.sign ) { if( left.sign == 0 ) return left.Sub(right); else return right.Sub(left); } else return left.Add(right); } bigInt operator*(bigInt& left, bigInt& right) { bigInt zero(0); if( left == zero || right == zero) return zero; return left.Mul(right); } bigInt operator/(bigInt& left, bigInt& right) { bigInt quotient; if( !left.Div(right,quotient,bigInt()) ) throw "error: Divisor is Zero!"; return quotient; } bool operator<(bigInt& left, bigInt& right) { if( left.compVal(right) == -1 ) return true; return false; } bool operator<=(bigInt& left, bigInt& right) { if(left.compVal(right) <= 0 ) return true; return false; } bool operator>(bigInt& left, bigInt& right) { if( left.compVal(right) == 1 ) return true; return false; } bool operator>=(bigInt& left, bigInt& right) { if( left.compVal(right) >= 0 ) return true; return false; } bool operator==(bigInt& left, bigInt& right) { if( left.compVal(right) == 0 ) return true; return false; } bool operator!=(bigInt& right, bigInt& left) { if( left.compVal(left) == 0 ) return false; return true; } bigInt operator+(bigInt& left, int& right) { return left + bigInt(right); } bigInt operator-(bigInt& left, int& right) { return left - bigInt(right); } bigInt operator*(bigInt& left, int& right) { return left * bigInt(right); } bigInt operator/(bigInt& left, int& right) { return left / bigInt(right); } bool operator>(bigInt& left, int& right) { return left > bigInt(right); } bool operator<(bigInt& left, int& right) { return left < bigInt(right); } bool operator==(bigInt& left, int& right) { return left == bigInt(right); } bool operator>=(bigInt& left, int& right) { return left >= bigInt(right); } bool operator<=(bigInt& left, int& right) { return left <= bigInt(right); } bool operator!=(bigInt& left, int& right) { return left != bigInt(right); }