功能强大的大数模板

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本文介绍了一个自定义的大数运算库，支持多种基本算术运算及高级运算如模幂运算和素数测试。该库能够处理非常大的整数，提供加、减、乘、除、取模等功能，并实现了RSA相关的算法，例如拉宾米勒素性测试。

#include <cstdio> #include <cstring> #include <cstdlib> //允许生成1120位（二进制）的中间结果 #define BI_MAXLEN 105 #define DEC 10 #define HEX 16 class CBigInt { public: //大数在0x100000000进制下的长度 unsigned m_nLength; //用数组记录大数在0x100000000进制下每一位的值 unsigned long m_ulValue[BI_MAXLEN]; CBigInt(); ~CBigInt(); /***************************************************************** 基本操作与运算 Mov，赋值运算，可赋值为大数或普通整数，可重载为运算符“=” Cmp，比较运算，可重载为运算符“==”、“!=”、“>=”、“<=”等 Add，加，求大数与大数或大数与普通整数的和，可重载为运算符“+” Sub，减，求大数与大数或大数与普通整数的差，可重载为运算符“-” Mul，乘，求大数与大数或大数与普通整数的积，可重载为运算符“*” Div，除，求大数与大数或大数与普通整数的商，可重载为运算符“/” Mod，模，求大数与大数或大数与普通整数的模，可重载为运算符“%” *****************************************************************/ void Mov(unsigned __int64 A); void Mov(CBigInt& A); CBigInt Add(CBigInt& A); CBigInt Sub(CBigInt& A); CBigInt Mul(CBigInt& A); CBigInt Div(CBigInt& A); CBigInt Mod(CBigInt& A); CBigInt Add(unsigned long A); CBigInt Sub(unsigned long A); CBigInt Mul(unsigned long A); CBigInt Div(unsigned long A); unsigned long Mod(unsigned long A); int Cmp(CBigInt& A); /***************************************************************** 输入输出 Get，从字符串按10进制或16进制格式输入到大数 Put，将大数按10进制或16进制格式输出到字符串 *****************************************************************/ void Get(char str[], unsigned int system=DEC); void Put(char str[], unsigned int system=DEC); /***************************************************************** RSA相关运算 Rab，拉宾米勒算法进行素数测试 Euc，欧几里德算法求解同余方程 RsaTrans，反复平方算法进行幂模运算 GetPrime，产生指定长度的随机大素数 *****************************************************************/ int Rab(); CBigInt Euc(CBigInt& A); CBigInt RsaTrans(CBigInt& A, CBigInt& B); void GetPrime(int bits); }; //小素数表 const static int PrimeTable[550]= { 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001 }; //构造大数对象并初始化为零 CBigInt::CBigInt() { m_nLength=1; for(int i=0;i<BI_MAXLEN;i++)m_ulValue[i]=0; } //解构大数对象 CBigInt::~CBigInt() { } /**************************************************************************************** 大数比较调用方式：N.Cmp(A) 返回值：若N<A返回-1；若N=A返回0；若N>A返回1 ****************************************************************************************/ int CBigInt::Cmp(CBigInt& A) { if(m_nLength>A.m_nLength)return 1; if(m_nLength<A.m_nLength)return -1; for(int i=m_nLength-1;i>=0;i--) { if(m_ulValue[i]>A.m_ulValue[i])return 1; if(m_ulValue[i]<A.m_ulValue[i])return -1; } return 0; } /**************************************************************************************** 大数赋值调用方式：N.Mov(A) 返回值：无，N被赋值为A ****************************************************************************************/ void CBigInt::Mov(CBigInt& A) { m_nLength=A.m_nLength; for(int i=0;i<BI_MAXLEN;i++)m_ulValue[i]=A.m_ulValue[i]; } void CBigInt::Mov(unsigned __int64 A) { if(A>0xffffffff) { m_nLength=2; m_ulValue[1]=(unsigned long)(A>>32); m_ulValue[0]=(unsigned long)A; } else { m_nLength=1; m_ulValue[0]=(unsigned long)A; } for(int i=m_nLength;i<BI_MAXLEN;i++)m_ulValue[i]=0; } /**************************************************************************************** 大数相加调用形式：N.Add(A) 返回值：N+A ****************************************************************************************/ CBigInt CBigInt::Add(CBigInt& A) { CBigInt X; X.Mov(*this); unsigned carry=0; unsigned __int64 sum=0; if(X.m_nLength<A.m_nLength)X.m_nLength=A.m_nLength; for(unsigned i=0;i<X.m_nLength;i++) { sum=A.m_ulValue[i]; sum=sum+X.m_ulValue[i]+carry; X.m_ulValue[i]=(unsigned long)sum; carry=(unsigned)(sum>>32); } X.m_ulValue[X.m_nLength]=carry; X.m_nLength+=carry; return X; } CBigInt CBigInt::Add(unsigned long A) { CBigInt X; X.Mov(*this); unsigned __int64 sum; sum=X.m_ulValue[0]; sum+=A; X.m_ulValue[0]=(unsigned long)sum; if(sum>0xffffffff) { unsigned i=1; while(X.m_ulValue[i]==0xffffffff){X.m_ulValue[i]=0;i++;} X.m_ulValue[i]++; if(m_nLength==i)m_nLength++; } return X; } /**************************************************************************************** 大数相减调用形式：N.Sub(A) 返回值：N-A ****************************************************************************************/ CBigInt CBigInt::Sub(CBigInt& A) { CBigInt X; X.Mov(*this); if(X.Cmp(A)<=0){X.Mov(0);return X;} unsigned carry=0; unsigned __int64 num; unsigned i; for(i=0;i<m_nLength;i++) { if((m_ulValue[i]>A.m_ulValue[i])||((m_ulValue[i]==A.m_ulValue[i])&&(carry==0))) { X.m_ulValue[i]=m_ulValue[i]-carry-A.m_ulValue[i]; carry=0; } else { num=0x100000000+m_ulValue[i]; X.m_ulValue[i]=(unsigned long)(num-carry-A.m_ulValue[i]); carry=1; } } while(X.m_ulValue[X.m_nLength-1]==0)X.m_nLength--; return X; } CBigInt CBigInt::Sub(unsigned long A) { CBigInt X; X.Mov(*this); if(X.m_ulValue[0]>=A){X.m_ulValue[0]-=A;return X;} if(X.m_nLength==1){X.Mov(0);return X;} unsigned __int64 num=0x100000000+X.m_ulValue[0]; X.m_ulValue[0]=(unsigned long)(num-A); int i=1; while(X.m_ulValue[i]==0){X.m_ulValue[i]=0xffffffff;i++;} X.m_ulValue[i]--; if(X.m_ulValue[i]==0)X.m_nLength--; return X; } /**************************************************************************************** 大数相乘调用形式：N.Mul(A) 返回值：N*A ****************************************************************************************/ CBigInt CBigInt::Mul(CBigInt& A) { if(A.m_nLength==1)return Mul(A.m_ulValue[0]); CBigInt X; unsigned __int64 sum,mul=0,carry=0; unsigned i,j; X.m_nLength=m_nLength+A.m_nLength-1; for(i=0;i<X.m_nLength;i++) { sum=carry; carry=0; for(j=0;j<A.m_nLength;j++) { if(((i-j)>=0)&&((i-j)<m_nLength)) { mul=m_ulValue[i-j]; mul*=A.m_ulValue[j]; carry+=mul>>32; mul=mul&0xffffffff; sum+=mul; } } carry+=sum>>32; X.m_ulValue[i]=(unsigned long)sum; } if(carry){X.m_nLength++;X.m_ulValue[X.m_nLength-1]=(unsigned long)carry;} return X; } CBigInt CBigInt::Mul(unsigned long A) { CBigInt X; unsigned __int64 mul; unsigned long carry=0; X.Mov(*this); for(unsigned i=0;i<m_nLength;i++) { mul=m_ulValue[i]; mul=mul*A+carry; X.m_ulValue[i]=(unsigned long)mul; carry=(unsigned long)(mul>>32); } if(carry){X.m_nLength++;X.m_ulValue[X.m_nLength-1]=carry;} return X; } /**************************************************************************************** 大数相除调用形式：N.Div(A) 返回值：N/A ****************************************************************************************/ CBigInt CBigInt::Div(CBigInt& A) { if(A.m_nLength==1)return Div(A.m_ulValue[0]); CBigInt X,Y,Z; unsigned i,len; unsigned __int64 num,div; Y.Mov(*this); while(Y.Cmp(A)>=0) { div=Y.m_ulValue[Y.m_nLength-1]; num=A.m_ulValue[A.m_nLength-1]; len=Y.m_nLength-A.m_nLength; if((div==num)&&(len==0)){X.Mov(X.Add(1));break;} if((div<=num)&&len){len--;div=(div<<32)+Y.m_ulValue[Y.m_nLength-2];} div=div/(num+1); Z.Mov(div); if(len) { Z.m_nLength+=len; for(i=Z.m_nLength-1;i>=len;i--)Z.m_ulValue[i]=Z.m_ulValue[i-len]; for(i=0;i<len;i++)Z.m_ulValue[i]=0; } X.Mov(X.Add(Z)); Y.Mov(Y.Sub(A.Mul(Z))); } return X; } CBigInt CBigInt::Div(unsigned long A) { CBigInt X; X.Mov(*this); if(X.m_nLength==1){X.m_ulValue[0]=X.m_ulValue[0]/A;return X;} unsigned __int64 div,mul; unsigned long carry=0; for(int i=X.m_nLength-1;i>=0;i--) { div=carry; div=(div<<32)+X.m_ulValue[i]; X.m_ulValue[i]=(unsigned long)(div/A); mul=(div/A)*A; carry=(unsigned long)(div-mul); } if(X.m_ulValue[X.m_nLength-1]==0)X.m_nLength--; return X; } /**************************************************************************************** 大数求模调用形式：N.Mod(A) 返回值：N%A ****************************************************************************************/ CBigInt CBigInt::Mod(CBigInt& A) { CBigInt X,Y; unsigned __int64 div,num; unsigned long carry=0; unsigned i,len; X.Mov(*this); while(X.Cmp(A)>=0) { div=X.m_ulValue[X.m_nLength-1]; num=A.m_ulValue[A.m_nLength-1]; len=X.m_nLength-A.m_nLength; if((div==num)&&(len==0)){X.Mov(X.Sub(A));break;} if((div<=num)&&len){len--;div=(div<<32)+X.m_ulValue[X.m_nLength-2];} div=div/(num+1); Y.Mov(div); Y.Mov(A.Mul(Y)); if(len) { Y.m_nLength+=len; for(i=Y.m_nLength-1;i>=len;i--)Y.m_ulValue[i]=Y.m_ulValue[i-len]; for(i=0;i<len;i++)Y.m_ulValue[i]=0; } X.Mov(X.Sub(Y)); } return X; } unsigned long CBigInt::Mod(unsigned long A) { if(m_nLength==1)return(m_ulValue[0]%A); unsigned __int64 div; unsigned long carry=0; for(int i=m_nLength-1;i>=0;i--) { div=m_ulValue[i]; div+=carry*0x100000000; carry=(unsigned long)(div%A); } return carry; } /**************************************************************************************** 从字符串按10进制或16进制格式输入到大数调用格式：N.Get(str,sys) 返回值：N被赋值为相应大数 sys暂时只能为10或16 ****************************************************************************************/ void CBigInt::Get(char str[], unsigned int system) { int len=strlen(str),k; Mov(0); for(int i=0;i<len;i++) { Mov(Mul(system)); if((str[i]>='0')&&(str[i]<='9'))k=str[i]-48; else if((str[i]>='A')&&(str[i]<='F'))k=str[i]-55; else if((str[i]>='a')&&(str[i]<='f'))k=str[i]-87; else k=0; Mov(Add(k)); } } /**************************************************************************************** 将大数按10进制或16进制格式输出为字符串调用格式：N.Put(str,sys) 返回值：无，参数str被赋值为N的sys进制字符串 sys暂时只能为10或16 ****************************************************************************************/ void CBigInt::Put(char str[], unsigned int system) { if((m_nLength==1)&&(m_ulValue[0]==0)){str="0";return;} char t[]="0123456789ABCDEF"; int a; char ch; CBigInt X; X.Mov(*this); int i = 0; while(X.m_ulValue[X.m_nLength-1]>0) { a=X.Mod(system); ch=t[a]; str[i++] = ch; X.Mov(X.Div(system)); } str[i] = 0x00; int len = strlen(str) - 1; int half_len = strlen(str) / 2; char tmp; for (i = 0; i<half_len; i++) { tmp = str[i]; str[i] = str[len-i]; str[len-i] = tmp; } } /**************************************************************************************** 求不定方程ax-by=1的最小整数解调用方式：N.Euc(A) 返回值：X,满足：NX mod A=1 ****************************************************************************************/ CBigInt CBigInt::Euc(CBigInt& A) { CBigInt M,E,X,Y,I,J; int x,y; M.Mov(A); E.Mov(*this); X.Mov(0); Y.Mov(1); x=y=1; while((E.m_nLength!=1)||(E.m_ulValue[0]!=0)) { I.Mov(M.Div(E)); J.Mov(M.Mod(E)); M.Mov(E); E.Mov(J); J.Mov(Y); Y.Mov(Y.Mul(I)); if(x==y) { if(X.Cmp(Y)>=0)Y.Mov(X.Sub(Y)); else{Y.Mov(Y.Sub(X));y=0;} } else{Y.Mov(X.Add(Y));x=1-x;y=1-y;} X.Mov(J); } if(x==0)X.Mov(A.Sub(X)); return X; } /**************************************************************************************** 求乘方的模调用方式：N.RsaTrans(A,B) 返回值：X=N^A MOD B ****************************************************************************************/ CBigInt CBigInt::RsaTrans(CBigInt& A, CBigInt& B) { CBigInt X,Y; int i,j,k; unsigned n; unsigned long num; k=A.m_nLength*32-32; num=A.m_ulValue[A.m_nLength-1]; while(num){num=num>>1;k++;} X.Mov(*this); for(i=k-2;i>=0;i--) { Y.Mov(X.Mul(X.m_ulValue[X.m_nLength-1])); Y.Mov(Y.Mod(B)); for(n=1;n<X.m_nLength;n++) { for(j=Y.m_nLength;j>0;j--)Y.m_ulValue[j]=Y.m_ulValue[j-1]; Y.m_ulValue[0]=0; Y.m_nLength++; Y.Mov(Y.Add(X.Mul(X.m_ulValue[X.m_nLength-n-1]))); Y.Mov(Y.Mod(B)); } X.Mov(Y); if((A.m_ulValue[i>>5]>>(i&31))&1) { Y.Mov(Mul(X.m_ulValue[X.m_nLength-1])); Y.Mov(Y.Mod(B)); for(n=1;n<X.m_nLength;n++) { for(j=Y.m_nLength;j>0;j--)Y.m_ulValue[j]=Y.m_ulValue[j-1]; Y.m_ulValue[0]=0; Y.m_nLength++; Y.Mov(Y.Add(Mul(X.m_ulValue[X.m_nLength-n-1]))); Y.Mov(Y.Mod(B)); } X.Mov(Y); } } return X; } /**************************************************************************************** 拉宾米勒算法测试素数调用方式：N.Rab() 返回值：若N为素数，返回1，否则返回0 ****************************************************************************************/ int CBigInt::Rab() { unsigned i,j,pass; for(i=0;i<550;i++){if(Mod(PrimeTable[i])==0)return 0;} CBigInt S,A,I,K; K.Mov(*this); K.m_ulValue[0]--; for(i=0;i<5;i++) { pass=0; A.Mov(rand()*rand()); S.Mov(K); while((S.m_ulValue[0]&1)==0) { for(j=0;j<S.m_nLength;j++) { S.m_ulValue[j]=S.m_ulValue[j]>>1; if(S.m_ulValue[j+1]&1)S.m_ulValue[j]=S.m_ulValue[j]|0x80000000; } if(S.m_ulValue[S.m_nLength-1]==0)S.m_nLength--; I.Mov(A.RsaTrans(S,*this)); if(I.Cmp(K)==0){pass=1;break;} } if((I.m_nLength==1)&&(I.m_ulValue[0]==1))pass=1; if(pass==0)return 0; } return 1; } /**************************************************************************************** 产生随机素数调用方法：N.GetPrime(bits) 返回值：N被赋值为一个bits位（0x100000000进制长度）的素数 ****************************************************************************************/ void CBigInt::GetPrime(int bits) { unsigned i; m_nLength=bits; begin: for(i=0;i<m_nLength;i++)m_ulValue[i]=rand()*0x10000+rand(); m_ulValue[0]=m_ulValue[0]|1; for(i=m_nLength-1;i>0;i--) { m_ulValue[i]=m_ulValue[i]<<1; if(m_ulValue[i-1]&0x80000000)m_ulValue[i]++; } m_ulValue[0]=m_ulValue[0]<<1; m_ulValue[0]++; for(i=0;i<550;i++){if(Mod(PrimeTable[i])==0)goto begin;} CBigInt S,A,I,K; K.Mov(*this); K.m_ulValue[0]--; for(i=0;i<5;i++) { A.Mov(rand()*rand()); S.Mov(K.Div(2)); I.Mov(A.RsaTrans(S,*this)); if(((I.m_nLength!=1)||(I.m_ulValue[0]!=1))&&(I.Cmp(K)!=0))goto begin; } } int main() { int t; int i, j; CBigInt big_a, big_b, big_ans; char ans[2005], a[1005], b[1005]; while (scanf("%d", &t) != EOF) { for (i = 0; i<t; i++) { if (i != 0) printf("/n"); scanf("%s%s", a, b); big_a.Get(a); big_b.Get(b); big_ans = big_a.Add(big_b); big_ans.Put(ans); printf("Case %d:/n%s + %s = %s/n", i+1, a, b, ans); } } return 0; }