数学基础与密码学部分实验内容文档格式.docx
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for(k=2;
k<
=1000000;
k++)
if(prime(k))printf("
%d"
k);
return0;
}
程序运行截图:
题目二:
使用VC++编程语言设计实现一个算法程序库,要求包括以下部分:
1)欧几里德算法求a,b的最大公倍数;
2)扩展的欧几里德算法,求出gcd(a,b)和满足gcd(a,b)=ax+by的整数x和y;
3)求解模线性方程ax≡b(modn)其中n>
0;
4)求解模线性方程组(中国余数定理);
5)模取幂运算,计算abmodn(a,b>
1032);
6)Miller-Rabin随机性素数测试算法(要求判定n>
1016);
1)欧几里德算法求a,b的最小公倍数;
程序设计思想:
采用欧几里得思想的推论,两个整数的积相乘并除以他们的最大公约数可以求出这两个整数的最小公倍数。
程序流程图:
程序源代码:
#include"
stdio.h"
intmain()
{
inta,b,c;
printf("
请依次输入两个数:
\n"
);
scanf("
%d%d"
&
a,&
b);
inta_cup,b_cup,res;
/*被除数,除数,余数*/
if(a>
0&
&
b>
0)
{
a_cup=a;
b_cup=b;
res=a_cup%b_cup;
while(res!
=0)
{
a_cup=b_cup;
b_cup=res;
res=a_cup%b_cup;
}
c=a*b/b_cup;
printf("
%d\n"
c);
运行结果:
2)扩展的欧几里德算法,求出gcd(a,b)和满足gcd(a,b)=ax+by的整数x和y;
intx,y,q;
voidextend_Eulid(inta,intb)
if(b==0)
x=1;
y=0;
q=a;
}
else
{
extend_Eulid(b,a%b);
inttemp=x;
x=y;
y=temp-a/b*y;
intmain()
inta,b;
Pleaseinputtwonumbers:
"
if(a<
b)
intm=a;
a=b;
b=m;
elseextend_Eulid(a,b);
%d=(%d)*%d+(%d)*%d\n"
q,x,a,y,b);
3)求解模线性方程ax≡b(modn)其中n>
源代码:
iostream>
string>
time.h>
usingnamespacestd;
BigNum.h"
BigNumgcd(BigNumx,BigNumy)//递归法求最大公约数
BigNumz;
z="
0"
;
if(y==z)
returnx;
//递归出口
else
returngcd(y,x%y);
//递归调用
BigNumQiuNi(BigNuma,BigNumm)//求逆
BigNumr0,r1,r2,s0,s1,s2,t0,t1,t2,q,x,y;
x="
y="
1"
r0=a,r1=m;
s0="
t1="
s1="
t0="
while(r1!
=x)
q=r0/r1;
r2=r0%r1;
s2=s0-q*s1;
t2=t0-q*t1;
s0=s1,s1=s2;
t0=t1,t1=t2;
r0=r1,r1=r2;
if(r0==y)
if(!
a.s&
s0.s)
{
s0=m+s0;
returns0;
}
elseif(!
!
s0.s)returns0;
elseif(a.s&
s0.s)
s0=s0-m;
elsereturnx;
voidmain()
BigNuma,b,m,p,q,s,t;
charx[100],y[100],z[100];
while
(1)//控制多次输入
cout<
<
inputa,b,m:
endl;
cin>
>
x>
y>
z;
s="
t="
a=x;
b=y;
m=z;
p=gcd(a,m);
//求(a,m)
q=QiuNi(a/(p),m/(p));
//求(a/((a,m)))的逆(a/((a,m)))^(-1)
if(b%p==s)//判断是否有解,有解输出解的情况
cout<
解得:
x≡"
((b/p*q)%(m/p)).CoutBigNum();
+"
(m/p).CoutBigNum();
t(mod"
m.CoutBigNum();
)t=0,...,"
(p-t).CoutBigNum();
."
elsecout<
无解!
//无解显示说明
结果为有解和无解两种情况
s.huH^,bB!
V5)模取幂运算,计算abmodn(a,b>
string.h>
#defineMAX100
main()
BigNumf,r,s,b2,c1,c2,b1[500],c3,c4;
chara[MAX]={'
\0'
},b[MAX]={'
},m[MAX]={'
};
inti=0,j=0,t=0;
while
(1)
cout<
inputa,b:
cin>
a>
b;
inputn:
m;
b2=b;
s=m;
f="
c1="
2"
c2="
c4="
r=a;
for(i=0;
500;
i++)b1[i]="
//十进制转换成二进制
i=0;
while(b2!
=c2)
b1[i]=b2%c1;
b2=b2/c1;
i++;
t=i;
for(j=0;
j<
(t-1)/2;
j++)
c3=b1[j];
b1[j]=b1[t-1-j];
b1[t-1-j]=c3;
t;
i++)//求余
f=(f*f)%s;
if(b1[i]==c4)
f=(f*r)%s;
f.CoutBigNum();
2的3次模6等于2
6)Miller-Rabin随机性素数测试算法(要求判定n>
iostream.h"
time.h"
stdlib.h"
math.h"
classjudge_prime
private:
public:
intBtest(inta,intn);
intMillRab(intn);
intRepeatMillRab(intn,intk);
intjudge_prime:
:
Btest(inta,intn)
ints=0;
intt=n-1;
inti=1;
intx=1;
inty;
do
s++;
t=t/2;
}while((t%2)!
=1);
while(i<
=t)
x=(x*a)%n;
i++;
if((x==1)||(x==n-1))return1;
for(intj=1;
=s-1;
y=1;
for(intk=1;
=j;
y=2*y;
i=1;
x=1;
=(y*t))
if(x==n-1)return1;
MillRab(intn)
inta;
srand((unsigned)time(0));
a=rand()%(n-3)+2;
returnBtest(a,n);
RepeatMillRab(intn,intk)
inti;
for(i=1;
=k;
if(MillRab(n)==0)return0;
intn=10000;
intresult=0;
2<
"
3<
for(i=5;
)
judge_primeP;
if(P.RepeatMillRab(i,(int)log10(i)))
i=i+2;
测试结果:
二、密码学基础实验部分
一、使用VC++编程语言设计实现一个200位以上十进制数运算的程序库:
能够完成大整数加、减、乘、除、求模、与、或、非、异或等运算。
二、使用VC++编程语言编写一个程序,实现DES算法,包括以下环节,密钥通
过随机函数产生,并应用密文反馈(CFB)工作模式和输出反馈(OFB)工
作模式进行加密、解密和输出。
#include"
memory.h"
#definePLAIN_FILE_OPEN_ERROR-1
#defineKEY_FILE_OPEN_ERROR-2
#defineCIPHER_FILE_OPEN_ERROR-3
#defineOK1
typedefcharElemType;
//下面是变量的声明
//初始置换表IP
intIP_Table[64]={57,49,41,33,25,17,9,1,
59,51,43,35,27,19,11,3,
61,53,45,37,29,21,13,5,
63,55,47,39,31,23,15,7,
56,48,40,32,24,16,8,0,
58,50,42,34,26,18,10,2,
60,52,44,36,28,20,12,4,
62,54,46,38,30,22,14,6};
//逆初始置换表IP^-1
intIP_1_Table[64]={39,7,47,15,55,23,63,31,
38,6,46,14,54,22,62,30,
37,5,45,13,53,21,61,29,
36,4,44,12,52,20,60,28,
35,3,43,11,51,19,59,27,
34,2,42,10,50,18,58,26,
33,1,41,9,49,17,57,25,
32,0,40,8,48,16,56,24};
//扩充置换表E
intE_Table[48]={31,0,1,2,3,4,
3,4,5,6,7,8,
7,8,9,10,11,12,
11,12,13,14,15,16,
15,16,17,18,19,20,
19,20,21,22,23,24,
23,24,25,26,27,28,
27,28,29,30,31,0};
//置换函数P
intP_Table[32]={15,6,19,20,28,11,27,16,
0,14,22,25,4,17,30,9,
1,7,23,13,31,26,2,8,
18,12,29,5,21,10,3,24};
//S盒
intS[8][4][16]=//S1
{{{14,4,13,1,2,15,11,8,3,10,6,12,5,9,0,7},
{0,15,7,4,14,2,13,1,10,6,12,11,9,5,3,8},
{4,1,14,8,13,6,2,11,15,12,9,7,3,10,5,0},
{15,12,8,2,4,9,1,7,5,11,3,14,10,0,6,13}},
//S2
{{15,1,8,14,6,11,3,4,9,7,2,13,12,0,5,10},
{3,13,4,7,15,2,8,14,12,0,1,10,6,9,11,5},
{0,14,7,11,10,4,13,1,5,8,12,6,9,3,2,15},
{13,8,10,1,3,15,4,2,11,6,7,12,0,5,14,9}},
//S3
{{10,0,9,14,6,3,15,5,1,13,12,7,11,4,2,8},
{13,7,0,9,3,4,6,10,2,8,5,14,12,11,15,1},
{13,6,4,9,8,15,3,0,11,1,2,12,5,10,14,7},
{1,10,13,0,6,9,8,7,4,15,14,3,11,5,2,12}},
//S4
{{7,13,14,3,0,6,9,10,1,2,8,5,11,12,4,15},
{13,8,11,5,6,15,0,3,4,7,2,12,1,10,14,9},
{10,6,9,0,12,11,7,13,15,1,3,14,5,2,8,4},
{3,15,0,6,10,1,13,8,9,4,5,11,12,7,2,14}},
//S5
{{2,12,4,1,7,10,11,6,8,5,3,15,13,0,14,9},
{14,11,2,12,4,7,13,1,5,0,15,10,3,9,8,6},
{4,2,1,11,10,13,7,8,15,9,12,5,6,3,0,14},
{11,8,12,7,1,14,2,13,6,15,0,9,10,4,5,3}},
//S6
{{12,1,10,15,9,2,6,8,0,13,3,4,14,7,5,11},
{10,15,4,2,7,12,9,5,6,1,13,14,0,11,3,8},
{9,14,15,5,2,8,12,3,7,0,4,10,1,13,11,6},
{4,3,2,12,9,5,15,10,11,14,1,7,6,0,8,13}},
//S7
{{4,11,2,14,15,0,8,13,3,12,9,7,5,10,6,1},
{13,0,11,7,4,9,1,10,14,3,5,12,2,15,8,6},
{1,4,11,13,12,3,7,14,10,15,6,8,0,5,9,2},
{6,11,13,8,1,4,10,7,9,5,0,15,14,2,3,12}},
//S8
{{13,2,8,4,6,15,11,1,10,9,3,14,5,0,12,7},
{1,15,13,8,10,3,7,4,12,5,6,11,0,14,9,2},
{7,11,4,1,9,12,14,2,0,6,10,13,15,3,5,8},
{2,1,14,7,4,10,8,13,15,12,9,0,3,5,6,11}}};
//置换选择1
intPC_1[56]={56,48,40,32,24,16,8,
0,57,49,41,33,25,17,
9,1,58,50,42,34,26,
18,10,2,59,51,43,35,
62,54,46,38,30,22,14,
6,61,53,45,37,29,21,
13,5,60,52,44,36,28,
20,12,4,27,19,11,3};
//置换选择2
intPC_2[48]={13,16,10,23,0,4,2,27,
14,5,20,9,22,18,11,3,
25,7,15,6,26,19,12,1,
40,51,30,36,46,54,29,39,
50,44,32,46,43,48,38,55,
33,52,45,41,49,35,28,31};
//对左移次数的规定
intMOVE_TIMES[16]={1,1,2,2,2,2,2,2,1,2,2,2,2,2,2,1};
//下面是函数的声明:
intByteToBit(ElemTypech,ElemTypebit[8]);
intBitToByte(ElemTypebit[8],ElemType*ch);
intChar8ToBit64(ElemTypech[8],ElemTypebit[64]);
intBit64ToChar8(ElemTypebit[64],ElemTypech[8]);
intDES_MakeSubKeys(ElemTypekey[64],ElemTypesubKeys[16][48]);
intDES_PC1_Transform(ElemTypekey[64],ElemTypetempbts[56]);
intDES_PC2_Transform(ElemTypekey[56],ElemTypetempbts[48]);
intDES_ROL(ElemTypedata[56],inttime);
intDES_IP_Transform(ElemTypedata[64]);
intDES_IP_1_Transform(ElemTypedata[64]);
intDES_E_Transform(ElemTypedata[48]);
intDES_P_Transform(ElemTypedata[32]);
intDES_SBOX(ElemTypedata[48]);
intDES_XOR(ElemTypeR[48],ElemTypeL[48],intcount);
intDES_Swap(ElemTypeleft[32],ElemTyperight[32]);
intDES_EncryptBlock(ElemTypeplainBlock[8],ElemTypesubKeys[16][48],ElemTypecipherBlock[8]);
intDES_DecryptBlock(ElemTypecipherBlock[8],ElemTypesubKeys[16][48],ElemTypeplainBlock[8]);
intDES_Encrypt(char*plainFile,char*keyStr,char*cipherFile);
intDES_Decrypt(char*cipherFile,char*keyStr,char*plainFile);
//字节转换成二进制
intByteToBit(ElemTypech,ElemTypebit[8]){
intcnt;
for(cnt=0;
cnt<
8;
cnt++){
*(bit+cnt)=(ch>
cnt)&
1;
}
}
//二进制转换成字节
intBitToByte(ElemTypebit[8],ElemType*ch){
*ch|=*(bit+cnt)<
cnt;
//将长度为8的字符串转为二进制位串
intChar8ToBit64(ElemTypech[8],ElemTypebit[64]){
c