• [BZOJ 1409] Password


    贴一发题面

    1409: Password

    Time Limit: 5 Sec  Memory Limit: 64 MB
    Submit: 242  Solved: 82
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    Description

    Rivest是密码学专家。近日他正在研究一种数列E = {E[1],E[2],……,E[n]},
    且E[1] = E[2] = p(p为一个质数),E[i] = E[i-2]*E[i-1] (若2<i<=n)。
    例如{2,2,4,8,32,256,8192,……}就是p = 2的数列。在此基础上他又设计了一种加密算法,该算法可以通过一个密钥q (q < p)将一个正整数n加密成另外一个正整数d,计算公式为:d = E[n] mod q。现在Rivest想对一组数据进行加密,但他对程序设计不太感兴趣,请你帮助他设计一个数据加密程序。

    Input

    第一行读入m,p。其中m表示数据个数,p用来生成数列E。 以下有m行,每行有2个整数n,q。n为待加密数据,q为密钥。 数据范围: 0 < p n< 2^31 0 < q < p 0 < m <= 5000。

    Output

    将加密后的数据按顺序输出到文件 第i行输出第i个加密后的数据。 输入样例1 2 7 4 5 4 6 输入样例2 4 7 2 4 7 1 6 5 9 3

    Sample Input

    输入样例1
    2 7
    4 5
    4 6
    输入样例2
    4 7
    2 4
    7 1
    6 5
    9 3

    Sample Output

    输出样例1
    3
    1

    输出样例2
    3
    0
    1
    1

    首先我们注意到数列$E$中的数都是$p$的幂,然后我们发现幂指数是一个$Fibonacci$ $Sequence$.由于最后要对$q$取模所以考虑欧拉定理,可以计算出$Finbonacci$ $Sequence$中的第$n$项然后对取模再快速幂走起w

    然后问题转化为如何快速计算$Fibonacci$ $Number$取模,数据范围这么大线性推和推表都别想了w,考虑矩阵快速幂,使用这个矩阵跑快速幂即可:

    然后是求的问题,因为数据范围刚好是$int$的最大值所以就不要想着筛了,这时可以选择先筛一个大小的素数表($u$为数据最大可能值)然后在线计算$varphi$.

    最后注意中间结果可能炸$int$以及其他一些细节(考试的时候$varphi$函数算错了结果惨烈滚粗正解被卡成暴力分QAQ)

    丢代码(懒癌发作所以直接丢了个考试的时候拿来试验的筛好的素数表上去qwq)

     1 #include <cmath>
     2 #include <cstdio>
     3 #include <cstring>
     4 #include <cstdlib>
     5 #include <iostream>
     6 #include <algorithm>
     7  
     8 const long long prime[]={2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997,1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,1063,1069,1087,1091,1093,1097,1103,1109,1117,1123,1129,1151,1153,1163,1171,1181,1187,1193,1201,1213,1217,1223,1229,1231,1237,1249,1259,1277,1279,1283,1289,1291,1297,1301,1303,1307,1319,1321,1327,1361,1367,1373,1381,1399,1409,1423,1427,1429,1433,1439,1447,1451,1453,1459,1471,1481,1483,1487,1489,1493,1499,1511,1523,1531,1543,1549,1553,1559,1567,1571,1579,1583,1597,1601,1607,1609,1613,1619,1621,1627,1637,1657,1663,1667,1669,1693,1697,1699,1709,1721,1723,1733,1741,1747,1753,1759,1777,1783,1787,1789,1801,1811,1823,1831,1847,1861,1867,1871,1873,1877,1879,1889,1901,1907,1913,1931,1933,1949,1951,1973,1979,1987,1993,1997,1999,2003,2011,2017,2027,2029,2039,2053,2063,2069,2081,2083,2087,2089,2099,2111,2113,2129,2131,2137,2141,2143,2153,2161,2179,2203,2207,2213,2221,2237,2239,2243,2251,2267,2269,2273,2281,2287,2293,2297,2309,2311,2333,2339,2341,2347,2351,2357,2371,2377,2381,2383,2389,2393,2399,2411,2417,2423,2437,2441,2447,2459,2467,2473,2477,2503,2521,2531,2539,2543,2549,2551,2557,2579,2591,2593,2609,2617,2621,2633,2647,2657,2659,2663,2671,2677,2683,2687,2689,2693,2699,2707,2711,2713,2719,2729,2731,2741,2749,2753,2767,2777,2789,2791,2797,2801,2803,2819,2833,2837,2843,2851,2857,2861,2879,2887,2897,2903,2909,2917,2927,2939,2953,2957,2963,2969,2971,2999,3001,3011,3019,3023,3037,3041,3049,3061,3067,3079,3083,3089,3109,3119,3121,3137,3163,3167,3169,3181,3187,3191,3203,3209,3217,3221,3229,3251,3253,3257,3259,3271,3299,3301,3307,3313,3319,3323,3329,3331,3343,3347,3359,3361,3371,3373,3389,3391,3407,3413,3433,3449,3457,3461,3463,3467,3469,3491,3499,3511,3517,3527,3529,3533,3539,3541,3547,3557,3559,3571,3581,3583,3593,3607,3613,3617,3623,3631,3637,3643,3659,3671,3673,3677,3691,3697,3701,3709,3719,3727,3733,3739,3761,3767,3769,3779,3793,3797,3803,3821,3823,3833,3847,3851,3853,3863,3877,3881,3889,3907,3911,3917,3919,3923,3929,3931,3943,3947,3967,3989,4001,4003,4007,4013,4019,4021,4027,4049,4051,4057,4073,4079,4091,4093,4099,4111,4127,4129,4133,4139,4153,4157,4159,4177,4201,4211,4217,4219,4229,4231,4241,4243,4253,4259,4261,4271,4273,4283,4289,4297,4327,4337,4339,4349,4357,4363,4373,4391,4397,4409,4421,4423,4441,4447,4451,4457,4463,4481,4483,4493,4507,4513,4517,4519,4523,4547,4549,4561,4567,4583,4591,4597,4603,4621,4637,4639,4643,4649,4651,4657,4663,4673,4679,4691,4703,4721,4723,4729,4733,4751,4759,4783,4787,4789,4793,4799,4801,4813,4817,4831,4861,4871,4877,4889,4903,4909,4919,4931,4933,4937,4943,4951,4957,4967,4969,4973,4987,4993,4999,5003,5009,5011,5021,5023,5039,5051,5059,5077,5081,5087,5099,5101,5107,5113,5119,5147,5153,5167,5171,5179,5189,5197,5209,5227,5231,5233,5237,5261,5273,5279,5281,5297,5303,5309,5323,5333,5347,5351,5381,5387,5393,5399,5407,5413,5417,5419,5431,5437,5441,5443,5449,5471,5477,5479,5483,5501,5503,5507,5519,5521,5527,5531,5557,5563,5569,5573,5581,5591,5623,5639,5641,5647,5651,5653,5657,5659,5669,5683,5689,5693,5701,5711,5717,5737,5741,5743,5749,5779,5783,5791,5801,5807,5813,5821,5827,5839,5843,5849,5851,5857,5861,5867,5869,5879,5881,5897,5903,5923,5927,5939,5953,5981,5987,6007,6011,6029,6037,6043,6047,6053,6067,6073,6079,6089,6091,6101,6113,6121,6131,6133,6143,6151,6163,6173,6197,6199,6203,6211,6217,6221,6229,6247,6257,6263,6269,6271,6277,6287,6299,6301,6311,6317,6323,6329,6337,6343,6353,6359,6361,6367,6373,6379,6389,6397,6421,6427,6449,6451,6469,6473,6481,6491,6521,6529,6547,6551,6553,6563,6569,6571,6577,6581,6599,6607,6619,6637,6653,6659,6661,6673,6679,6689,6691,6701,6703,6709,6719,6733,6737,6761,6763,6779,6781,6791,6793,6803,6823,6827,6829,6833,6841,6857,6863,6869,6871,6883,6899,6907,6911,6917,6947,6949,6959,6961,6967,6971,6977,6983,6991,6997,7001,7013,7019,7027,7039,7043,7057,7069,7079,7103,7109,7121,7127,7129,7151,7159,7177,7187,7193,7207,7211,7213,7219,7229,7237,7243,7247,7253,7283,7297,7307,7309,7321,7331,7333,7349,7351,7369,7393,7411,7417,7433,7451,7457,7459,7477,7481,7487,7489,7499,7507,7517,7523,7529,7537,7541,7547,7549,7559,7561,7573,7577,7583,7589,7591,7603,7607,7621,7639,7643,7649,7669,7673,7681,7687,7691,7699,7703,7717,7723,7727,7741,7753,7757,7759,7789,7793,7817,7823,7829,7841,7853,7867,7873,7877,7879,7883,7901,7907,7919,7927,7933,7937,7949,7951,7963,7993,8009,8011,8017,8039,8053,8059,8069,8081,8087,8089,8093,8101,8111,8117,8123,8147,8161,8167,8171,8179,8191,8209,8219,8221,8231,8233,8237,8243,8263,8269,827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,99623,99643,99661,99667,99679,99689,99709,99713,99719,99721,99733,99761,99767,99793,99809,99817,99823,99829,99833,99839,99859,99871,99877,99881,99901,99907,99923,99929,99961,99971,99991,100003};
     9  
    10 long long Fibonacci(long long,long long);
    11 long long Phi(long long);
    12 long long Pow(long long,long long,long long);
    13  
    14 int main(){
    15     int m,p,q,n;
    16     scanf("%d%d",&m,&p);
    17     while(m--){
    18         scanf("%d%d",&n,&q);
    19         printf("%lld
    ",Pow(p,Fibonacci(n-1,Phi(q))/*+Phi(q)*/,q));
    20     }
    21     return 0;
    22 }
    23  
    24 long long Phi(long long x){
    25     long long ans=x;
    26     long long k=ceil(sqrt(x));
    27     for(long long i=0;prime[i]<=k;i++){
    28         if(x%prime[i]==0){
    29             ans=ans*(prime[i]-1)/prime[i];
    30             while(x%prime[i]==0)x/=prime[i];
    31         }
    32     }
    33     if(x>1)
    34         ans=ans*(x-1)/x;
    35     return ans;
    36 }
    37  
    38 long long Pow(long long a,long long n,long long p){
    39     int ans=1;
    40     while(n!=0){
    41         if((n&1)!=0){
    42             ans=ans*a%p;
    43         }
    44         n>>=1;
    45         a=a*a%p;
    46     }
    47     ans%=p;
    48     return ans;
    49 }
    50  
    51 long long Fibonacci(long long x,long long p){
    52     long long a[2][2],b[2][2],c[2][2];
    53     memset(a,0,sizeof(a));
    54     memset(b,0,sizeof(b));
    55     a[0][0]=a[1][0]=a[0][1]=1;
    56     b[0][0]=b[1][1]=1;
    57     while(x!=0){
    58         if((x&1)!=0){
    59             memset(c,0,sizeof(c));
    60             for(int i=0;i<2;i++)
    61                 for(int j=0;j<2;j++)
    62                     for(int k=0;k<2;k++)
    63                         c[i][j]=(c[i][j]+b[i][k]*a[k][j]%p)%p;
    64             memcpy(b,c,sizeof(c));
    65         }
    66         x>>=1;
    67         memset(c,0,sizeof(c));
    68         for(int i=0;i<2;i++)
    69             for(int j=0;j<2;j++)
    70                 for(int k=0;k<2;k++)
    71                     c[i][j]=(c[i][j]+a[i][k]*a[k][j]%p)%p;
    72         memcpy(a,c,sizeof(c));
    73     }
    74     return b[0][0];
    75 }
    Backup

    然后丢图跑w

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  • 原文地址:https://www.cnblogs.com/rvalue/p/7259292.html
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