BSGS (big small step algorithm)
Known \ (A, B, C \) , find \ (X \) . Order \ (A ^ X \ equiv B \ PMOD C \) .
step
\ [m = \ lceil \ sqrtc \ \ rceil \] \ [X = I * MJ \ \ (I \ in [. 1, m], J \ in [0, m]) \] \ [A ^ {I * } MJ \ equiv B \ PMOD C \] \ [A ^ {m} * I \ * A B equiv ^ J \ PMOD C \]
enum \ (a ^ j (j \ in [0, m]) \) Add \ (the hash \) table inside, and then enumerate \ (a ^ {m} * I \) , in \ (the hash \) table look inside the same there, the same as if it exists.
If the interrogation in the case of more, can be \ (BM = \ lceil {C ^ \ FRAC. 3} {2} {} \ rceil \) , \ (GM = \ lceil {C ^ \ FRAC. 1 {{}. 3 }} \ rceil \) \ ((I \ in [. 1, GM], J \ in [0, BM]) \)
Implementation code
struct Hash{
int head[maxm], cnt, mod;
struct Node{
int v, id, next;
}node[maxn];
void init(){
mes(head, -1);
cnt = 0;
mod = 1333331;
}
void insert(int x, int id){
int u = x%mod;
node[++cnt].v = x;
node[cnt].id = id;
node[cnt].next = head[u];
head[u] = cnt;
}
int find(int x){
int u = x%mod;
for(int i = head[u]; ~i; i = node[i].next){
if(node[i].v == x)
return node[i].id;
}
return -1;
}
}hs;
ll qpow(ll a, ll b, int mod){
ll ans = 1;
while(b){
if(b&1)
ans = ans*a%mod;
a = a*a%mod;
b /= 2;
}
return ans;
}
struct BSGS{
int bm, gm, a, b, x0, p;
void init(){
bm = (int)ceil(pow(p, 2.0/3));
gm = (int)ceil(pow(p, 1.0/3));
ll ans = 1;
hs.init();
for(int i = 0; i <= bm; i++){ //a^(i*bm - j)%p = b;
hs.insert(ans, i);
ans = ans*a%p;
}
}
ll get(ll v){
v = qpow(v, p-2, p);
ll num = qpow(a, bm, p);
for(int i = 1; i <= gm; i++){
v = v*num%p;
int ans = hs.find(v);
if(ans != -1)
return 1ll*i*bm - ans;
}
return -1;
}
}bsgs;
example
Cattle off more school training camp fifth - generator 2
\ (x_n = (A * X_ {n--. 1} + B) \% P \) , you \ (n-, x_0, A, B, P \) , \ (Q \) th ask \ (x_0, x_1, x_2 ..., x_ {n -1} \) finding the minimum compliance \ (x_i = v \) a \ (I \) , the output is not present \ (- 1 \) .
Thinking
\ [x_n = A * X_ {n--. 1} + B \] \ [x_n = A ^ n-* x_0 + B * \ FRAC {. 1-A ^ n-} {. 1-A} \] \ [A ^ n-= \ FRAC {(A-. 1) _ * x_n + B} {(A-. 1) * x_0 + B} \] \ [x_n = V, so that B = \ frac {(a- 1) _ * v + b} {(a-1) * x_0 + b} \] it is seeking \ [^ n-a \ B equiv \ PMOD P \]
\ (a = 0, a =. 1 \) case to consider the special
AC Code
#include <map>
#include <set>
#include <list>
#include <ctime>
#include <cmath>
#include <stack>
#include <queue>
#include <cfloat>
#include <string>
#include <vector>
#include <cstdio>
#include <bitset>
#include <cstdlib>
#include <cstring>
#include <iostream>
#include <algorithm>
#define lowbit(x) x & (-x)
#define mes(a, b) memset(a, b, sizeof a)
#define fi first
#define se second
#define pb push_back
#define pii pair<int, int>
typedef unsigned long long int ull;
typedef long long int ll;
const int maxn = 1e6 + 10;
const int maxm = 2e6 + 10;
const ll INF = 1e18 + 100;
const int inf = 0x3f3f3f3f;
const double pi = acos(-1.0);
const double eps = 1e-8;
using namespace std;
int n, m;
int cas, tol, T;
struct Hash{
int head[maxm], cnt, mod;
struct Node{
int v, id, next;
}node[maxn];
void init(){
mes(head, -1);
cnt = 0;
mod = 1333331;
}
void insert(int x, int id){
int u = x%mod;
node[++cnt].v = x;
node[cnt].id = id;
node[cnt].next = head[u];
head[u] = cnt;
}
int find(int x){
int u = x%mod;
for(int i = head[u]; ~i; i = node[i].next){
if(node[i].v == x)
return node[i].id;
}
return -1;
}
}hs;
ll qpow(ll a, ll b, int mod){
ll ans = 1;
while(b){
if(b&1)
ans = ans*a%mod;
a = a*a%mod;
b /= 2;
}
return ans;
}
struct BSGS{
int bm, gm, a, b, x0, p;
void init(){
bm = (int)ceil(pow(p, 2.0/3));
gm = (int)ceil(pow(p, 1.0/3));
ll ans = 1;
hs.init();
for(int i = 0; i <= bm; i++){ //a^(i*bm - j)%p = b;
hs.insert(ans, i);
ans = ans*a%p;
}
}
ll get(ll v){
v = qpow(v, p-2, p);
ll num = qpow(a, bm, p);
for(int i = 1; i <= gm; i++){
v = v*num%p;
int ans = hs.find(v);
if(ans != -1)
return 1ll*i*bm - ans;
}
return -1;
}
}bsgs;
int main() {
scanf("%d", &T);
while(T--){
ll n,x0, a, b, p;
scanf("%lld%lld%lld%lld%lld", &n, &x0, &a, &b, &p);
bsgs.x0 = x0; bsgs.a = a; bsgs.b = b; bsgs.p = p;
bsgs.init();
int q;ll v;
scanf("%d", &q);
ll inv = ((a-1)*x0%p+b+p)%p;
if(inv == 0){
while(q--){
scanf("%lld", &v);
if(x0 == v)
printf("0\n");
else
printf("-1\n");
}
continue;
}
if(a == 0){
while(q--){
scanf("%lld", &v);
if(v == x0) //x = x0;
printf("0\n");
else if(v == b) //xn = b
printf("1\n");
else
printf("-1\n"); //无解
continue;
}
continue;
}
if(a == 1){ //xn = x0 + nb;
ll invb = qpow(b, p-2,p);
while(q--){
scanf("%lld", &v);
if(b == 0){
if(v == x0)
printf("0\n");
else
printf("-1\n");
}
else{ // n = (v - x0)/b;
ll ans = ((v-x0+p) % p * invb%p + p)%p;
if(ans < n)
printf("%lld\n", ans);
else
printf("-1\n");
}
}
continue;
}
inv = qpow(inv, p-2, p);
while(q--){
scanf("%lld", &v);
v = ((a-1)*v%p+b+p)%p*inv%p;
ll ans = bsgs.get(v);
if(ans == -1 || ans >= n)
printf("-1\n");
else
printf("%lld\n", ans);
}
}
return 0;
}