Skaitļu teorijā, ja dots vesels skaitlis A un pozitīvs vesels skaitlis N ar gcd( A N) = 1, moduļa N reizināšanas secība ir mazākais pozitīvais veselais skaitlis k ar A^k( mod N ) = 1. ( 0< K < N )
Piemēri:
Input : A = 4 N = 7 Output : 3 explanation : GCD(4 7) = 1 A^k( mod N ) = 1 ( smallest positive integer K ) 4^1 = 4(mod 7) = 4 4^2 = 16(mod 7) = 2 4^3 = 64(mod 7) = 1 4^4 = 256(mod 7) = 4 4^5 = 1024(mod 7) = 2 4^6 = 4096(mod 7) = 1 smallest positive integer K = 3 Input : A = 3 N = 1000 Output : 100 (3^100 (mod 1000) == 1) Input : A = 4 N = 11 Output : 5
Ja mēs to rūpīgi aplūkojam, mēs redzam, ka mums nav katru reizi jāaprēķina jauda. mēs varam iegūt nākamo jaudu, reizinot “A” ar iepriekšējo moduļa rezultātu.
Explanation : A = 4 N = 11 initially result = 1 with normal with modular arithmetic (A * result) 4^1 = 4 (mod 11 ) = 4 || 4 * 1 = 4 (mod 11 ) = 4 [ result = 4] 4^2 = 16(mod 11 ) = 5 || 4 * 4 = 16(mod 11 ) = 5 [ result = 5] 4^3 = 64(mod 11 ) = 9 || 4 * 5 = 20(mod 11 ) = 9 [ result = 9] 4^4 = 256(mod 11 )= 3 || 4 * 9 = 36(mod 11 ) = 3 [ result = 3] 4^5 = 1024(mod 5 ) = 1 || 4 * 3 = 12(mod 11 ) = 1 [ result = 1] smallest positive integer 5
Palaidiet cilpu no 1 līdz N-1 un atgrieziet A mazāko +ve jaudu saskaņā ar moduli n, kas ir vienāda ar 1.
Zemāk ir iepriekš minētās idejas īstenošana.
C++// C++ program to implement multiplicative order #include
// Java program to implement multiplicative order import java.io.*; class GFG { // function for GCD static int GCD(int a int b) { if (b == 0) return a; return GCD(b a % b); } // Function return smallest +ve integer that // holds condition A^k(mod N ) = 1 static int multiplicativeOrder(int A int N) { if (GCD(A N) != 1) return -1; // result store power of A that raised to // the power N-1 int result = 1; int K = 1; while (K < N) { // modular arithmetic result = (result * A) % N; // return smallest +ve integer if (result == 1) return K; // increment power K++; } return -1; } // driver program to test above function public static void main(String args[]) { int A = 4 N = 7; System.out.println(multiplicativeOrder(A N)); } } /* This code is contributed by Nikita Tiwari.*/
# Python 3 program to implement # multiplicative order # function for GCD def GCD (a b ) : if (b == 0 ) : return a return GCD( b a % b ) # Function return smallest + ve # integer that holds condition # A ^ k(mod N ) = 1 def multiplicativeOrder(A N) : if (GCD(A N ) != 1) : return -1 # result store power of A that raised # to the power N-1 result = 1 K = 1 while (K < N) : # modular arithmetic result = (result * A) % N # return smallest + ve integer if (result == 1) : return K # increment power K = K + 1 return -1 # Driver program A = 4 N = 7 print(multiplicativeOrder(A N)) # This code is contributed by Nikita Tiwari.
// C# program to implement multiplicative order using System; class GFG { // function for GCD static int GCD(int a int b) { if (b == 0) return a; return GCD(b a % b); } // Function return smallest +ve integer // that holds condition A^k(mod N ) = 1 static int multiplicativeOrder(int A int N) { if (GCD(A N) != 1) return -1; // result store power of A that // raised to the power N-1 int result = 1; int K = 1; while (K < N) { // modular arithmetic result = (result * A) % N; // return smallest +ve integer if (result == 1) return K; // increment power K++; } return -1; } // Driver Code public static void Main() { int A = 4 N = 7; Console.Write(multiplicativeOrder(A N)); } } // This code is contributed by Nitin Mittal.
// PHP program to implement // multiplicative order // function for GCD function GCD ( $a $b ) { if ($b == 0 ) return $a; return GCD( $b $a % $b ) ; } // Function return smallest // +ve integer that holds // condition A^k(mod N ) = 1 function multiplicativeOrder($A $N) { if (GCD($A $N ) != 1) return -1; // result store power of A // that raised to the power N-1 $result = 1; $K = 1 ; while ($K < $N) { // modular arithmetic $result = ($result * $A) % $N ; // return smallest +ve integer if ($result == 1) return $K; // increment power $K++; } return -1 ; } // Driver Code $A = 4; $N = 7; echo(multiplicativeOrder($A $N)); // This code is contributed by Ajit. ?> <script> // JavaScript program to implement // multiplicative order // function for GCD function GCD(a b) { if (b == 0) return a; return GCD(b a % b); } // Function return smallest +ve integer that // holds condition A^k(mod N ) = 1 function multiplicativeOrder(A N) { if (GCD(A N) != 1) return -1; // result store power of A that raised to // the power N-1 let result = 1; let K = 1; while (K < N) { // modular arithmetic result = (result * A) % N; // return smallest +ve integer if (result == 1) return K; // increment power K++; } return -1; } // Driver Code let A = 4 N = 7; document.write(multiplicativeOrder(A N)); // This code is contributed by chinmoy1997pal. </script>
Izvade:
3
Laika sarežģītība: O(N)
Kosmosa sarežģītība: O(1)
Atsauce: https://en.wikipedia.org/wiki/Multiplicative_order