Added Fast Power in Number Theory

NumberTheory/FastPower.cpp

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+/*
+    Fast Power
+    ----------------
+    Fast Power is an optimized algorithm to compute exponentiation in a short time.
+    It calculates the power by squaring, meaning we divide the power by 2 on each step and multiply
+    the base by itself. We keep doing this until the power is 0 where we return 1.
+
+
+    Time Complexity
+    -----------------
+    O(log(N)) where N is the exponent.
+
+    Space Complexity
+    -----------------
+    O(log(N)) where N is the exponent.
+
+*/
+#include <iostream>
+#include <cmath>
+
+using namespace std;
+typedef unsigned long long ull;
+
+//Function that returns base raised to the exponent
+ull fast_power (ull base,ull exponent,ull mod = ULLONG_MAX)
+{
+    if (exponent==0) return 1;
+
+    if (exponent%2==1)  //If the power is odd
+    {
+        return (fast_power(base,exponent-1,mod) * base)%mod;
+    }
+    else    //If the power is even
+    {
+        base = fast_power(base,exponent/2,mod);
+        return (base*base)%mod;
+    }
+}
+
+#ifndef FAST_POWER_TEST
+
+int main()
+{
+    //Testing the function
+    ull base,exponent;
+    cout<<"Enter the number and its exponent:"<<endl;
+
+    cin>>base>>exponent;
+
+    if (base == 0 && exponent == 0)
+    {
+        cout<<"undefined"<<endl;
+    }
+    else
+    {
+        int digits_required = floor(exponent * log10(base)) + 1;
+
+        if (digits_required > 19)
+        {
+            cout<<endl<<fast_power(base,exponent,1000000007)<<endl;
+            cout<<"*The output is modulo 10^9+7"<<endl;
+        }
+        else
+        {
+            cout<<endl<<fast_power(base,exponent)<<endl;
+        }
+    }
+
+    return 0;
+}
+
+#endif

Test/NumberTheory/FastPower.test.cpp

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+#define FAST_POWER_TEST
+#define CATCH_CONFIG_MAIN
+
+#include "../catch.hpp"
+#include "../../NumberTheory/FastPower.cpp"
+
+
+TEST_CASE("Base cases", "[fast_power]") {
+    //REQUIRE(fast_power(0,0) == undefined);
+    REQUIRE(fast_power(0,1,ULLONG_MAX) == 0);
+    REQUIRE(fast_power(1,0,ULLONG_MAX) == 1);
+    REQUIRE(fast_power(1,1,ULLONG_MAX) == 1);
+}
+
+TEST_CASE("Normal cases", "[fast_power]") {
+    REQUIRE(fast_power(2,2,ULLONG_MAX) == 4);
+    REQUIRE(fast_power(2,4,ULLONG_MAX) == 16);
+    REQUIRE(fast_power(3,4,ULLONG_MAX) == 81);
+    REQUIRE(fast_power(7,9,ULLONG_MAX) == 40353607);
+    REQUIRE(fast_power(15,10,ULLONG_MAX) == 576650390625);
+}
+
+TEST_CASE("Overflow cases", "[fast_power]") {
+    REQUIRE(fast_power(2,100,1000000007) == 976371285);
+    REQUIRE(fast_power(10,99,1000000007) == 22673271);
+}
+
+#undef FAST_POWER_TEST