A ^ B mod C's divide and governance Thoughts

A ^ B mod C

Assume 0 <a, B, c <n

1. The original method is to first obtain a ^ B and finally mod C.

However, this method is inefficient, and the time complexity is O (B), and a ^ B must be less than N to avoid overflow. It has great limitations.

 

2. Improvement Method 1:

Assume a> C, there is a ^ B mod c = (a mod c) ^ (B mod C)

This situation is applicable to a> C, but cannot be used when a <C.

In the worst case, a ^ B must be less than N to avoid overflow. This method is not suitable either.

 

Method 2:

We can break down B in a ^ B into (2a + 2B + 2C ...)

For example, 12 ^ 36 = 12 ^ (22 + 25)

12 ^ 36 = 12 ^ 22*12 ^ 25

12 ^ 36 mod 35 = (12 ^ 22 mod 35) * (12 ^ 25 mod 35) mod 35

 

We can know

(12 ^ 21 mod 35)

(12 ^ 22 mod 35)

(12 ^ 23 mod 35)

(12 ^ 24 mod 35)

(12 ^ 25 mod 35)

There is a relationship

(12 ^ 2n mod 35) = (12 ^ 2n-1 mod 35) * 2mod 35

Therefore, the above statements can be obtained in sequence.

 

Final table query (12 ^ 22 mod 35) * (12 ^ 25 mod 35) mod 35

 

However, this method also has a drawback: A * A must be less than N, otherwise it will also cause overflow of results.

Although this method is larger than the preceding two methods, it still cannot meet our requirements.

 

 

Method 3:

Since the threshold for overflow in the previous step is a * A <n.

Then we will try to break down a * A and make the final value of A * a mod C less than N.

 

Here we need to mention a formula a * B mod c = (a mod c) * (B mod C)

 

Assume that in a * a, the first a is X, the second a is Y, and

Y = 2a + 2B + 2C...

So x * Y = x * (2a + 2B + 2C ...)

So x * y MOD c = (x * 2a mod c) + (x * 2B mod C) mod c) + (x * 2C mod C )) moD c ......

 

As we can know

(X ^ 21 mod C)

(X ^ 22 mod C)

(X ^ 23 mod C)

(X ^ 24 mod C)

(X ^ 25 mod C)

The following relationships also exist:

(X ^ 2n mod c) = (x ^ 2n-1 mod c) * 2mod C

Therefore, the above statements can be obtained in sequence.

 

The final value of X * x mod C can also be obtained.

Finally, use method 2 to obtain a ^ B mod C.

 

This method breaks down a again. Therefore, the scope is further expanded.

The range of this method can be obtained when a * 2 <n.