Can you describe an algorithm to efficiently multiply two large integers that exceed the normal integer range?

To efficiently multiply two large integers that exceed the normal integer range, one common algorithm used is the Karatsuba algorithm. This algorithm utilizes a divide-and-conquer approach to break down the multiplication problem into smaller sub-problems, making the overall computation more efficient.

Here is a brief description of the Karatsuba algorithm:

1. Divide the input numbers, let's say x and y, into two equal halves.
2. Express each number as a sum of two parts: x = a * 10^(n/2) + b, y = c * 10^(n/2) + d.
3. Compute three products recursively:
- Compute ac
- Compute bd
- Compute (a + b)(c + d) = ac + ad + bc + bd
4. Calculate the final product using the formulas:
- Result = ac * 10^n + (a*d + b*c) * 10^(n/2) + bd

By breaking down the multiplication into smaller parts and using a recursive approach, the Karatsuba algorithm achieves efficiency when multiplying large integers that exceed the normal integer range. It reduces the number of required multiplications and trades them off for additions and subtractions, leading to a more optimized multiplication operation.