How Can We Square Large Numbers Quickly and Accurately Without Precision Loss?

Calculating y = x^2 Quickly Without Precision Loss

Problem:
Given an input bignum x represented as an array of unsigned 32-bit integers, calculate y = x^2 as quickly as possible without losing precision due to multiplication.

Initial Approach:
The initial approach proposed by the question author involves computing y = x*x to eliminate multiple multiplications. However, this has several disadvantages, including:

Karatsuba Multiplication:
Karatsuba multiplication is a divide-and-conquer algorithm that speeds up multiplication operations. It has three recursive steps:

• Divide the inputs into smaller chunks
• Apply the recursive step to each chunk
• Combine the results

This approach can significantly improve the performance of multiplication, as it reduces the time complexity from O(n^2) to O(n^log2(3)).

Modified Schönhage-Strassen Multiplication (NTT):
The Schönhage-Strassen algorithm, when modified using the NTT (Number Theoretic Transform), can further speed up multiplication operations. It relies on performing the multiplication in the frequency domain.

However, there are limitations to using NTT due to overflow issues. The NTT input/output vector size is constrained by the maximum allowable size of the input bignum. In the implementation provided by the question author, NTT is used for both multiplication and squaring, with varying thresholds depending on the size of the operands.

Conclusion:

For small numbers, the author's fast squaring approach is the best option. For larger numbers, Karatsuba or NTT multiplication becomes more efficient. Through various optimizations, NTT multiplication has become faster than Karatsuba after a certain threshold.

Outstanding Questions:

The author acknowledges that there may be a more efficient algorithm that has been overlooked. Further research and experimentation are necessary to identify the best approach for each specific use case and data size range.

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