On squaring and multiplying large integers

Methods of squaring large integers are discussed. The obvious O(n ²) method turns out to be best for small numbers. The existing ≈ O(n^1.585) method becomes better as the numbers get bigger. New methods that are ≈ O(n^1.465) and ≈ O(n ^2.404) are presented. All of these methods can be generalized to multiplication and turn out to be faster than a fast Fourier transform (FFT) multiplication for numbers that can be quite large (>3,000,000 b). Squaring seems to be fundamentally faster than multiplication, but it is shown that T_mult ⩽ 2T_sq + O(n)