When performing 8-bit multiplication, the result can overflow and produce a 16-bit value. This overflow issue arises when multiplying two 8-bit numbers, resulting in a 16-bit product. To handle this overflow in the Karatsuba algorithm:
- We decompose the input numbers X and Y into their respective 8-bit components: ``` X = x0 + x1 << 8 Y = y0 + y1 << 8 ``` - We compute the partial products: ``` z2 = x1 * y1 -> 16 bit z1 = x1 * y0 + x0 * y1 -> 17 bit z0 = x0 * y0 -> 16 bit ``` - Note that z1 has 17 bits because the maximum value of x1 * y0 + x0 * y1 is 65025 + 65025 = 130050. - To handle the overflow, we perform the following steps: ``` z1 += z0 >> 8; z0 &= 255; z2 += z1 >> 8; z1 &= 255; z3 = z2 >> 8; Z = z0 + z1 << 8 + z2 << 16 + z3 << 24 ``` - This approach ensures that the final result Z is a 32-bit value, accommodating the potential overflow from the partial products. - It's worth noting that modern hardware implementations of multiplication often handle this overflow automatically, providing a 32-bit result instead of a single 16-bit value. - When adding the 8-bit sub-results, we require at least 10 bits to accommodate the addition of three numbers or 9 bits (or 8 bits + 1 carry) for sequential addition and carry propagation. - If signed values are involved, an additional bit is required for the sign. To avoid this, the signs of the operands can be remembered, absolute values can be used, and the sign of the result can be set based on the original signs.