When utilizing the Karatsuba Algorithm with 8-bit numbers, the multiplication of x0*y0 results in a 16-bit product. However, subsequent calculations in the algorithm can lead to intermediate results exceeding 8 bits, causing overflow.
To handle this overflow, we need to split the intermediate results into multiple segments and apply bitwise operations to correctly propagate the carry and maintain the accuracy of the final product.
// Calculate intermediate results z2 = x1*y1; // 16 bits z1 = x1*y0 + x0*y1; // 17 bits z0 = x0*y0; // 16 bits // Handle overflow z1 += z0 >> 8; // Propagate carry from z0 to z1 z0 &= 255; // Mask z0 to 8 bits z2 += z1 >> 8; // Propagate carry from z1 to z2 z1 &= 255; // Mask z1 to 8 bits z3 = z2 >> 8; // Extract carry from z2
These operations ensure that each segment of the intermediate results is properly adjusted to fit within the 8-bit boundary, allowing us to obtain the final product Z as the sum of these segments.
// Construct the final result Z = z0 + (z1 << 8) + (z2 << 16) + (z3 << 24);
It's important to note that hardware implementations of multiplication typically handle this overflow internally, producing a result in multiple words (e.g., two 32-bit words for a 64-bit product).
When extending the Karatsuba Algorithm to handle signed values, an additional bit is necessary to represent the sign. To avoid this, it's common practice to work with absolute values and determine the sign of the final result based on the original signs of the operands.
For further details, refer to the following resources: