When utilizing the Karatsuba algorithm for multiplication of large integers, we encounter the possibility of overflow in the intermediate results. To handle this, we employ a technique known as "propagation of carry."
Understanding Overflow
- In traditional 8-bit multiplication, the result of multiplying two 8-bit numbers can produce a 16-bit result.
- The highest value that can be represented in 8 bits is 255 (FFh), and multiplying two such values results in a 16-bit value, FE01h (65025).
Karatsuba Algorithm and Overflow
- In the Karatsuba algorithm, we divide the operands into smaller parts and perform multiplications recursively.
- The intermediate results, denoted as z0, z1, and z2, can potentially exceed the 8-bit limit, leading to overflow.
- z1, in particular, can have a bit width of 17 bits due to the addition of the products x1*y0 and x0*y1, which can result in a maximum value of 130050.
Handling Overflow with Propagation of Carry
- To manage overflow, we employ a technique called "propagation of carry."
- The least significant 8 bits of each intermediate result (z0, z1, z2) are retained, and the remaining bits are propagated to the higher-order digits.
- This is analogous to how we handle carries in traditional addition, where the carry is propagated to the next digit.
z1 += z0 >> 8; z0 &= 255; z2 += z1 >> 8; z1 &= 255; z3 = z2 >> 8; Z = z0 + z1 << 8 + z2 << 16 + z3 << 24;
Hardware Implementation
- Typically, hardware implementations of multiplication handle overflow automatically, providing the result as two words instead of one.
- This allows for efficient handling of large multiplication results without the need for manual propagation of carry.
Considerations for Signed Values
- When dealing with signed values, an additional bit is required to represent the sign.
- To avoid this, it's advisable to temporarily treat the values as absolute values, remembering their original signs, and then apply the appropriate sign to the final result.
Further Resources