Consider an eight-bit ripple-carry adder for computing the sum of $A$ and $B$, where $A$ and $B$ are integers represented in 2’s complement form. If the decimal value of $A$ is one, the decimal value of $B$ that leads to the longest latency for the sum to stabilize is __________.

In an 8-bit ripple-carry adder, the longest latency occurs during the worst-case carry propagation, where a carry is generated at the least significant bit (bit 0) and propagates through all bit positions up to the most significant bit (bit 7).

1. The 8-bit 2's complement representation of $A = 1$ is:

   $$A = 00000001_2$$

2. For the least significant stage (bit 0):

   • $A_0 = 1$

   • To generate a carry ($C_1 = 1$) with initial carry $C_0 = 0$, we must have $B_0 = 1$.

3. For stages 1 through 7:

   • To propagate the carry without killing it ($C_{i+1} = C_i$), the propagate condition requires $A_i \oplus B_i = 1$.

   • Since $A_i = 0$ for all $i \in \{1, 2, \dots, 7\}$, we must have $B_i = 1$ for all $i \in \{1, 2, \dots, 7\}$.

Thus, the binary value of $B$ is:

In 8-bit 2's complement representation, $11111111_2 = -1$.

Therefore, the decimal value of $B$ is -1.