Properties of Carry Value Transformation

Table 1

Shows the contributions in calculating the sum in different cases.

𝑎𝑘	𝑏𝑘	Sum of contributions of 𝑎𝑘 and 𝑏𝑘 in 𝑎+𝑏	𝑐𝑘=𝑎𝑘∧𝑏𝑘	Contribution of 𝑐𝑘 in CVT(𝑎,𝑏)	Contribution of 𝑎𝑘⊕𝑏𝑘 in (𝑎⊕𝑏)	Sum of contributions of 𝑐𝑘 and 𝑎𝑘⊕𝑏𝑘 in CVT(𝑎,𝑏)+(𝑎⊕𝑏)
0	1	2^𝑘−1	0	0	2^𝑘−1	2^𝑘−1
1	0	2^𝑘−1	0	0	2^𝑘−1	2^𝑘−1
0	0	0	0	0	0	0
1	1	2^𝑘	1	2^𝑘	0	2^𝑘