Properties of Carry Value Transformation

Table 2

Shows the contributions in calculating the sum for two possible cases.

Cases	Conditions	Sum of contributions of 𝑎𝑘 and 𝑏𝑘 in 𝑎+𝑏	𝑐𝑘	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)
Case 1	𝑎𝑘+𝑏𝑘<𝛽	(𝑎𝑘+𝑏𝑘)𝛽^𝑘−1	0	0	(𝑎𝑘+𝑏𝑘)𝛽^𝑘−1	(𝑎𝑘+𝑏𝑘)𝛽^𝑘−1
Case 2	𝑎𝑘+𝑏𝑘≥𝛽	(𝑎𝑘+𝑏𝑘)𝛽^𝑘−1	1	𝛽^𝑘	(𝑎𝑘+𝑏𝑘)𝛽^𝑘−1−𝛽^𝑘	(𝑎𝑘+𝑏𝑘)𝛽^𝑘−1