# Differential probability of modular subtraction $xdp^-$

The modular addition has been a main component in block cipher. The use of modular subtraction in block cipher is not common but there are some cipher such as Bel-T cipher which uses both modular addition and subtraction.

To find the xor differential probability of modular addition, the following equation is used :

$$xdp^+ (\alpha,\beta,\gamma)= 2^{-\sum_{i=0}^{n-2}¬eq(\alpha[i],\beta[i],\gamma[i])}$$

$$$$eq(\alpha[i],\beta[i],\gamma[i])=\begin{cases} 1, & \text{if \alpha[i]=\beta[i]=\gamma[i]}.\\ 0, & \text{otherwise}. \end{cases}$$$$

Iam interested to find the xor differential probability equation of modular subtraction $$xdp^-$$

Q: How to derive the $$xdp^-$$ equation? or $$xdp^+$$ and $$xdp^-$$ are same

• Define the operator $eq(.,.,.)$ and the symbols! – kodlu Oct 12 '18 at 23:16
• eq() is added , $\alpha$ , $\beta$ are input differences , $\lambda$ is output difference – hardyrama Oct 13 '18 at 5:03
• Still unclear. What is the symbol in front of $eq()$? If its a NOT how is it represented in the reals? Is there a modular inverse in the exponent? – kodlu Oct 14 '18 at 7:08
• Same as addition. Probability that $(a \oplus \alpha) - (b \oplus \beta) = (a - b) \oplus \gamma$ is the same as that of $(a \oplus \alpha) + (-b \oplus \beta) = (a + -b) \oplus \gamma$. – Samuel Neves Oct 15 '18 at 13:10