I heard that random masking method requires a random number whose length is as long as a message to be sent, but I’m not sure that. So, I'd like to ask about that.
1. Boolean masking : $ x_i \oplus r_i$
Assume that a message is $ (x_7, x_6, \cdots , x_0)$ where $x_i$=0 or 1. Then, a random value of 8 bits is required! That is $(r_7, r_6, \cdots , r_0)$ where $r_i$=0 or 1 and the Boolean masking is $(x_7 \oplus r_7, x_6 \oplus r_6, \cdots , x_0 \oplus r_0)$
2. Arithmetic masking : $x + r$
Assume a message $x$ of 8 bits where $x \in \Bbb Z_{2^8}$. Then, a random value $r$ of 8 bits where $r \in \Bbb Z_{2^8}$ is required!
2-1. Arithmetic masking
Assume that the length of all message $x$ to be sent is 4 bits (that is, $0 \le x < 2^4$) where $x \in \Bbb Z_{2^8}$.
(Question 1) Is it possible to hide the message $x$ with random number $r$ of 4 bits using arithmetic masking method $x+r$ in $\Bbb Z_{2^8}$?
3. Multiplicative masking : $x*r$
Assume that the length of all message $x$ to be sent is 4 bits (that is, $0 \le x < 2^4$) where $x \in \Bbb Z_{2^8}$.
(Question 2) Is it possible to hide the message $x$ with random number $r$ of 4 bits using multiplicative masking method $x*r$ in $\Bbb Z_{2^8}$?
(Question 3) Where can I find the related references?
Thank you in advance.