# One time pad (OTP) perfect secrecy with different key space

Let say
$K_{0} = \left \{ 0,1\right \}^n$
$K_{1} = K=\left \{ 0,1\right \}^n$ \ $0^n$
$[b\leftarrow \left \{0,1 \right\}, k \leftarrow K_{b}:b=1|k \neq 0^n]$ --- (1)
Key is chosen using constraints (1).
Is this scheme perfect secret assuming OTP is used as encryption scheme.

I thought of using Shannon theory (|K|>=|M|) of perfect secrecy but the key distribution is different. Therefore I could not come up with any solution.

Also: What's the probability of constraints 1? If we can find also we can say that given scheme is perfectly secret or not.

It cannot be perfectly secret : if $C = M \oplus K$ is your encrypted message, all the possible values of $M$ do not happen with equal probability. Indeed, the case $M = C$ (corresponding to $K = 0$) is twice less liquely to happen. You cannot have $H(M) = H(M|C)$; if e.g. $M$ is drawn uniformly at random, the distribution of $M$ given $C$ is clearly not the uniform distribution.
The probability distribution of $M$ given $C$ is $1/2(1/2^n + 1/(2^n-1))$ for every $M \neq C$, and $1/2^{n+1}$ for $M = C$. Hence the distribution of $M$ given $C$ is clearly not uniform.