Alice is using one$-$time pad and notices that when her key is the all$-$zeroes string k $=0\backslash$lambda $,$ then Enc$($k$,$m$)=$ m and her message is sent in the clear! To avoid this problem, she decides to modify KeyGen to exclude the all$-$zeroes key. She modifies KeyGen to choose a key uniformly from $\{0,1\}\backslash$lambda $\backslash \{0\backslash$lambda $\},$ the set of all $\backslash$lambda $-$bit strings except $0\backslash$lambda $.$ In this way, she guarantees that her plaintext is never sent in the clear.

Is it still true that the eavesdropper$$s ciphertext distribution is uniformly distributed on $\{0,1\}\backslash$lambda $?$ Justify your answer.