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Purpose of modulus in one-time pad?

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user97210

I have the following question and don't really understand it. I thought OTP offers perfect secrecy, why do we need modulus? Can somebody please help me answer the question?

$Z_n$ denotes the ring of integers $\pmod n$. Alice and Bob share a random key $k \leftarrow Z_n$. Alice wants to send a bit $b \in \{0, 1\}$ securely to Bob (so that Eve cannot learn any information about $b$). She computes $b + k$ in order to use $k$ as a one-time pad. Unfortunately she forgets to compute the modulo operation - i.e. she computes $c = b + k$ over the integers as opposed to computing $c = b + k \pmod n$ and sends $c$ to Bob. Suppose $n = 4$.

Is this a secure encryption scheme? If not, then why not? If yes, what type of security does it get, does it for instance have computational, statistical, perfect security? What if $n = 2κ$ , where $κ$ is a security parameter?

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kelalaka avatar
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kelalaka
Welcome to Cryptography.SE. This is a dump of HW that we don't answer and you don't show any work at all. Hint: start considering that what happens when $k = n-1$
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Paul Uszak avatar
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Paul Uszak
Alternatively, what exactly would Persona Alpha send if $ b + k > n $?
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