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2

What you need for this is something called an $n$-wise independent hash function (like "pairwise independent" but $n$ instead). Such a hash function has the property that when applied to at most $n$ different inputs, its outputs are completely random. These can be constructed efficiently; e.g., a random polynomial of the appropriate degree works. What you ...

1

Perfect secrecy requires that your key be as long as the plaintext you are encrypting. If you want to use the same key to encrypt multiple messages, you can use each part of the key once as a one-time pad. Your idea (assuming you fix the issue raised by poncho) seems to be just a more complicated way to achieve the same thing. If you remove the ...

1

Both constructions are not perfectly secure! In an attempt to express things a bit more mathematically, I'd say you can implement a one-time-pad with a message $m$ and a key $r$, when $m$ and $r$ are elements of $\mathbb{Z}/n\mathbb{Z}$, the additive group of integers modulo n, or when $m$ and $r$ are elements of a group $G$ isomorphic to ...

0

After those very helpful hints by poncho and yyyyyyy the answers are now obvious and I'll briefly argue why each of the construction is perfectly secure or not. Concerning construction 1: It can't be perfectly secure because it has been proven that perfect secrecy (=perfect security) requires the keyspace $\mathcal K$ to be as large as the message space ...

2

Assume that $P_1$ contains " the ". In that case you can get the key stream by XOR'ing " the " with $C_1$, lets call this key stream $K^1$. If this key stream is correct then $P_3^1$ should make sense, where $P_3^1 = K^1 \oplus C_3$. If $P_3^1$ doesn't make sense then you can create $K^2$ and $P_3^2$ from $C_2$ in using an identical calculation and check ...

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