Be this the Experiment for multiple COA-security:

  • $PrivK_{\mathcal{A},\Pi}^{mult}(n)$:

  • $(m_0^1 , ... , m_0^t,m_1^1 , ... , m_1^t) \leftarrow \mathcal{A}(1^n), |m_0^i|=|m_1^i| \forall i \in [1,t]$

  • $k\leftarrow Gen(1^n)$

  • $b \leftarrow \{0,1\}$

  • $C = (c_b^1 , ... , c_b^t) \leftarrow (Enc_k(m_b^1) , ... , Enc_k(m_b^t))$

  • $b' \leftarrow \mathcal{A}(C)$

  • if $b' = b$ return 1 else return 0

If $PrivK_{\mathcal{A},\Pi}^{mult}(n) = 1$ $\mathcal{A}$ wins. For a cryptosystem to have that security, there should not exist an adversary that wins that experiment better than $1/2 + negl(n)$, where $negl(n)$ is a negigible function.

Now I want to construct a cryptosystem that has this security but not KPA- or CPA- or CCA-security. My idea:

  • $Gen(1^n)$: Creates a uniform random key $k \leftarrow \{0,1\}^n$
  • $Enc_k(m)$: Create a uniform random number $r \leftarrow \{0,1\}^n$ and create $c = m \oplus PRG(k \oplus r) $. Output $(c,r)$
  • $Dec_k((c,r))$: Create $m = c \oplus PRG(k \oplus r)$ and output $m$

Assume that PRG is a secure pseudo random generator, then this cryptosystem should be multiple COA-secure (or mult-EAV-IND-secure from Katz & Lindell's textbook (2nd edition))

Is that right or did I overlook something?


1 Answer 1


I think that your design will pass the experiment provided that $t$ is small enough that collisions in $r$ occur with negligible probability.

However, I also think that your design is KPA, CPA and CCA secure unless I have misunderstood.

  • $\begingroup$ I just rethought that idea, and it resulted in a question: Is the given construction the idea behind the pseudorandom functions? $\endgroup$
    – Titanlord
    Nov 24, 2021 at 15:27

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