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Let $n$ be an integer (the motivating context had $n\approx2^{27}$). All other lowercase variables are non-negative integers less than $n$ (elements of $\mathbb Z_n$). All uppercase variables are vectors of $n$ distinct such elements, or equivalently permutations of $n$ elements.

We want to efficiently distinguish among oracles, with as few queries as possible, in an experiment where an oracle initializes, then at query $j$ outputs a permutation $V_j$, with the additional property that: $\forall i,\forall j,\forall k,\;j\ne k\implies V_j[i]\ne V_k[i]$.

  1. Oracle 1 initializes with an empty memory of earlier outputs, then at each query outputs a random permutation uniformly distributed among those still allowed by the additional property.
  2. Oracle 2 initializes by choosing 3 independent random uniformly distributed permutations $P$, $Q$, $R$; then computes $V_j[i]$ as $R[(P[i]+Q[j])\bmod n]$.

Both oracles meet their duty and can answer $n$ queries. Each of their output, taken in isolation, is indistinguishable from a random permutation. The scheme of oracle 2 and standard Format-Preserving Encryption techniques allow to build a PRP generator meeting the additional property, with $O(\ln n)$ memory, and direct access to any output value.

By a counting argument, for $n>4$, a distinguisher is possible with 3 queries if we disregard efficiency. But how can we build an efficient distinguisher?

Any proposal for an efficiently implementable oracle 3, harder to distinguish from oracle 1?

Additions:

  • As pointed by David Cary, what's constructed is known as a latin square. Terry Ritter has a literature survey, and discussions on their uses in cryptography. See also Smile Markovski Design of crypto primitives based on quasigroups in Quasigroups and Related Systems, 2015.
  • Possible application: decide how to lend to participant $i$ out of $n$, on week $j$, sample number $V_j[i]$ out of $n$, in a randomized manner additionally such that no participant gets the same sample twice. In that practice, permutations can be implemented as ciphers using Format Preserving Encryption.
  • Proof that Oracle 2 has the additional property: If $j\ne k$, then (since $Q$ is a permutation, thus injective) it holds that $Q[j]\ne Q[k]$, thus $((P[i]+Q[j])\bmod n)\ne((P[i]+Q[k])\bmod n)$, thus $R[(P[i]+Q[j])\bmod n]\ne R[(P[i]+Q[k])\bmod n]$, thus $V_j[i]\ne V_k[i]$.
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fgrieu
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1 Answers1

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a distinguisher is possible with 3 queries if we disregard efficiency. But how can we build an efficient distinguisher?

I'm not sure exactly what you mean by efficient (since $n \approx 2^{27}$ is a small number), but here is a sketch of a possible constructive attack. I'll let you tell me whether this satisfies your idea of efficiency. I figure that if you have enough time to accept a long ($n$ items) output from the oracle, you have time to do the attack I have in mind. To keep notation cleaner, I won't explicitly write the modular reductions in $R[ \alpha \bmod n ]$.

Query twice, to obtain (assuming oracle #2): $V_1[i] = R[ P[1] + Q[j] ]$ and $V_2[i] = R[ P[2] + Q[j] ]$. Let $\Delta = P[2] - P[1] \bmod n$, which is unknown to the attacker. Still, if $V_1[i] = R[\alpha]$ then $V_2[i] = R[\alpha + \Delta]$. So the function $F$ that maps all $V_1[i]$ to $V_2[i]$ (which the attacker can compute) is a function that maps all $R[\alpha]$ to $R[\alpha + \Delta]$.

I think the cycle structure of $F$ already leads to an attack! For example, if $n$ is prime then $F$ must be a simple cycle. This is because the sequence $\alpha, \alpha+\Delta, \alpha+2\Delta, \ldots$ generates all numbers mod $n$ before repeating (so $F$ visits all positions in $R$ before repeating). But I would guess that under oracle #1, the attacker would get $F$ to be a random involution. So this already gives an attack with reasonable bias, with just 2 queries.

In general, all cycles in $F$ must have cycle length that divides $n$ (in the presence of oracle #2). Again, this seems much less likely to happen in the presence of oracle #1.

OK, but suppose that reasoning doesn't work out for some reason. In that case, make another query. The attacker can compute a new $F$ again now using $\{V_2,V_3\}$ this time instead of $\{V_1, V_2\}$. If we got lucky, and $P[3]-P[2] = \Delta$ (which happens with probability $1/n$), then the attacker will obtain the same $F$ as before! Again I have not worked out the probabilities, but in the presence of oracle #1 this event seems much less likely than probability $1/n$.

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