Indifferentiability of the Sum of Random Permutations Toward Optimal Security

Abstract

A natural way of constructing a pseudorandom function from multiple pseudorandom permutations is to simply add the permutations. In the indistinguishability model, it is known that even two random permutations are sufficient to provide security up to $2^{n}$ queries. On the other hand, the indifferentiability has been proved only up to $2^{({2n}/{3})}$ queries for any number of summands. In thispaper, we improve this bound up to $2^{(({(l-1)n})/{l})}$ queries for an even integer $l\geq 4$ . This is the first result that shows the indifferentiablity of the sum of random permutations is strengthened towards the optimal bound $2^{n}$ as the number of summands increases.