Full Indifferentiable Security of the Xor of Two or More Random Permutations Using the χ2 Method

The construction \(\mathsf {XORP}\) (bitwise-xor of outputs of two independent n-bit random permutations) has gained broad attention over the last two decades due to its high security. Very recently, Dai et al. (CRYPTO’17), by using a method which they term the Chi-squared method (\(\chi ^2\) method), have shown n-bit security of \(\mathsf {XORP}\) when the underlying random permutations are kept secret to the adversary. In this work, we consider the case where the underlying random permutations are publicly available to the adversary. The best known security of \(\mathsf {XORP}\) in this security game (also known as indifferentiable security) is \(\frac{2n}{3}\)-bit, due to Mennink et al. (ACNS’15). Later, Lee (IEEE-IT’17) proved a better \(\frac{(k-1)n}{k}\)-bit security for the general construction \(\mathsf {XORP}[k]\) which returns the xor of k (\(\ge 2\)) independent random permutations. However, the security was shown only for the cases where k is an even integer. In this paper, we improve all these known bounds and prove full, i.e., n-bit (indifferentiable) security of \(\mathsf {XORP}\) as well as \(\mathsf {XORP}[k]\) for any k. Our main result is n-bit security of \(\mathsf {XORP}\), and we use the \(\chi ^2\) method to prove it.