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We consider a problem of constructing a secure block cipher from a tweakable block cipher (TBC) with long tweaks. Given a TBC with n-bit blocks and τn-bit tweaks for τ ≥ 1, one of the constructions by Minematsu in DCC 2015 shows that a simple iteration of the TBC for 3d rounds yields a block cipher with dn-bit blocks that is secure up to 2dn/2 queries, where d = τ + 1. In this paper, we show three results. 1. Iteration of 3d − 2 rounds is enough for the security up to 2dn/2 queries, i.e., the security remains the same even if we reduce the number of rounds by two. 2. When the number of queries is limited to 2, d+ 1 rounds are enough, and with d+ ` rounds for 1 ≤ ` ≤ d− 1, the security bound improves as ` grows. 3. A d-round construction gives a block cipher secure up to 2n/2 queries, i.e., it achieves the classical birthday-bound security. Our results show that a block cipher with beyond-birthday-bound (BBB) security (with respect to n) is obtained as low as d + 1 rounds, and we draw the security spectrum of d+ ` round version in the range of 1 ≤ ` ≤ d− 1 and ` = 2d− 2 for BBB security, and ` = 0 for birthday-bound security.

Iterative Block Ciphers from Tweakable Block Ciphers with Long Tweaks.