Bit-Based Division Property and Application to Simon Family

Ciphers that do not use S-boxes have been discussed for the demand on lightweight cryptosystems, and their round functions consist of and, rotation, and xor. Especially, the Simon family is one of the most famous ciphers, and there are many cryptanalyses again the Simon family. However, it is very difficult to guarantee the security because we cannot use useful techniques for S-box-based ciphers. Very recently, the division property, which is a new technique to find integral characteristics, was shown in Eurocrypt 2015. The technique is powerful for S-box-based ciphers, and it was used to break, for the first time, the full MISTY1 in CRYPTO 2015. However, it has not been applied to non-S-box-based ciphers like the Simon family effectively, and only the existence of the 10-round integral characteristic on Simon32 was proven. On the other hand, the experimental characteristic, which possibly does not work for all keys, covers 15 rounds, and there is a 5-round gap. To fill the gap, we introduce a bit-based division property, and we apply it to show that the experimental 15-round integral characteristic always works for all keys. Though the bit-based division property finds more accurate integral characteristics, it requires much time and memory complexity. As a result, we cannot apply it to symmetric-key ciphers whose block length is over 32. Therefore, we alternatively propose a method for designers. The method works for ciphers with large block length, and it shows “provable security” against integral cryptanalyses using the division property. We apply this technique to the Simon family and show that Simon48, 64, 96, and 128 probably do not have 17-, 20-, 25-, and 29-round integral characteristics, respectively.