Improved Related-Key Boomerang Distinguisher for Full-Round FUTURE

FUTURE is a lightweight block cipher with a 64-bit block, a 128-bit key and $10$ rounds, proposed at AFRICACRYPT 2022 for low-latency hardware. This study analyzes FUTURE in the related-key setting with a bit-level constraint model that carries the exact weights of the differential distribution table and the exact entries of the boomerang connectivity table, and whose objective function $2w_0 + 2w_1 + w_{\mathrm{bct}} = -\log_2(p^2q^2r)$ optimizes both sub-ciphers and the middle layer of the sandwich framework together. The model returns a full-round distinguisher whose objective function value $36$ is optimal over all switching rounds and whose probability is $P = \hat{p}^2\,\bar{r}\,\hat{q}^2 = 11\cdot 2^{-39} \approx 2^{-35.54}$, at a cost of $2^{37.54}$ queries under four related keys and $2^{37.54}$ XOR operations. Running it on the full cipher over $2^{44.34}$ quartets returns $341$ right quartets and an experimental probability of $2^{-35.92}$, in $63.9$ hours on an ordinary personal computer; at the previous full-round probability $2^{-45.8}$ the same computer would take about $6.8$ years. The distinguisher is therefore a practical one, and improves the best previously known full-round related-key boomerang distinguisher of FUTURE by a factor of $2^{10.26}$ in probability, in data and in time. An $8$-round distinguisher placed into the unified key recovery framework further gives a full-round attack with $2^{51.05}$ data, $2^{64}$ time and $2^{64}$ memory, whose time complexity attains the optimum of the framework at any memory within the codebook and improves the best previously known attack by a factor of $2^{6}$.