This paper proposes tweakable block cipher (TBC) based modes
and
that are efficient in threshold implementations (TI). Let
t
be an algebraic degree of a target function, e.g.
(resp.
) for linear (resp. non-linear) function. The
d
-th order TI encodes the internal state into
shares. Hence, the area size increases proportionally to the number of shares. This implies that TBC based modes can be smaller than block cipher (BC) based modes in TI because TBC requires
s
-bit block to ensure
s
-bit security, e.g. PFB and Romulus, while BC requires 2
s
-bit block. However, even with those TBC based modes, the minimum we can reach is 3 shares of
s
-bit state with
and the first-order TI (
).
Our first design
aims to break the barrier of the 3
s
-bit state in TI. The block size of an underlying TBC is
s
/2 bits and the output of TBC is linearly expanded to
s
bits. This expanded state requires only 2 shares in the first-order TI, which makes the total state size 2.5
s
bits. We also provide rigorous security proof of
. Our second design
further increases a parameter
: a ratio of the security level
s
to the block size of an underlying TBC. We prove security of
for any
under some assumptions for an underlying TBC and for parameters used to update a state. Next, we show a concrete instantiation of
for 128-bit security. It requires a TBC with 64-bit block, 128-bit key and 128-bit tweak, while no existing TBC can support it. We design a new TBC by extending SKINNY and provide basic security evaluation. Finally, we give hardware benchmarks of
in the first-order TI to show that TI of
is smaller than that of PFB by more than one thousand gates and is the smallest within the schemes having 128-bit security.