  Lightweight cryptography is a key tool for building strong security solutions
for pervasive devices with limited resources. Due to the stringent cost
constraints inherent in extremely large applications (ranging from RFIDs and
smart cards to mobile devices), the efficient implementation of cryptographic
hardware and software algorithms is of utmost importance to realize the vision
of generalized computing.
  In CRYPTO 2016, Beierle, Kranz and Leander have considered lightweight
multiplication in $\mathds{F}_{2^n}$. Specifically, they have considered the
fundamental question of optimizing finite field multiplications with one fixed
element and investigated which field representation, that is which choice of
basis, allows for an optimal implementation. They have left open a conjecture
related to two XOR-count. Using the theory of linear algebra, we prove in the
present paper that their conjecture is correct. Consequently, this proved
conjecture can be used as a reference for further developing and implementing
cryptography algorithms in lightweight devices.
