In the paper the variety \({\mathbf R}_2\) of commutative rings with unit, of characteristic two, with the square operation (as an additional operation) is studied. It is proved that this variety is generated by the finite Galois field \({\mathbf G}{\mathbf F}(2^k)\) and that this variety is equivalent to the variety of all algebras \((B,\wedge,\vee,-,\delta, \delta', 0,1)\), where \((B,\wedge,\vee,-,0,1)\) is a Boolean algebra, \(\delta\) is an automorphism of \(B\) and \(\delta'= \delta^{-1}\).