This paper provides an efficient algorithm for computing the ${eta}_T$ pairing on supersingular elliptic curves over fields of characteristic two. In the proposed algorithm, we deploy a modified multiplication in $F_{2^{4n}}$ using the Vandermonde matrix. For F, G ${in}$ $F_{2^{4n}}$ the proposed multiplication method computes ${eta}{cdot}F{cdot}G$ instead of $F{cdot}G$ with some ${eta}$ ${in}$ $F^*_{2n}$ because ${eta}$ is eliminated by the final exponentiation of the ${eta}_T$ pairing computation. The proposed multiplication method asymptotically requires only 7 multiplications in $F_{2^n}$ as n ${ ightarrow}$ ${infty}$, while the cost of the previously fastest Karatsuba method is 9 multiplications in $F_{2^n}$. Consequently, the cost of the ${eta}_T$ pairing computation is reduced by 14.3%.