Abstract
This work presents a new method to compute the GHASH function involved in the Galois/Counter Mode of operation for block ciphers. If $X= X_1\dots X_n$ is a bit string made of $n$ blocks of 128 bits each, then the GHASH function essentially computes $X_1H^n + X_2H^{n-1} + \cdots+ X_nH$, where $H$ is the hash key and an element of the binary field $\Fd_{2^{128}}$. This operation is usually computed by using $n$ successive multiply-and-add operations over $\Fd_{2^{128}}$. Our proposed method replaces all but a fixed number of those multiplications by additions on the field. This is achieved by using the characteristic polynomial of $H$. We present both how to use this polynomial to speed up the GHASH function and how to efficiently compute it for each session that uses a new $H$. We also show that the proposed technique can be parallelized to compute GHASH even faster. In order to try to completely eliminate the need for a field multiplication, we investigate a different set of bases for the field element representation and report their architectural and possible security impacts.