Fast Inversions in Small Finite Fields by Using Binary Trees

Abstract

Inversions in small finite fields are playing a key role in many areas. We present techniques to exploit binary trees for fast inversions in $GF(2^n)$ and $GF(p)$ , where $n$ is a positive integer and $p$ is a prime number. The non-pipelined versions of our design in $GF(2^n)$ and $GF(p)$ have the execution time of $(n-1)(T_{AND}+T_{XOR})$ and $\lfloor \log _2p\rfloor (T_{AND}+T_{XOR})$ , where $T_{AND}$ and ${T_{XOR}}$ are delays of AND and XOR gates, respectively. The pipelined version of our design has a throughput rate of one result per $T_{AND}$ (or $T_{XOR}$ ). The latency is the greater value between $T_{AND}$ and $T_{XOR}$ . In other words, the time complexities of non-pipelined and pipelined versions are $O(n)$ (or $O(log_2p)$ ) and $O(1)$ , respectively. Experimental results and comparisons show that our design provides significant reductions in both the execution time and time–area product, e.g. the execution time of inversion in $GF(2^{12})$ is reduced by 73 $\%$ and time–area product of inversion in $GF(2^6)$ is reduced by 77 $\%$ .