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(2n) and GF(p),where n is a positive integer and p is a prime number.The non-pipelined versions of our design in GF(2n) and GF(p) have the execution time of (n-1)(TAND + TXOR) and 「log2p」(TAND + TXOR),where TAND and TXOR are delays of AND and XOR gates,respectively.The pipelined version of our design has a throughput rate of one result per TAND (or TXOR).The latency is the greater value between TAND and TXOR.In other words,the time complexities of non-pipelined and pipefined versions are O(n)(or O(log2p)) 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(212) is reduced by 73% and time-area product of inversion in GF(26) is reduced by 77%.