ROM based methods for computing the squaring operation in modular rings

Signal Processing algorithms generally rely heavily on the convolution operation which in turn is multiplication intensive. However, more recently convolution algorithms based on the squaring operation as opposed to the multiplication operation have been developed. In this article we present two ROM based methods for performing the squaring operation modulo 2 ⁿ , modulo 2 ⁿ −1, or modulo 2 ⁿ +1. The performance, cost, and implementation issues of the two methods are analyzed in detail and compared against each other as well as with a traditional ROM based implementation. It is shown that both methods obtain ROM bit savings of 99.99%, for 32-bit word lengths, when compared with traditional techniques. However, one of the methods outperforms the other in all other respects such as overhead costs, of up to 99.48% savings, performance, up to about 20 times faster, and regularity and simplicity of hardware design.