This paper proposes design techniques for the efficient VLSI implementation of bit-serial multiplication over a modulus. These techniques reduce multiplication into simple cyclic shifts, where the number representation of the data is chosen appropriately. This representation will, in general, be highly redundant, implying a relatively poor throughput for the multiplier. It is then shown how, by splitting the multiplier into two pipelined multipliers, the throughput of the unit can be increased, whilst still retaining a cyclic-shift implementation. The split multiplier requires a mid-computation basis conversion, and the two number representations, used within the unit, are only moderately redundant. Thus, highthroughput, bit-serial multipliers are achieved, with most of the complexity contained within systolic basis converter modules. The multipliers are applicable to the VLSI implementation of high-throughput, signal processing operations performed over finite fields, in particular, transform and filter operations. 1 Introduction Recent advances in VLSI technology have suggested novel approaches to the implementation of arithmetic units over algebraic rings and fields other than real or complex [l-41. These new approaches have been spurred on by the need for fault-tolerant, systolic architectures, where throughput is maximised and design complexity minimised. Residue number systems (RNSs) perform arithmetic over a modulus, M where M can be expressed, M = f l ; : ; m i , and all arithmetic can be decomposed into a combination of smaller, parallel, arithmetic subcomputations, thus reducing the granular dimension of any consequent VLSI implementation [S, 61. However, some or all of these mi can still be quite large, hence the need for efficient implementations of modular arithmetic units [2, 4, 7, 81. The concept of basis and basis flow (BF) are defined, and it is shown how, for a specified BF, multiplication by an element q, modm, can be implemented using the exponent, e, as input, where (Be),,, = q, g is an element of the field/ring, and the basis a = g. (+), means the residue 0 IEE, 1994 Paper 1527E (C2, EIO), first &wd 15th July 1993 and in revised form
