Division and Overflow Detection in Residue Number Systems

Abstract

Residue arithmetic has some intriguing characterThe sign of an integer I will be represented implicistics which could possibly be exploited in a special purpose, or even itly: a general purpose, computer. However, simple mechanizations of the operations of division and overflow detection are not inherent in _ < I <-¢ I is ositive the structure of a residue number system. Methods of handling these 2 P operations are discussed in this paper. In the first section, a relatively straightforward division process is presented whose execution time M is comparable to those of existing binary machines. The next section -< I < M X I is negative. shows how the division process can be cut short under certain condi2 M-I. tions. The amount of equipment required for this is not insignificant; however, this equipment can also be used to speed up the normal diviThe integer value of a number W will be designated sion procedure and to detect multiplicative overflow. The multiplicaby [W]. tive overflow detection scheme proposed in the concluding section has Er o the following desirable features: E(I) = 1092(I 1) It does not require a redundant number system. T(I) -2E 2) It is compatible with the division process and requires no Y| X signifies that Y divides X evenly; i.e., X is special circuitry. divisible by Y. If X is not divisible by Y, this is 3) It is faster than the brute-force approaches which either redenoted by Y| /X. quire residue division or essentially check the residue multi(Y, mi) =1 signifies that Y and mi are relatively plication by a multiplication in a more conventional number v system. prime.