Zot-binary: a New Numbering System with an Application on Big-integer Multiplication

In this paper we present a new numbering system with an efficient application on Big-Integer multiplication. The paper starts with an introduction to a new redundant positional numbering system known as “Big-Digit Numbering System” (BDNS). With BDNS, a new non-redundant positional numbering system known as ZOT-Binary is proposed. ZOT-Binary has a low Hamming weight with an average of 23.8% nonzero symbols, and therefore is highly suitable for Big-Integer calculation, especially for BigInteger multiplication. To harvest such benefit from the ZOT-Binary representation, a new Big-Integer multiplication algorithm, ZOT-CM, which is based on the Classical multiplication algorithm, is proposed. Our result shows that when compared with the Classical multiplication algorithm, ZOT-CM is about 12 times faster for multiplying 128 bits numbers and at least 16 times faster for multiplying numbers that are bigger than 32,000 bits long. Our result also shows that ZOT-CM is about 20 to 3 times faster than Karatsuba multiplication algorithm, for multiplying numbers that are ranging from 128 bits to 32,000 bits long. From the findings, it is clear that ZOT-CM is a valuable addition to Big-Integer multiplication algorithm, and it is also believed that ZOT-Binary representation can benefit many other Big-Integer calculations.