Modification of Blum-Blum-Shub Generator (BBS) with a 2×2 Matrix and the First Digit Property of Generated Random Numbers and Bits

Abstract

The Blum-Blum Shub algorithm (BBS) that uses quadratic generator produces one output per iteration. For this reason, the proposed paper seeks to improve the efficiency of the BBS by increasing output, that is, longer sequence of random numbers and bits taken per iteration. To do so, this study proposes modification to the quadratic generator of the BBS with a matrix generator by squaring matrix of 2 × 2 which generates more outputs, namely, four outputs per iteration. Subsequently, occurrence difference of 0s and 1s of these random bits was investigated for both generators. Results show that the occurrence-difference of 0s and 1s of the matrix BBS generator is decreasing as number of iterations are increasing represented by its linear trendline with negative slope. Furthermore, the comparison was made for the first digits of such random numbers for pseudo-oscillation trend per iteration for both generators. Though random numbers obtained from different generators, pseudo-oscillation trends of the first digits for both are considerably quite similar.