Annales Univ. Sci. Budapest., Sect. Comp. 46 (2017) 255–273 INVERSIVE GENERATOR OF THE SECOND ORDER WITH A VARIABLE SHIFT FOR THE SEQUENCE OF PRN’S Pavel Varbanets and Sergey Varbanets (Odessa, Ukraine) Dedicated to the memory of Professor Antal Iv´anyi Communicated by Imre K´ atai (Received February 5, 2017; accepted March 30, 2017) Abstract. The inversive congruential generator of second order modulo a prime power is investigated. This generator generalizes the inversive congruential generator of the first order studied in the works of Eichenauer, Lehn, Topuzoˇ glu, Niederreiter, Shparlinski etc. Also we prove that the produced sequences of pseudo random numbers pass the s-dimensional test on the uniform distribution for s =1, 2, 3. 1. Introduction Uniform pseudorandom numbers (abbreviate., PRN’s) in the interval [0, 1] are basic ingredients of any stochastic simulation. Their quality is of fundamental importance for the success of the simulation, since the typical stochastic simulation essentially depends on the structural and statistical prop- erties of the producing pseudorandom number generators. In the cryptograph- ical applications of pseudorandom numbers the significant importance is of the availability of property of the unpredictability to generated sequence of pseudorandom numbers. The classical and most frequently used method for Key words and phrases : Inversive congruential generator, sequences of pseudorandom num- bers, exponential sum, discrepancy. 2010 Mathematics Subject Classification : 11K45, 11T23, 11T71.