R S Wikramaratna
June 2025
The Additive Congruential Random Number (ACORN) generator is a method for generating uniformly distributed pseudo-random numbers which is straightforward to implement for arbitrarily large order and modulus (where the modulus is a sufficiently large power of 2, typically up to 2120); it has been demonstrated in previous papers to give rise to a family of sequences with long period which, for a k-th order ACORN generator with modulus a power of 2, can be proven from theoretical considerations to approximate in a particular defined sense to the desired properties of uniformity in up to k dimensions.
In 2021 two conjectures were proposed in REAMC Report-003. These assert that for order 8 (or larger) and modulus 2120 almost every choice of odd seed together with an arbitrary set of initial values (including the case with all initial values set to zero) leads to a different sequence that can reasonably be expected to pass all the tests in the current Version 1.2.3 of the standard empirical test package known as TestU01. Supporting results were included for ACORN generators of modulus 2120 in Report-003 (for order 8, 9 and 10) and later extended in REAMC Report-004 (orders 11 to 15 inclusive) and in REAMC Report-006 (selected orders in the range 16 to 101 inclusive).
All these existing published results were for the set of cases R1-000 to R1-999 (each with a different randomly chosen seed and all initial values set to zero) and for the cases S1-xxx (each case S1-xxx having the same seed as the corresponding case R1-xxx, and each having a different specified set of non-zero initial values). The present REAMC Report-008, together with four Appendices A to D, provides further supporting results for ACORN generators of modulus 2120 and orders 8 to 10. Each Appendix includes results obtained using a different set of 1000 randomly chosen seed values to define the cases. It is anticipated that five further Appendices E to I may be published in the future containing results for additional sets of 1000 randomly chosen seeds; however, the report itself is expected to remain unchanged unless there are unexpected results that significantly change some aspect of the conclusions.
Each set of results provides additional support for the two conjectures originally proposed in Report-003. The author is not aware of any other family of pseudo-random number generators for which comparable results have been demonstrated for such a wide range of initialisations, now amounting to some five thousand cases per conjecture for each of the orders 8, 9 and 10.
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