Detecting and Recovering Integer Data Manipulated by Multiplication With a Nonintegral Real Number and a Rounding Operation

This paper presents a method for detecting and restoring integer datasets that have been manipulated by operations involving nonintegral real-number multiplication and rounding. As we discuss in the paper, detecting and restoring such manipulated integer datasets is not straightforward, nor are there any known solutions. We introduce the manipulation process, which was motivated by an actual case of fraud, and survey several areas of literature dealing with the possibility that manipulation may have happened or might occur. From our mathematical analysis of the manipulation process, we can prove that the nonintegral real number (
$\alpha $
) used in the multiplication exists not as a single real number but as an interval containing infinitely many real numbers, any of which could have been used to produce the same manipulation result. Based on these analytic findings, we provide an algorithm that can detect and restore manipulated integer datasets. To validate our algorithm, we applied it to 40,000 test datasets that were randomly generated using controllable parameters that matched the real fraud case. Our results indicated that the algorithm detected and perfectly restored all datasets for which the value of the nonintegral real number was at least 16 (
$\alpha \geq 16$
) and the number of data entries was at least 40 (
$n\geq 40$
).