Summary: We analyze a variant of the well-known Miller-Rabin test, that may be useful in preventing side-channel attacks to the random prime generation on smart cards: In the Miller-Rabin primality test for a positive integer $n$, one computes repeatedly the expression $a^ω\pmod n$ for random bases $a\in \Bbb N$ and exponents $ω$ such that $ω$ divides $n - 1$ and $(n - 1)/ω$ is a power of 2. In each round one chooses, at random, a different base a, and uses binary exponentiation to compute $a^ω\pmod n$. ‘Listening’ to many rounds, it seems at least plausible that an outside spy could retrieve the integer $n - 1$. In the variant we consider, one chooses in each round two positive random integers $a$ and $ρ$ and applies the test with base $a$ and exponents $ωρ$, $ω$ as above. This increases the safety against side-channel attacks. However at the same time, it decreases the performance of the Miller-Rabin test. In this article we use elementary means to analyze this variant. We will not be able to obtain results as strong as those by {\it I. Damgård, P. Landrock} and {\it C. Pomerance} on prime generation using the original Miller-Rabin test [Math. Comput. 61, No. 203, 177‒194 (1993; Zbl 0788.11059)]. However by imposing restrictions on the random parameter $ρ$, we obtain satisfactory estimates on the variant described here which justify practical implementation.