An Extended Quadratic Frobenius Primality Test with Average- and Worst-Case Error Estimate

Evaluation of the Effectiveness of the Frobenius Primality Test

“…Choice with the base z = 2 + √ c or z = 1 + √ c is not random. For some n may exist "bad" bases, or in the terminology of the works [4,5] "liars". The smallest example is n = 7 • 19 • 43 = 5719.…”

“…(5, 53, 1613), (13,109,1549), (29, 37, 1549), (13,37,1549), (29, 109, 1429), (13,37,853), (5,53,677), (13,37,613), (13,37,541), (13,61,397), (5,53,397), (13,37,397), (5,197,389)…”

“…In addition, in [4,5] an upper bound on the error probability of the method (≈ 1/1300) is proved. This is much less than the estimate for the Miller-Rabin (1/4) method, but still the error probability looks very significant.…”

“…The Frobenius method, that is, the method based on the Frobenius automorphism of the field GF (p 2 ) for prime p, has been known for a long time ( [3,4,5,8,7] etc.). In ( [4,12]), even some amplifications of this test are suggested.…”

“…In the usual definition of the Frobenius test (see, for example, [4,5]), it is also suggested to make a pseudorandom choice of the "base" z = a + b √ c. In this paper, we propose to fix this choice in the form 2 + √ c or 1 + √ c depending on c (for details see the definition (2.1)). This is much more convenient and, most importantly, quite enough.…”