Talk:Inversive congruential generator

198.142.19.205's Ruminations
Points in favour: generally well behaved. no obvious bias or correlation. not bitwise linear. Points against: slower than commonly used methods. With a 32 bit modulus as commonly used, period is too short, and may have insufficient resolution for some uses. 64 bit modulus should fix this, but is even slower and rarely seen. Misc notes: Prime modulus seems better behaved but slower than power of 2 modulus. In any case not useful for cryptography. TODO: find these points in some wikipedia approved secondary source and write them into the article. 198.142.19.205 (talk) 03:24, 19 April 2009 (UTC)  Also Explicit inversive congruential generators. 198.142.19.85 (talk) 10:51, 28 April 2009 (UTC)

Non-primitive maximal-period parameters
Prior to my recent edit, the text suggested that only primitive polynomials can lead to full period. This is incorrect; Chou gives the exact conditions here.

If we take $$q=13,a=-5,c=-3$$, then the ICG has the maximal period of $$q=13$$, but the polynomial $$f(x)=x^2-cx-a=x^2+3x+5$$ is not primitive over $$\mathbb F_{13}$$; we have $$x^{(q^2-1)/3}\equiv1\mod{(13,~f(x))}$$.

(If the modulus is a Fermat prime, as it is in the $$q=5$$ example currently in the article, then (if I'm not mistaken) every maximal-period polynomial will also be primitive. This property is specific to the Fermat primes.) E1a12bf1 (talk) 17:23, 29 October 2022 (UTC)