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Old 2022-03-11, 15:54   #1
paulunderwood
 
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Sep 2002
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Lightbulb Strengthening Damg ̊ard and Frandsen, and Khashin

Strengthening Damg ̊ard and Frandsen[1], and Khashin[2]

I propose the test:

{tst(n,c)=
Mod(c,n)!=-1&&
kronecker(c,n)==-1&&
Mod(c,n)^((n-1)/2)==-1&&
Mod(Mod(x+1,n),x^2-c)^(n+1)==1-c;}

The Frobenius part's squaring during left-right binary exponentiation can be computed in 2 Selfridges for small c [2:2.2]:

(s*x+t)^2 == 2*s*t*x + (t+c*s)*(t+s) - (c+1)*s*t, where s and t are intermediate values.

Multiplication by the base x+1 is:

(s*x+t)*(x+1) = (t+s)*x + c*s+t.

Therefore the test is 1+2 Selfridges and has no known counterexamples so far... Do any of the authors mention the addition of the Euler PRP test?

[1] An Extended Quadratic Frobenius Primality Test with Average Case Error Estimates & Ivan Bjerre Damg ̊ard Gudmund Skovbjerg Frandsen, https://eprint.iacr.org/2001/102.pdf

[2] Counterexamples for Frobenius primality test, Sergey Khashin, https://arxiv.org/pdf/1307.7920.pdf

Edit: In order to prevent cyclotomic polynomials I add Mod(c,n)!=-3 and Mod(c,n)!=-1/3, the latter where gcd(3,n)==1

However a counterexample is [n, c]=[79786523, 2982522].

Last fiddled with by paulunderwood on 2022-03-11 at 20:29
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