mersenneforum.org Strengthening Damg ̊ard and Frandsen, and Khashin
 Register FAQ Search Today's Posts Mark Forums Read

 2022-03-11, 15:54 #1 paulunderwood     Sep 2002 Database er0rr 102348 Posts 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

 Similar Threads Thread Thread Starter Forum Replies Last Post paulunderwood Computer Science & Computational Number Theory 9 2020-06-30 19:28

All times are UTC. The time now is 18:28.

Sun Aug 7 18:28:57 UTC 2022 up 31 days, 13:16, 1 user, load averages: 1.10, 1.28, 1.33

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.

≠ ± ∓ ÷ × · − √ ‰ ⊗ ⊕ ⊖ ⊘ ⊙ ≤ ≥ ≦ ≧ ≨ ≩ ≺ ≻ ≼ ≽ ⊏ ⊐ ⊑ ⊒ ² ³ °
∠ ∟ ° ≅ ~ ‖ ⟂ ⫛
≡ ≜ ≈ ∝ ∞ ≪ ≫ ⌊⌋ ⌈⌉ ∘ ∏ ∐ ∑ ∧ ∨ ∩ ∪ ⨀ ⊕ ⊗ 𝖕 𝖖 𝖗 ⊲ ⊳
∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ 𝓟
¬ ∨ ∧ ⊕ → ← ⇒ ⇐ ⇔ ∀ ∃ ∄ ∴ ∵ ⊤ ⊥ ⊢ ⊨ ⫤ ⊣ … ⋯ ⋮ ⋰ ⋱
∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ
𝛢𝛼 𝛣𝛽 𝛤𝛾 𝛥𝛿 𝛦𝜀𝜖 𝛧𝜁 𝛨𝜂 𝛩𝜃𝜗 𝛪𝜄 𝛫𝜅 𝛬𝜆 𝛭𝜇 𝛮𝜈 𝛯𝜉 𝛰𝜊 𝛱𝜋 𝛲𝜌 𝛴𝜎𝜍 𝛵𝜏 𝛶𝜐 𝛷𝜙𝜑 𝛸𝜒 𝛹𝜓 𝛺𝜔