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35
Existence of primitive divisors of Lucas and Lehmer numbers
 J. Reine Angew. Math
, 2001
"... We prove that for n ? 30, every nth Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor. Whether the mathematicians like it or not, the computer is here to stay. ..."
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We prove that for n ? 30, every nth Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor. Whether the mathematicians like it or not, the computer is here to stay.
LUC: A New Public Key System
"... We describe public key cryptosystems and analyse the RSA cryptosystem, pointing out a weakness (already known) of the RSA system. We define Lucas functions and derive some of their properties. Then we introduce a public key system based on Lucas functions instead of exponentiation. The computational ..."
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We describe public key cryptosystems and analyse the RSA cryptosystem, pointing out a weakness (already known) of the RSA system. We define Lucas functions and derive some of their properties. Then we introduce a public key system based on Lucas functions instead of exponentiation. The computational requirements of the new system are only a little greater than those for the RSA system, and we prove that the new system is cryptographically stronger than the RSA system. Finally, we present a Lucas function equivalent of the DiffieHellman key negotiation method. Keyword Codes: E.3; K.4.2; K.6.5 Keywords: Data Encryption; Social Issues; Security and Protection 1. Public Key Encryption Publickey encryption was first discussed by Diffie and Hellman [1] as a general principle. The new concept which they introduced was the use of trapdoor functions for cryptography. A trapdoor function is a computable function whose inverse can be computed in a reasonable amount of time only if a (small) amou...
On two classes of simultaneous Pell equations with no solutions
 Math. Comp
, 1999
"... Abstract. In this paper we describe two classes of simultaneous Pell equations of the form x 2 − dy 2 = z 2 − ey 2 = 1 with no solutions in positive integers x, y, z. The proof is elementary and covers the case (d, e) =(8,5), which was solved by E. Brown using very deep methods. 1. ..."
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Abstract. In this paper we describe two classes of simultaneous Pell equations of the form x 2 − dy 2 = z 2 − ey 2 = 1 with no solutions in positive integers x, y, z. The proof is elementary and covers the case (d, e) =(8,5), which was solved by E. Brown using very deep methods. 1.
On the Iteration of Certain Quadratic Maps over GF(p)
"... We consider the properties of certain graphs based on iteration of the quadratic maps x ! x and x ! x 2 over a finite field GF(p). ..."
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We consider the properties of certain graphs based on iteration of the quadratic maps x ! x and x ! x 2 over a finite field GF(p).
On the quadratic character of quadratic units
 J. Number Theory
"... Abstract. Let p ≡ 1 (mod 4) be a prime. Let a, b ∈ Z with p ∤ a(a2 + b2 √ a2 +b2 In the paper we mainly determine ( b+ 2) p−1 2 (mod p) by assuming As an p = c2 + d2 or p = Ax2 + 2Bxy + Cy2 with AC − B2 = a2 + b2. application we obtain simple criteria for εD to be a quadratic residue (mod p), where ..."
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Abstract. Let p ≡ 1 (mod 4) be a prime. Let a, b ∈ Z with p ∤ a(a2 + b2 √ a2 +b2 In the paper we mainly determine ( b+ 2) p−1 2 (mod p) by assuming As an p = c2 + d2 or p = Ax2 + 2Bxy + Cy2 with AC − B2 = a2 + b2. application we obtain simple criteria for εD to be a quadratic residue (mod p), where D> 1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the quadratic field Q ( √ D) with negative norm. We also establish the congruences for U (p±1)/2 (mod p) and obtain a general criterion for p  U (p−1)/4, where {Un} is the Lucas sequence defined by U0 = 0, U1 = 1 and Un+1 = bUn + k2Un−1 (n ≥ 1).
Building Pseudoprimes With A Large Number Of Prime Factors
, 1995
"... We extend the method due originally to Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong Dicks ..."
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We extend the method due originally to Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong Dickson pseudoprimes.
Efficient SIMD arithmetic modulo a Mersenne number
 In IEEE Symposium on Computer Arithmetic – ARITH20
, 2011
"... Abstract—This paper describes carryless arithmetic operations modulo an integer 2 M − 1 in the thousandbit range, targeted at single instruction multiple data platforms and applications where overall throughput is the main performance criterion. Using an implementation on a cluster of PlayStation ..."
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Abstract—This paper describes carryless arithmetic operations modulo an integer 2 M − 1 in the thousandbit range, targeted at single instruction multiple data platforms and applications where overall throughput is the main performance criterion. Using an implementation on a cluster of PlayStation 3 game consoles a new record was set for the elliptic curve method for integer factorization.
Lower Bounds for Lucas Chains
 ISSN 0097–5397. URL: http://www.mpisb.mpg.de/~mkutz/ publications.html. Citations in this paper
, 2002
"... LRT hain ar s ecia y o additio hain satisfyin a extr condition for th represe tation a k = a j + a i o ea eleme t a k i th hain th di#erence a j a i us also co taine i th hain I analog t th relatio e ee additio hain an ex one tiation, Lat hain yiel computatio sequence foLmF functions s ecia kin o ..."
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LRT hain ar s ecia y o additio hain satisfyin a extr condition for th represe tation a k = a j + a i o ea eleme t a k i th hain th di#erence a j a i us also co taine i th hain I analog t th relatio e ee additio hain an ex one tiation, Lat hain yiel computatio sequence foLmF functions s ecia kin o linea recurrences. sh tha th grea m jori o natura u ers n d e no hL'6 hain shorte than # log # n fo a y 0 where # i th golde ratio. eteL Mo tgomer a th firs t consideLns hains i th earl eig ties H disc ered decom ositio theore foLmF hain an l e oun o thei lengt i term o Fi onacci u ers Hi or a no published Therefor se era o Mo tgomery' origina idea ar represe te i thi pa er. Ke ords.LsR hain additio hain Lin function l e ound Fi onacc u er, golde ratio sm ot u er AM su je classifications. 11Y55 11Y16 11B39 11N25 68Q25 P I. S0097539700379255 1 I tr duction. A increasinseqeas = a 0 1 r = n of i teger i calle an additio chain for n i fo ea index k ther exist i a k = a i + a j . (1) Thi notio i moti ate th proble o computing x from x wit fe ultiplications s on i primaril i tereste i hain o smal length r fo gi en n Sinc their firs ap earanc i [12] additio hain h ee i tensi el studied See fo example S honhage' l e oun i [13 o Bergeron Berstel an Brlek' pa e [1 on ad ance meth d fo th constructio o shor additio hains refe t Section 4.6. o K uth' classi [5 fo broade sur e . I thi pa er i estigateLu a chains aria o additio hain i tr duced ete L Mo tgomer [9] Thos ar hain fo whi th indices i j i (1 can hose su tha either i = j o th di#erence a j a i i als par o th hain. Th ter "Luca hain i du t th obser atio tha su hain yiel computation seqputa fo Luca functions s ecia kin o linea recurrences. Mo tgomery' pa e [9] writte i 1983 ha ne e ee published fo se eral ear n furt...
Digital signature schemes based on Lucas functions
, 1995
"... In 1993 Lennon and Smith proposed to use Lucas functions instead of the exponentiation function as a oneway function in cryptographic mechanisms. Recently Smith and Skinner presented an ElGamal signature scheme based on Lucas functions. In this paper we point out the weakness in this approach and p ..."
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In 1993 Lennon and Smith proposed to use Lucas functions instead of the exponentiation function as a oneway function in cryptographic mechanisms. Recently Smith and Skinner presented an ElGamal signature scheme based on Lucas functions. In this paper we point out the weakness in this approach and present our version of an ElGamal signature scheme based on Lucas functions. Furthermore we outline how to apply the ideas of the MetaElGamal signature scheme to Lucas functions. As a result we get various new signature schemes. Unfortunately the new schemes are slightly less efficient than the schemes in finite fields and additionally  in contradiction to a conjecture by Smith and Skinner  the security of the schemes isn't increased: It can be proved that a variant of the signature schemes based on Lucas functions can be universally forged iff a related signature scheme in GF(p) can be universally forged.
Last revised: 30 December 2003
"... As for (1.6), noting that L r = F r + 2F r1 and then applying (1.5) we get ks L js+n F js+n + 2 F js+n1 s F km+n + 2F s F km+n1 = F s L km+n . This completes the proof. In the special case s = 1 and n = 0, (1.5) is due to H.Siebeck ([D,p.394]), and the gene ..."
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As for (1.6), noting that L r = F r + 2F r1 and then applying (1.5) we get ks L js+n F js+n + 2 F js+n1 s F km+n + 2F s F km+n1 = F s L km+n . This completes the proof. In the special case s = 1 and n = 0, (1.5) is due to H.Siebeck ([D,p.394]), and the general case s = 1 of (1.5) is due to Z.W.Sun. Taking m = 1 in (1.5) and (1.6) we get (1.7) F s F k+n = F k F n+s F ks F n , F s L k+n = F k L n+s F ks L n . From this we have the following wellknown results (see [D],[R1] and [R2]): (1.8) (Catalan) F k+n F kn = F kn n , (1.9) F 2n = F n L n , F 2n+1 = F n + F n+1 , L 2n = L . Putting n = 1 in (1.8) we find F k1 F k+1 and so F k1 is prime to F k . For m 1 it follows from (1.5) that s F km+n (s1)m ks F n + (1) (s1)(m1) ks F n+s (mod F k ). So (1.11) F km+n k1 F n +mF k F k1 F n+1 (mod F k ) and hence (1.12) F km (mod F k ). Let (a, b) be the greatest common divisor of a an