Explicit Maximal and Minimal Curves over Finite Fields of Odd Characteristics

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1 Explicit Maximal ad Miimal Curves over Fiite Fields of Odd Characteristics Ferruh Ozbudak, Zülfükar Saygı To cite this versio: Ferruh Ozbudak, Zülfükar Saygı. Explicit Maximal ad Miimal Curves over Fiite Fields of Odd Characteristics. Pascale Charpi, Nicolas Sedrier, Jea-Pierre Tillich. WCC2015-9th Iteratioal Workshop o Codig ad Cryptography 2015, Apr 2015, Paris, Frace. 2016, Proceedigs of the 9th Iteratioal Workshop o Codig ad Cryptography 2015 WCC2015. <hal > HAL Id: hal Submitted o 18 Feb 2016 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

2 Explicit Maximal ad Miimal Curves over Fiite Fields of Odd Characteristics Ferruh Özbudak1,2 ad Zülfükar Saygı 3 1 Departmet of Mathematics, Middle East Techical Uiversity, Dumlupıar Bul., No:1, 06800, Akara, Turkey 2 Istitute of Applied Mathematics, Middle East Techical Uiversity, Dumlupıar Bul., No:1, 06800, Akara, Turkey 3 Departmet of Mathematics, TOBB Uiversity of Ecoomics ad Techology, Söğütözü 06530, Akara, Turkey ozbudak@metu.edu.tr,zsaygi@etu.edu.tr Abstract. I this work we preset explicit classes of maximal ad miimal Arti Schreier type curves over fiite fields havig odd characteristics. Our results iclude the proof of Cojecture 5.9 give i [1] as a very special subcase. We use some techiques developed i [2], which were ot used i [1] at all. Keywords: Algebraic curves, Ratioal poits, Maximal curves, Miimal curves. 1 Itroductio Algebraic curves over fiite fields have various applicatios i codig theory, cryptography, quasiradom umbers ad related areas see, for example, [6, 7, 11, 12]). For these applicatios it is importat to kow the umber of ratioal poits of the curve. Throughout this paper by a curve we mea a smooth, geometrically irreducible ad projective curve over a fiite field of odd characteristic. Let p be a odd prime, e be a positive iteger, q = p e ad be a positive iteger. Let F q deote the fiite field with q elemets. Let h 0 ad Sx) = s 0 x + s 1 x q + + s h x qh F q [x] be a F q -liearized polyomial of degree q h i F q [x]. We cosider the Arti-Schreier type curves χ give by y q y = xsx) = These curves are related with the quadratic forms h s i x qi +1. 1) i=0 Qx) = TrxSx)) 2) where Tr deote the trace map from F q to F q. Let NQ) deote the cardiality NQ) = {x F q Tr xsx)) = 0} ad let Nχ) be the umber of F q ratioal poits of the curve χ. The usig Hilbert s Theorem 90 we have Nχ) = 1 + qnq), ad hece determiig Nχ) is the same as determiig NQ). Note that i geeral, it is difficult to determie Nχ). For the umber Nχ), the Hasse-Weil iequality states that q + 1 2gχ) q Nχ) q gχ) q

3 where gχ) is the geus of χ. We kow that there exist curves attaiig the Hasse-Weil bouds. If the upper boud is attaied the the curve is called a maximal curve ad if the lower boud is attaied the the curve is called a miimal curve. Here we ote that usig [11, Propositio ] the geus q 1)qh of the curve χ i 1) is gχ) =. 2 Usig the relatios betwee the curve χ i 1) ad the quadratic form Q i 2) some characterizatios ad classificatio results o maximal ad miimal curves are obtaied i [3, 4, 8 10] for the curves over fiite fields with eve characteristics. Also usig similar relatios some results are obtaied for the curves over fiite fields with odd characteristics i [1]. Furthermore for all itegers 0 mod 12 ad for all primes p, 5 p 29 with gcdp, ) = 1 the followig cojecture is give i [1, Cojecture 5.9]. Cojecture 1. [1] Let p > 3 be a odd prime ad let be a iteger relatively prime to p ad divisible by 12. The the curve χ over F p defied by 12 1 y p y = x x p4 x p3 + x p2) p 6j is a miimal curve. Similar observatios ad discussios are also give i the last sectio of [1]. I this paper we prove a much more stroger versio of the cojecture above. We also give explicit classes of may other maximal ad miimal curves. We use some techiques developed i [2], which were ot used i [1] at all. 2 Prelimiaries I this sectio we recall basic defiitios ad some facts that we use i this paper. A quadratic form o F q over F q is a map such that Qαx) = α 2 Qx) for all α F q ad x F q, ad The related map Bx, y) o F q F q defied by is a biliear map over F q. Bx, y) = Qx + y) Qx) Qy) The radical W of Q is a F q -liear subspace of F q give by W = {x F q : Bx, y) = 0 for all y F q }. Let w be the F q -dimesio of W. By codimesio of the radical we mea the differece w. The followig result was prove i [5] usig some tools from algebraic geometry. Here we give a differet proof usig oly elemetary tools. We will use the followig propositio later. Propositio 1. Let q be a prime power ad m 1 be a iteger. Cosider the curve χ over F q 2m defied by y q y = x a 0 x + a 1 x q + + a m x qm). Assume that a m 0 ad χ is maximal over F q 2m. The a 0 = a 1 = = a m 1 = 0 ad a m +a m q m = 0. The coverse holds as well. Proof. As a m 0, the geus of χ is q 1)qm 2 ad hece it has 1 + q 2m+1 may ratioal poits i F q 2m. By Hilbert s Theorem 90 this meas that Tr x a 0 x + a 1 x q + + a m x qm)) = 0 for all x F q 2m,

4 where Tr is the trace from F q 2m to F q. Let I {0, 1,..., m 1} be the subset cosistig of 0 i m 1 with a i 0. Assume that I. For each i I ad 0 l 2m 1 let i,l be the iteger 0 i,l q 2m 1 with i,l q i+1)l mod q 2m 1). Note that 1, q + 1,..., q m 1 + 1, q m + 1 are i distict q-cyclotomic cosets modulo q 2m 1). Hece there exists a polyomial HT ) F q 2m[T ] defied as HT ) = q a m + a m) m T qm m 1 a ql i T i,l, i I such that Hx) = 0 for all x F q 2m. If HT ) is ot the zero polyomial we get a cotradictio as HT ) has degree strictly less that q 2m 1 ad has q 2m distict zeroes. This implies that I = ad a m + a m q m = 0. The coverse holds trivially. l=0 3 Mai results I this sectio we state ad prove our results. Note that i the proof of the followig theorem we use the otio of fuctio fields. The theory of algebraic curves is essetially equivalet to the theory of fuctio fields. For a brief survey of the relatios betwee algebraic curves ad fuctio fields we refer to [11, Appedix B]. Theorem 1. Let q be a power of a odd prime ad m 2 be a iteger. Let Sx) = a 1 x q + + a m 1 x qm 1 F q 2m[x] with a 1 a m 1 0. Assume that the radical of the quadratic form TrxSx)) has dimesio 2m 2 over F q. The the curve y q y = xsx) is a miimal curve over F q 2m. Proof. Let E 1 = F q 2mx, y) with y q y = xsx) be the fuctio field of χ. As the dimesio of the radical is 2m 2 ad a m 1 0, it is rather well kow that E 1 or equivaletly χ) is either maximal or miimal over F q 2m see, for example, [2]). Usig [2, Propositio 5.1] we obtai that a extesio field E 2 of E 1 such that E 2 is maximal miimal) E 1 is maximal miimal). Moreover a affie equatio for E 2 is also give: E 2 = F q 2mz, t) with t q t = zrz). Here [2, Propositio 5.1] proves existece of c F q 2m such that Dx) q = Sx q + cx) ad Rx) = cdx) q + Dx) 3) i the polyomial rig F q 2m[x]. Put Usig 3) we obtai that Rx) = b 0 x + b 1 x q + + b m x qm. b 0 = c q a 1 ) 1/q 0. If E 2 is maximal, the b 0 = 0 by Propositio 1. Hece E 2 ad E 1 are miimal, which completes the proof.

5 Usig a similar idea we prove the followig Theorem 2. Let q be a power of a odd prime ad m 4 be a iteger. Let Sx) = a 2 x q2 + + a m 2 x qm 2 F q 2m[x] with a 2 a m 2 0. Assume that the radical of the quadratic form TrxSx)) has dimesio 2m 4 over F q. The the curve y q y = xsx) is a miimal curve over F q 2m. Geeralizig the techique i the above theorems we have the followig result. Theorem 3. Let q be a power of a odd prime ad k, m be positive itegers with m 2k. Let Sx) = a k x qk + a k+1 x qk a m k x qm k F q 2m[x] with a k a m k 0. Assume that the radical of the quadratic form TrxSx)) has dimesio 2m 2k over F q. The the curve y q y = xsx) is a miimal curve over F q 2m. Usig the above theorems we obtai the followig explicit class of miimal curves. Note that this result icludes [1, Cojecture 5.9]. Theorem 4. Let q be a power of a odd prime ad let be a iteger divisible by 12. The the curve χ over F q defied by 12 1 y q y = x x q4 x q3 + x q2) q 6j is a miimal curve. Proof. We have Sx) = x q4 x q3 + x q2) q 6j The the radical W of the quadratic form TrxSx)) becomes 12 1 W = x F q x q4 x q3 + x q2) q 6j + x q 4 x q 3 + x q 2) ) q 6j = = x F q ) q 6j x q2 x q + x = 0. The the correspodig q-associate of 6 1 ) q 6j x q2 x q + x becomes x 2 x + 1 ) 1 + x 6 + x x 6). As 12 ad x 1 = x 6 1) 1 + x 6 + x x 6) we have deg gcd x 1, x 2 x + 1 ) 1 + x 6 + x x 6))) = 4, which meas that dim Fq W = 4. The we complete the proof usig Theorem 2.

6 Usig Theorem 2 ad similar observatio as i the proof of Theorem 4 we obtai the followig result. Theorem 5. Let q be a power of a odd prime ad let be a iteger divisible by 12. The the curve χ over F q defied by is a miimal curve y q y = x x q4 + x q3 + x q2) q 6j I the followig theorems we obtai maximal ad miimal curves depedig o the characteristic of the fiite fields. We observe that the results are true for all odd primes of the form p 1 mod 6 ad p = 3. Theorem 6. Let = 6 ad p = 3 or p is a prime satisfyig p 1 mod 6. The the curve χ over F p defied by y p y = x 2x p + x) is a miimal curve if p 1 mod 4 ad a maximal curve if p 3 mod 4. Proof. Let Sx) = 2x p + x ad c F p is a root of c 2 c + 1 = 0. Note that c 2 c + 1 = 0 has roots c 1 = ad c 2 = i F p, sice -3 is a quadratic residue as p 1 mod 6. The defie Dx) p = Sx p + cx) cx The we have Dx) = 2x p + 1 2c)x. Now defie = 2x p c)x p. Rx) = csx p + cx) + Dx) + cx p = 2cx p2 cx. The the radical of TrxRx)) becomes { } W R = x F p 6 2c) p2 x p4 2cx p2 2cx = 0 { } = x F p 6 x p4 + x p2 + x = 0. Correspodig p-associate becomes t 4 + t ad as deg gcd t 6 1, t 4 + t )) = 4 we obtai that dim W R = 4. Now we will use the same sceario. Assume that S 1 x) = 2x p2 + x. The defie D 1 x) p = S 1 x p + c 1 x) c 1 x The we have D 1 x) = 2x p2 + 2c p 1 xp + x. Now defie = 2x p3 + 2c p2 1 xp2 + x p. R 1 x) = c 1 S 1 x p + c 1 x) + D 1 x) + c 1 x p ) = 2c 1 x p3 + 2 c p x p2 + 2 c p 1 + c 1) x p + c )x. The we kow that the curve y p y = xr 1 x) is maximal if ad oly if the followig system has a solutio i F p 6 see Propositio 1): 2c 1 ) + 2c 1 ) p3 = 0 c p = 0 c p 1 + c 1 = 0 c = 0. Note that this system has a solutio i F p 6 if ad oly if 1 is a quadratic oresidue modulo p, that is, p 3 mod 4.

7 Similarly we obtai the followig result. Theorem 7. Let = 6 ad p = 3 or p is a prime satisfyig p 1 mod 6. The the curve χ over F p defied by y p y = x 2x p + x) is a miimal curve if p 1 mod 4 ad a maximal curve if p 3 mod 4. Remark 1. I the proof of Theorem 1 we have used [2, Propositio 5.1], which guaratees the existece of c F q such that 2m Dx) q = Sx q + cx) ad Rx) = cdx) q + Dx). Now we wat to give explicit c s for our Theorem 4. Assume that q = p 1 mod 6 ad = 12. Let The let us defie which gives Also defie Sx) = x p4 x p3 + x p2. Dx) p = Sx p + cx) = x p5 x p4 + x p3 + cx) p4 cx) p3 + cx) p2 Dx) = x p4 x p3 + x p2 + cx) p3 cx) p2 + cx) p. Rx) = cdx) p + Dx) = c x p5 x p4 + x p3 + cx) p4 cx) p3 + cx) p2) + x p4 x p3 + x p2 + cx) p3 cx) p2 + cx) p ) ) ) = cx p5 + c p4 +1 c + 1 x p4 + 2c c p x p3 + c p2 +1 c + 1 x p2 + c p x p. We kow that x 2 x + 1 = 0 has roots i F p, sice -3 is a quadratic residue as p 1 mod 6. If we take c F p as a root of x 2 x + 1 = 0 the c p4 +1 c + 1 = c p2 +1 c + 1 = 0. Therefore we have ) Rx) = cx p5 + 2c c p x p3 + c p x p = c x p5 + x p3 + x p) sice c p = c ad c p3 +1 = c 2. The the radical of TrxRx)) becomes Correspodig p-associate becomes { } W = x F q 12 x p5 + x p3 + x p + x p 5 + x p 3 + x p 1 = 0 { } = x F q 12 x p10 + x p8 + x p6 + x + x p2 + x p4 = 0. t 10 + t 8 + t 6 + t 4 + t = t 4 + t ) t ), which meas that dim W = 10. Note that the same techique is also works for > 12 for the case q = p 1 mod 6.

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