亚洲免费av电影一区二区三区,日韩爱爱视频,51精品视频一区二区三区,91视频爱爱,日韩欧美在线播放视频,中文字幕少妇AV,亚洲电影中文字幕,久久久久亚洲av成人网址,久久综合视频网站,国产在线不卡免费播放

        ?

        A New Class of p-Ary Quadratic Bent Functions

        2011-01-23 02:23:26CollegeofMathematicsandStatisticsSouthCentralUniversityforNationalitiesWuhan430074China
        關(guān)鍵詞:函數(shù)

        (College of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China)

        1 Introduction

        Bent functions were first introduced by Rothaus in 1976 as an interesting combinatorial object[1]and they have been extensively studied for their important applications in coding theory, cryptography and sequence designs. In 1985, Kumar, Scholtz and Welch generalized bent functions to the case of an arbitrary finite field[2]. Precisely, letf(x) be a function mapping Fpnto Fp. The Walsh transform off(x) is defined by

        (1)

        Recently, weakly regular bent functions were shown to be useful for constructing certain combinatorial objects such as partial difference sets, strongly regular graphs and association schemes (see Ref[4,5] ). This justifies why the classes of (weakly) regular bent functions are of independent interest. The quadratic bent functions have been comprehensively studied and they are shown to be weekly regular in Ref[6].

        are studied in this paper, and they are proved to bep-ary bent functions whenmis odd ormis even buta(pn-1)/(p+1)≠1. Notef(x) is weekly regular since it is quadratic, and hence it can be used to construct partial difference sets, strongly regular graphs and association schemes[4,5].

        The remainder of this paper is organized as follows. Section 2 gives some definitions and preliminaries. Sections 3 proves the main result and section 4 concludes the studies.

        2 Preliminaries

        whereaij∈Fp. The rankrof the quadratic formf(x) is defined as the codimension of the Fp-vector space

        W={z∈Fpn|f(x+z)=f(x) for allx∈Fpn},

        namely, |W|=pn-r. The quadratic formf(x) mapping Fpnto Fpis nondegenerate if its rank is equal ton. For more details about quadratic form over finite fields, the reader is referred to Ref[3].

        The following lemma is the proposition 1 in Ref[6] and it will be used to prove the main result in this paper.

        Lemma1[6]Any quadratic formf(x) mapping Fpnto Fpis bent if and only if it is nondegenerate. Moreover, all quadraticp-ary bent functions are weakly regular.

        3 Main theorem and its proof

        (2)

        Theorem1 Letf(x) be the function defined in Eq. (2). Then,f(x) is a bent function ifmis odd ormis even buta(pn-1)/(p+1)≠1.

        ProofLetrbe the rank off(x). Then,pn-ris the number of the variablez∈Fpnsuch that

        f(x+z)=f(x)

        (3)

        for allx∈Fpn. Eq. (3) holds if and only if

        i.e.,

        (4)

        If Eq.(4) holds for allx∈Fpn, then

        apm+1zp2-(γdp+γdpm+1)zp+az=0.

        (5)

        and

        (6)

        By Eq.(5), one has

        i.e.,

        which leads to

        Thus, Eq.(5) implies Eq.(6) and the number of solutionszto Eq.(3) is the same as that to Eq.(5). Note:

        apm+1zp2+az=0.

        (7)

        Ifz≠0, Eq.(7) is equivalent to:

        zp2-1=-(a-1)pm+1-1.

        (8)

        Letαbe a primitive element of Fpnand

        a-1=αi0.

        (9)

        for somei0∈{0,1,…,pn-2}. Then, Eq.(8) is equivalent to the equation of the unknownt∈{0,1,…,pn-2}:

        which holds if and only if

        (10)

        The congruence (10) has solutions intif and only if

        (11)

        1(mod 2).

        Thus, in this case Eq.(11) can not hold and Eq.(10) has no solutions int.

        gcd(p2-1,pm+1-1)=p-1.

        Thus

        Moreover,a(pn-1)/(p+1)≠1 implies thatp+1 does not dividei0, wherei0is defined in Eq.(9). Therefore,

        i.e.,p2-1 does not dividei0(pm+1-1). Thus, Eq.(11) can not hold and Eq.(10) has no solutions int.

        Combining the above two cases, one can conclude that ifmis odd ormis even buta(pn-1)/(p+1)≠1,z=0 is the only one solution of Eq.(5) in Fpn. Therefore, the rank off(x) isnandf(x) is nondegenerate. By Lemma 1,f(x) is a bent function and the proof is finished.

        4 Conclusion

        Sf(b)Frequency9339(-1+3i)2249(-1-3i)224

        Example2 Letn=6,m=3,p=3 andαbe a primitive element of F36with the minimal polynomialx6+x5+2. Leta=1,γ=α. The bent function isf(x)=

        表2 函數(shù)的Walsh變換Sf(b),b∈F36Tab.2 Walsh Transform Sf(b) of f(x)=t(x82-α14x28),b∈F36

        [1] Rothaus O S. On bent functions[J]. J Comb Theory A, 1976, 20(3): 300-305.

        [2] Kumar P V, Scholts R A, Welch L R. Generalized bent function and their properties[J]. J Comb Theory A, 1985, 40(1): 90-107.

        [3] Lidl R, Niederreiter H. Finite fields[M]. Cambridge : Cambridge University Press, 1994.

        [4] Pott A, Tan Y, Feng T, et al. Association schemes arising from bent functions[J]. Des Codes Cryptography, 2011, 59(1-3): 319-331.

        [5] Tan Y, Pott A, Feng T. Strongly regular graphs associat-ed with ternary bent functions[J]. J Combin Theory Ser A, 2010, 117(6): 668-682.

        [6] Helleseth T, Kholosha A. Monomial and quadratic bent functions over the finite fields of odd characteristic[J]. IEEE Trans Inf Theory, 2006, 52(5): 2018-2032.

        猜你喜歡
        函數(shù)
        第3講 “函數(shù)”復(fù)習(xí)精講
        二次函數(shù)
        第3講 “函數(shù)”復(fù)習(xí)精講
        涉及Picard例外值的亞純函數(shù)正規(guī)族
        求解一道抽象函數(shù)題
        二次函數(shù)
        函數(shù)備考精講
        第3講“函數(shù)”復(fù)習(xí)精講
        話說函數(shù)
        第3講 “函數(shù)”復(fù)習(xí)精講
        亚欧AV无码乱码在线观看性色| 熟女体下毛荫荫黑森林| 最近中文字幕免费完整版| 午夜天堂av天堂久久久| 99久久精品国产一区二区三区| 亚洲成人免费网址| av网站可以直接看的| 上海熟女av黑人在线播放| 中文字幕无线码| 久久天天爽夜夜摸| 色偷偷亚洲女人的天堂| 日本av一区二区三区在线| 精品少妇人妻av无码专区| 另类一区二区三区| 亚洲女同av一区二区在线观看| 婷婷色婷婷开心五月四| 亚洲精品无码不卡在线播放he| 精品国产免费久久久久久| 精品一区二区三区老熟女少妇| 图片小说视频一区二区| 国产成人亚洲精品无码mp4| 日本高清一区二区不卡视频| 三级网站亚洲三级一区| 成 人 免 费 黄 色| 人体内射精一区二区三区| 亚洲熟女国产熟女二区三区 | 亚洲高清三区二区一区 | 亚洲日韩成人av无码网站| 2021国内精品久久久久精免费| 成年男女免费视频网站点播| 真人抽搐一进一出视频| 久久亚洲中文字幕无码| 亚洲午夜无码久久久久软件 | 初尝人妻少妇中文字幕| 国产免费一区二区三区在线观看| 久久国产精品男人的天堂av | 亚洲自偷自拍熟女另类| 国产高清一级毛片在线看| 久久精品熟女亚洲av麻豆永永| 亚洲精品乱码8久久久久久日本 | 69精品人妻一区二区|