6054 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 9, SEPTEMBER 2012

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1 6054 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 9, SEPTEMBER 2012 A Class of Binomial Bent Functions Over the Finite Fields of Odd Characteristic Wenjie Jia, Xiangyong Zeng, Tor Helleseth, Fellow, IEEE, Chunlei Li Abstract This paper studies a class of binomial functions over the finite fields of odd characteristic characterizes their bentness in terms of the Kloosterman sums Numerical results show that the proposed class contains bent functions that are affinely inequivalent to all known monomial binomial ones Index Terms Exponential sum, Kloosterman sum, -ary bent function, regular bent function, Walsh transform I INTRODUCTION B OOLEAN bent functions, introduced by Rothaus [24], exist only for an even number of variables achieve the maximum Hamming distance to the set of all affine functions This class of functions is an interesting combinatorial object, they are also extensively discussed for their signicant relation to the topics in sequences, cryptography coding theory [2], [11], [22] The notion of Boolean bent functions was generalized to the case of functions over an arbitrary finite field by Kumar et al [16] Complete classication of bent functions seems hopeless in the Boolean generalized cases However, a lot of interesting results on bent functions were found through studying monomial, binomial, quadratic functions (see [1], [3], [5], [6], [9], [10], [18], [19], [23], references therein) Throughout this paper, it is always assumed that is an odd prime For a positive integer, let denote the finite field with elements The function : represents the absolute trace function over, namely, for In this paper, for even, we study the bentness of the binomial functions where, the integer is coprime to We characterize the bentness of these functions in terms of Kloosterman sums on the coefficient The algebraic degree normality of the proposed functions are also considered Manuscript received December 20, 2011; revised April 18, 2012; accepted April 28, 2012 Date of publication May 16, 2012; date of current version August 14, 2012 The work of W Jia X Zeng was supported by the National Natural Science Foundation of China under Grant The work of T Helleseth C Li was supported by the Norwegian Research Council W Jia X Zeng are with the Faculty of Mathematics Computer Science, Hubei University, Wuhan , China ( xzeng@hubueducn; xiangyongzeng@yahoocomcn) T Helleseth C Li are with the Department of Informatics, University of Bergen, N-5020 Bergen, Norway ( TorHelleseth@iiuibno; chunlei li@iiuibno) Communicated by N Y Yu, Associate Editor for Sequences Digital Object Identier /TIT (1) The numerical results show that there exist bent functions obtained in this paper affinely inequivalent to all known monomial binomial ones Moreover, the proposed family is the first proven class containing 5-ary regular bent functions with algebraic degree satisfying the upper bound in [12] To prove the main result, ie, Theorem 1, we first prove that the bentness of the proposed functions only depends on its Walsh transform value at the point zero, then establish a relationship between the bentness of these functions some partial exponential sums (see (3) in Section II for their definitions) Inspired by the technique developed in [10, Proposition 2], we consider a dferent counting problem reduce it to a situation similar to that studied in [10, Proposition 1], then establish a relationship between Kloosterman sums some partial exponential sums in (3), which leads to Theorem 1 The remainder of this paper is organized as follows Section II introduces some preliminaries In Section III, the relationship between Kloosterman sums some partial exponential sums is determined A family of -ary binomial functions their properties are considered in Section IV The proof of Theorem 1 is completed in Section V Section VI discusses an improved version of Theorem 1 in some special cases II PRELIMINARIES For a -ary function from to, the direct inverse Walsh transform operations on are defined as follows: where is the complex primitive th root of unity the elements of are considered as integers modulo is called a -ary bent function (or generalized bent function) for all A bent function is called regular for each, for some -ary function from to A bent function is called weakly regular there is a complex with unit magnitude such that The function is called the dual of Further, the dual of a (weakly) regular bent function is again a (weakly) regular bent function /$ IEEE

2 JIA et al: A CLASS OF BINOMIAL BENT FUNCTIONS OVER THE FINITE FIELDS OF ODD CHARACTERISTIC 6055 The Walsh spectrum of a -ary bent function satisfies [16] For, three exponential sums are defined as The finite field can be identied as an -dimensional linear space over under a chosen basis over When is even, a -ary function from to is called normal there exists an -dimensional flat of such that the restriction of to this flat is constant Moreover, is called weakly normal there is an -dimensional flat such that the restriction of to this flat is affine Note that each function from to can be uniquely determined by a polynomial of variables over This polynomial is called the algebraic normal form of its degree is called the algebraic degree of, denoted by Itis well known that the maximal algebraic degree of a Boolean bent function on is equal to For a -ary bent function, some estimates on the upper bound are established in [12] [17] Up to now, the best upper bound is given in [12, Proposition 45] as follows Proposition 1 ([12]): Let be a -ary bent function on, then Moreover, is a (weakly) regular bent function, then Let denote the general linear group of order on Two -ary functions are called affinely equivalent there exist some,, such that It is well known that algebraic degree bentness of a -ary function are affine invariants, it is an interesting topic to find -ary bent functions affinely inequivalent to the known ones Let denote a primitive element of An element of is called a square for some element Otherwise, is called a nonsquare For each square, let denote an element such that Let denote the sets of squares nonsquares in, respectively, ie, where For an even, let represent the cyclic subgroup of with elements as (2) Thus, Let for a positive integer, define a set as Lemma 1: For each element, there exists a pair such that for with with Thus, runs through exactly once as run through, respectively For, the Kloosterman sum is defined as where for Remark 1: It is well known that the value of a Kloosterman sum is a real number, satisfies the Weil bound [20], namely Furthermore, the Kloosterman sum always takes an integer value or 3 When, the set is exactly the set of integers divisible by 3 in the Weil range [13] Let for be a mapping from to itself For each, let indicate that (3) (4) (5) The following result is a slight generalization of [7, Th 14] their proofs are similar For the reader s convenience, we still give a proof here Proposition 2: For, is even for each only is a nonsquare, or is a square with Proof: If is even for each, is a square with, then (6) Two subsets of are defined as Thus, For an element in the set,wehave then The equality implies That is to say, the elements in occur in pairs except the element Thus

3 6056 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 9, SEPTEMBER 2012 TABLE I VALUE OF N FOR SOME VALUES OF THE PARAMETERS p AND m is odd this leads to a contradiction Consequently, is a nonsquare, or a square with On the other h, is a nonsquare or is a square with, then must be even Indeed, for each in,wehave Otherwise, implies then for each Let, then we have Assume, then By (12), we have where This implies for each It can be veried that which contradicts the fact Therefore, the elements in occur in pairs, ie, is an even number for each The aforementioned analysis finishes the proof Proposition 3: For, where denotes the secant function then is a nonsquare, or a square with Proof: By (8), without loss of generality, we can assume that For a fixed, can be abbreviated as for each Note that Equalities (7) (8) imply By (9) Then, the coefficients in equality (10) satisfy (7) (8) (9) (10) That is to say, there are integers taking the value ones taking the value Thus ie,, then both are even By Proposition 2, the element is a nonsquare, or a square with Remark 2: Let for For some values of the parameters, with the help of computer, we examine the values of for all, the numerical results show that takes the same value for each Table I lists the value of for,,, it indicates that the Kloosterman sum can not always take the values in III KLOOSTERMAN SUMS AND SOME PARTIAL EXPONENTIAL SUMS In this section, it is always assumed that A relationship between the Kloosterman sums the exponential sums defined by (3) is established by the third author Kholosha as follows Lemma 2 (see [9]): For,wehave (11) since is the minimal polynomial of over the rational numbers Equalities (6) (11) indicate (12) The following result is a straightforward generalization of [10, Proposition 1], their proofs are similar Thus, we omit the proof here Lemma 3: For, define (13)

4 JIA et al: A CLASS OF BINOMIAL BENT FUNCTIONS OVER THE FINITE FIELDS OF ODD CHARACTERISTIC 6057 Then where otherwise otherwise (14) Therefore, for For, define For, define (15) Then, by (3) Lemma 4: For any,wehave (16) otherwise Proof: If, then Thus, Therefore, by (15) By the technique developed in [10, Proposition 2], we obtain the following lemma Lemma 5: For,,wehave (17) then where is defined by (14) Proof: Note that Then (18) since is a balanced function from to Then If,, ie, Then, Therefore where in we replace with Similarly, we have Let, then we have (19)

5 6058 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 9, SEPTEMBER 2012 Equalities (18) (19) imply (20) For, (respectively, ) only (respectively, ), ie, (respectively, ) By Lemma 3 (20) otherwise where is defined by (14) This finishes the proof The following result directly follows from Lemma 2 (23) It establishes a relationship between the Kloosterman sum the partial exponential sums Lemma 7: For otherwise (24) otherwise (25) where (21) (21) is defined by (14) Thus, equality (17) follows from In the rest of this paper, for fixed, an element, we always use the notations, as (22) where in the notation the element is required to be a square Lemma 6: For otherwise Proof: By (16), (17), Lemma 4, we have (23) IV NEW CLASS OF -ARY BINOMIAL FUNCTIONS For,, with, define a -ary function from to as in (1), namely The bentness of the function has already been investigated in [9] as follows Proposition 4 (see [9]): For, with, the -ary function (26) is a regular bent function only Remark 3: The Kloosterman sum cannot take the value zero for (see [9] [14]) That is to say, there exist bent functions only for We characterize the bentness of the binomial function defined by (1) for an arbitrary even integer an arbitrary odd prime Theorem 1: For,, with, the functions defined by (1) are both regular bent functions only Furthermore, is a regular bent function, then the dual function is given by otherwise where denotes the unique solution of the equation in defined by (4)

6 JIA et al: A CLASS OF BINOMIAL BENT FUNCTIONS OVER THE FINITE FIELDS OF ODD CHARACTERISTIC 6059 When, Theorem 1 also holds it is exactly Proposition 4 We claim that a function given by (1) is regular bent only the function is regular bent According to this statement, which will be explained in Proposition 6 (2) in Section V, we only need to study the bentness of instead In the following, we always let be short for for the sake of convenience The proof of Theorem 1 will be presented in the next section Note that is a square (respectively, nonsquare) only is a square (respectively, nonsquare) in The following conclusion directly follows from Theorem 1 Proposition 3 Corollary 1: For,, with, are regular bent functions, then is a nonsquare in or a square in with The value of (8) can be easily computed for 5 Thus, we can obtain the following corollaries as two special cases of Theorem 1 Corollary 2: For,, with, both are regular bent functions only By Remark 1, there is an element such that Let be a primitive element of ; then is a primitive element of Consequently, for some integer with Thus, the functions are regular bent, ie, for, there are always bent functions having the form as in (1) Furthermore, some of these bent functions can be affinely inequivalent to the ones having the form as in (26) This will be illustrated by the following example Example 1: Let be generated by the primitive polynomial be a primitive element of By computer, we find 30 binomial bent functions with the form where These functions can be classied into two equivalence classes whose representatives are, respectively On the other h, for each positive integer with, there are 20 monomial bent functions with the form, where It can be veried that all these bent functions are affinely equivalent to the function in this case Moreover, it also can be checked that are affinely inequivalent, while are affinely equivalent Corollary 3: For,,,, with, both are regular bent functions only are bent A total of 52 specic are found they are all nonsquares in with We also find a total of 52 pairs of bent functions with the form, where all the specic are nonsquares in with Moreover, by some easy calculation, we find that all these bent functions have algebraic degree 8, which reaches the upper bound of algebraic degree for 5-ary (weakly) regular bent functions Indeed, this is the first proven class that contains 5-ary regular bent functions having the largest algebraic degree By Remark 3, there are no monomial bent functions having the form as in (26) for Consequently, these bent functions obtained in Example 2 are affinely inequivalent to any monomial function with the form By Table I Corollary 1, there do not exist the bent function pairs with the form as in (1) when In the sequel, we discuss the algebraic degree normality of the functions with the form as in (1) It can be veried that by Proposition 3 in [9] Note that the two exponents are not in the same cyclotomic coset modulo Consequently, for any, the function satisfies For an odd prime, some known classes of -ary monomial binomial bent functions are listed in Tables II III, where is a positive integer,, r (respectively, wr ) means that the bent function is regular (respectively, weakly regular) Some notations in the tables are explained in order: ar is short for arbitrary, Deg for algebraic degree, K-M for Kumar Moreno, C-M for Coulter Matthews, H-K for Helleseth Kholosha Recall that algebraic degree is an affine invariant From Tables II, III Example 1, there exist bent functions with the form as in (1) which are affinely inequivalent to all known ones listed in Tables II III Proposition 5: Let A regular bent function defined as in (1) is normal Proof: For each regular bent function, define Then, Thus, Note that is the minimal polynomial of over the rational numbers Consequently, we have Example 2: Let be a finite field generated by the primitive polynomial We made an exhaustive search for all such that both of the two binomial functions That is to say,, ie, there exists such that Then, is an -dimensional flat of For an arbitrary, we

7 6060 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 9, SEPTEMBER 2012 TABLE II SOME KNOWN CLASSES OF p-ary MONOMIAL BENT FUNCTIONS TABLE III SOME KNOWN CLASSES OF p-ary BINOMIAL BENT FUNCTIONS have, where belongs to satisfying or 1 If, then If,wehave That is to say, is constant on then is normal V PROOF OF THEOREM 1 In this section, we discuss the relationship of bentness between then characterize the bentness of in terms of the values, respectively The same analysis as in the proof of [9, Proposition 4] leads to the following result Lemma 8 (see [9]): Let be a positive integer : be a regular bent function satisfying, then Proposition 6: Let be defined by (1) Then (1) is a regular bent function only ; (2) is a regular bent function only is regular bent Furthermore, is a regular bent function, then the dual function is given by otherwise where denotes the unique solution of the equation in defined by (4) Proof: (1) If is a regular bent function, then by the definition of a regular bent function Lemma 8 On the other h,, then by Lemma 1 which is equivalent to For any (27) (28) It is easy to very that is the unique solution of the equation in Asa consequence, equality (28) indicates (29) where the last step follows from (27) Then is a regular bent function, the dual function is also given by (29) (2) According to the previous statement, is a regular bent function only equality (27) holds Note that

8 JIA et al: A CLASS OF BINOMIAL BENT FUNCTIONS OVER THE FINITE FIELDS OF ODD CHARACTERISTIC 6061 the positive integer satisfies is a permutation on As a consequence, we have That is to say, for any with, have the same bentness (30) Proof of Theorem 1: Due to Proposition 6 (2), we can assume that without loss of generality By Lemma 1, for each, there are unique such that Then where,, are given by (22) Assume that, then, ie, By Proposition 3, is a nonsquare in or is a square in with Note that is a nonsquare (respectively, square) in only (respectively, ), then by (34) By Proposition 6, both are regular bent functions If both are regular bent functions, by Proposition Suppose that, thus by (34) then It is in contradiction with That is to say, or with Again by (34), one has This finishes the proof VI IMPROVEMENT ON THEOREM 1 In this section, we give an improved version of Theorem 1 in two special cases Especially, for or, we prove that only the function (not the pair of functions ) is regular bent Proposition 7: Let,, with Let be given by (22), be given by (14), the function defined by (1) be a regular bent function Define If, then A similar deduction as in (31) shows (31) As- Proof: By Proposition 6, we have sume that, then by (34) (32) (35) By (31) (32), we have where are given by (22) is given by (14) Note that As a consequence, by (35) (33) Equalities (24), (25), (33) imply then (34) (36) By (36)

9 6062 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 9, SEPTEMBER 2012 It is in contradiction with (5), so or with It follows from (34) that With the preparation above, we have the following result Theorem 2: Let or For, with, defined by (1) is a regular bent function only Proof: First, the sufficient condition directly follows from Theorem 1 In the following, we will prove the necessary condition According to Proposition 6 (2), we can assume that without loss of generality If is a regular bent function, then by Proposition 6 Let, be given by (22) By (34) (37) If or with, then by (37), ie, Consequently, it suffices to consider the case with In the case of, only is odd Note that is a real number while is a pure imaginary number in the case of odd Thus, cannot be equal to 1, which contradicts equality (37) Thus, for the case of, we have or with, then the conclusion has been proved as previously In the case of, we only need to consider even When, ie, By computer, for every bent function with the form as in (1), satisfies Note that, ie, or 2 Then, for any,wehave When is even By Proposition 7, for even As a consequence, This finishes the proof For the cases other than,we do not know how to improve Theorem 1 it is left to the reader as an open problem REFERENCES [1] A Canteaut, P Charpin, G Kyureghyan, A new class of monomial bent functions, Finite Fields Appl, vol 14, no 1, pp , Jan 2008 [2] C Carlet, Boolean Functions for Cryptography Error Correction Codes, in Boolean Medels Methods in Mathematics Computer Science, Engineering, Y Crama P Hammer, Eds Cambridge, UK: Cambridge Univ Press, 2010, pp [3] P Charpin G Kyureghyan, Cubic monomial bent functions: A subclass of M, SIAM J Discr Math, vol 22, no 2, pp , Mar 2008 [4] R S Coulter R W Matthews, Planar functions planes of Lenz-Barlotti class II, Des Codes Cryptogr, vol 10, no 2, pp , Feb 1997 [5] J F Dillon, Elementary Hadamard dference sets, PhD dissertation, Univ Maryl, Collage Park, 1974 [6] H Dobbertin, G Leer, A Canteaut, C Carlet, P Felke, P Gaborit, Construction of bent functions via Niho power functions, J Combin Theory, ser A, vol 113, no 5, pp , Jul 2006 [7] K Garaschuk P Lisoněk, On ternary Kloosterman sums modulo 12, Finite Fields Appl, vol 14, no 4, pp , Nov 2008 [8] T Helleseth, H D L Hollmann, A Kholosha, Z Y Wang, Q Xiang, Proofs of two conjectures on ternary weakly regular bent functions, IEEE Trans Inf Theory, vol 55, no 11, pp , Nov 2009 [9] T Helleseth A Kholosha, Monomial quadratic bent functions over the finite fields of odd characteristic, IEEE Trans Inf Theory, vol 52, no 5, pp , May 2006 [10] T Helleseth A Kholosha, New binomial bent functions over the finite fields of odd characteristic, IEEE Trans Inf Theory, vol 56, no 9, pp , Sep 2010 [11] T Helleseth P V Kumar, Sequences with low correlation, in Hbook of Coding Theory, V S Pless W C Huffman, Eds Amsterdam, The Netherls: Elsevier Science, 1998, vol II, pp [12] X D Hou, p-ary q-ary versions of certain results about bent functions resilient functions, Finite Fields Appl, vol 10, no 4, pp , Oct 2004 [13] N Katz R Livné, Sommes Kloosterman et courbes elliptiques universelles en caract eristiques 2 et 3, C R Acad Sci Paris Ser I Math, vol 309, no 11, pp , 1989 [14] K Kononen, M Rinta-aho, K Väänänen, On integer values of Kloosterman sums, IEEE Trans Inf Theory, vol 56, no 8, pp , Aug 2010 [15] P V Kumar O Moreno, Prime-phase sequences with periodic correlation properties better than binary sequences, IEEE Trans Inf Theory, vol 37, no 3, pp , May 1991 [16] P V Kumar, R A Scholtz, L R Welch, Generalized bent functions their properties, J Combin Theory, ser A, vol 40, no 1, pp , Sep 1985 [17] P Langevin, On generalized bent functions, in Proc Eurocode 92 Int Symp (CISM Courses Lectures), Vienna, Austria, 1993, vol 339, pp [18] P Langevin G Leer, Monomial bent functions Stickelberger s theorem, Finite Fields Appl, vol 14, no 3, pp , Jul 2008 [19] N G Leer, Monomial bent functions, IEEE Trans Inf Theory, vol 52, no 2, pp , Feb 2006 [20] R Lidl H Niederreiter, Finite Fields (Encyclopedia of Mathematics Its Applications) London, UK: Addison-Wesley, 1983, vol 20 [21] S C Liu J J Komo, Nonbinary Kasami sequences over GF(p), IEEE Trans Inf Theory, vol 38, no 4, pp , Jul 1992 [22] F MacWilliams N Sloane, The Theory of Error-Correcting Codes Amsterdam, The Netherls: North Holl, 1977 [23] S Mesnager, Bent hyper-bent functions in polynomial form their link with some exponential sums Dickson polynomials, IEEE Trans Inf Theory, vol 57, no 9, pp , Sep 2011 [24] O Rothaus, On bent functions, J Combin Theory, ser A, vol 20, no 3, pp , May 1976 ACKNOWLEDGMENT The authors would like to thank the two anonymous referees for their helpful comments, which have improved the presentation of this paper Wenjie Jia received the BS degree in mathematics from Hubei University, Wuhan, China, in 2010 He is currently working toward the MS degree in Hubei University His research interest includes cryptography

10 JIA et al: A CLASS OF BINOMIAL BENT FUNCTIONS OVER THE FINITE FIELDS OF ODD CHARACTERISTIC 6063 Xiangyong Zeng received the BS degree from the Department of Mathematics, Hubei University, Wuhan, China in 1995, MS degree PhD degree from the Department of Mathematics, Beijing Normal University, Beijing, China in respectively From 2002 to 2004, he was a postdoctoral member in the Computer School of Wuhan University, Wuhan, China He is currently a professor of Hubei University His research interests include cryptography, sequence design coding theory Tor Helleseth (M 89 SM 96 F 97) received the C Real Dr Philos degrees in mathematics from the University of Bergen, Bergen, Norway, in , respectively From 1973 to 1980, he was a Research Assistant with the Department of Mathematics, University of Bergen, Norway From 1981 to 1984, he was with the Chief Headquarters of Defense in Norway Since 1984, he has been a Professor in the Department of Informatics, University of Bergen During the academic years , he was on sabbatical leave at the University of Southern Calornia, Los Angeles, during , he was a Research Fellow at the Eindhoven University of Technology, Eindhoven, The Netherls His research interests include coding theory cryptology Prof Helleseth served as an Associate Editor for Coding Theory for the IEEE TRANSACTIONS ON INFORMATION THEORY from 1991 to 1993 He was Program Chairman for Eurocrypt 93 for the Information Theory Workshop in 1997 in Longyearbyen, Norway He was a Program Co-Chairman for SETA04, Seoul, Korea, SETA06, Beijing, China He was also a Program Co-Chairman for the IEEE Information Theory Workshop in Solstr, Norway, in 2007 During , he served on the Board of Governors for the IEEE Information Theory Society In 1997 he was elected an IEEE Fellow for his contributions to coding theory cryptography In 2004, he was elected a member of Det Norske Videnskaps-Akademi Chunlei Li received the BS degree MS degree in mathematics from Faculty of Mathematics Computer Science, Hubei University, Wuhan, China in respectively He is currently pursuing the PhD degree in reliable communication with the Department of Informatics, University of Bergen, Norway His research interest lies in cryptographic properties of Boolean pseudo-boolean functions, sequences coding theory