On Generalizations of Maiorana-McFarland and $\mathcal{PS}_{ap}$ Functions

On Generalizations of Maiorana–McF arland and P S ap F unctions Sezel Alk an 1 , Nurdag ¨ ul An bar 1 , A thina Avran tini 2 , Erro xe Etxabarri-Alb erdi 3 , T ekg ¨ ul Kala ycı 4 , Beatrice T o esca 5 1 Sabancı Univ ersity , MDBF, Orhanlı, T uzla, 34956 ˙ Istan bul, T urk ey 2 Departmen t of Mathematics, Universit y of P ennsylv ania, Philadelphia, P A 19104-6395, USA 3 Basic Sciences Departmen t, Mondragon Unib ertsitatea, 20500 Arrasate, Gipuzk oa, Spain 4 Institut f ¨ ur Mathematik, Alp en-Adria-Univ ersit¨ at Klagenfurt, Austria 5 Institute of Mathematics, Univ ersity of Zuric h, Switzerland Email: sezel.alkan@sabanciuniv.edu Email: nurdagulanbar2@gmail.com Email: athina@sas.upenn.edu Email: eetxabarri@mondragon.edu Email: tekgulkalayci1@gmail.com Email: beatrice.toesca@math.uzh.ch Abstract W e study generalizations of tw o classical primary constructions of Bo olean b ent functions, namely the Maiorana–McF arland ( M M ) class and the (Desarguesian) partial spread ( P S ap ) class. The construction of b en t functions lying outside the completed M M class has attracted considerable atten tion in recen t y ears. In this direction, w e construct families of generalized Maiorana–McF arland b ent functions that are not equiv alen t to an y function in the classical M M or P S ap classes, and hence lie outside their completed classes. As a second contribution, we in vestigate the decomp osition of generalized P S ap functions. W e pro ve that when the degree is sufficiently small relative to the size of the underlying finite field, suc h functions do not, in general, admit a decomp osition in to b en t or semibent fun ctions. Consequen tly , they cannot be obtained from kno wn secondary constructions based on concatenation. Finally , we present a secondary construction of Bo olean b en t functions arising from the concatenation of comp onen ts of v ectorial generalized P S ap functions. Our constructions and pro ofs rely on classical results concerning second-order deriv atives of b en t functions and their duals. In addition, w e employ metho ds from the theory of algebraic curv es and their function fields. 1 Keyw ords: Boolean functions, Ben t functions, Concatenation, Decomposition, Maiorana– McF arland functions, Partial spread functions, Algebraic curv es, F unction fields. Mathematics Sub ject Classification (2010): 11T06, 94A60, 14H05. 1 In tro duction Let p b e a prime and let V ( p ) n b e an n -dimensional v ector space o v er the prime field F p . W e fix a non-degenerate inner pro duct ⟨· , ·⟩ n on V ( p ) n . If V ( p ) n = F n p is the v ector space of n -tuples o v er F p , we take the usual dot pro duct ⟨ b, x ⟩ n = b · x . If V ( p ) n = F p n is the finite field of order p n , w e define ⟨ b, x ⟩ n = T r n 1 ( bx ), where, for any divisor k of n , T r n k denotes the trace mapping from F p n to F p k . When n = 2 m , it is often conv enien t to identify V ( p ) n = F p m × F p m , in which case the inner pro duct is given by ⟨ ( u, v ) , ( x, y ) ⟩ n = T r m 1 ( ux + v y ). A function f : V ( p ) n → F p is called a p -ary function ; in the special case p = 2, it is called a Bo ole an function . The W alsh transform of f is the complex-v alued function defined b y W f ( b ) = X x ∈ V ( p ) n ζ f ( x ) −⟨ b,x ⟩ n p , b ∈ V ( p ) n , where ζ p = e 2 π i/p and i is a primitive fourth ro ot of unity . The extende d Walsh sp e ctrum of f is the m ultiset { | W f ( b ) | : b ∈ V ( p ) n } . The function f : V ( p ) n → F p is called b ent if | W f ( b ) | = p n/ 2 for all b ∈ V ( p ) n . Equiv a- len tly , f is b en t if and only if, for ev ery nonzero v ector a ∈ V ( p ) n , its first-order deriv ative D a f ( x ) = f ( x + a ) − f ( x ) is balanced, i.e., for each c ∈ F p , the equation D a f ( x ) = c has exactly p n − 1 solutions in V ( p ) n . In the Bo olean case p = 2, the W alsh transform of a b en t function f satisfies W f ( b ) = 2 n/ 2 ( − 1) f ∗ ( b ) , where f ∗ : V (2) n → F 2 is a Bo olean function, called the dual of f , whic h is itself bent. Since W f is in teger-v alued in this case, Bo olean bent functions exist only when n is even. In con trast, for o dd p , b ent functions f : V ( p ) n → F p exist for b oth ev en and o dd v alues of n . Their W alsh transform satisfies W f ( b ) =    ± ζ f ∗ ( b ) p p n/ 2 , if n is even, or if n is o dd and p ≡ 1 mo d 4 , ± i ζ f ∗ ( b ) p p n/ 2 , if n is o dd and p ≡ 3 mo d 4 , where f ∗ : V ( p ) n → F p is again called the dual of f ; see [19]. A function f : V ( p ) n → F p is called s -plate aue d if there exists an integer 0 ≤ s ≤ n such that | W f ( b ) | ∈ { 0 , p ( n + s ) / 2 } for all b ∈ V ( p ) n . In particular, f is b ent if and only if s = 0. Moreo v er, in the Bo olean case p = 2, since the W alsh transform W f tak es integer v alues, the in tegers n and s must hav e the same parity . A Bo olean function is called semib ent if s = 1 when n is o dd, or if s = 2 when n is even. 2 Let F : V ( p ) n → V ( p ) k b e a function. F or each α ∈ V ( p ) k \ { 0 } , the asso ciated c omp onent function of F is defined by F α ( x ) = ⟨ α, F ( x ) ⟩ k . The function F is called (ve ctorial) b ent if all its comp onen t functions F α , α ∈ V ( p ) k \ { 0 } , are bent functions. In this case, the set of comp onent functions of F , together with the zero function, forms a k -dimensional v ector space o ver F p of b en t functions. In terms of the W alsh transform, a function F : V ( p ) n → V ( p ) k is b en t if, for ever y α ∈ V ( p ) k \ { 0 } and ev ery b ∈ V ( p ) n , W F α ( b ) = X x ∈ V ( p ) n ζ ⟨ a,F ( x ) ⟩ k −⟨ b,x ⟩ n p , ζ p = e 2 π i/p , has absolute v alue p n/ 2 . Equiv alently , a function F : V ( p ) n → V ( p ) k is b ent if, for every nonzero a ∈ V ( p ) n , the deriv ativ e D a F ( x ) = F ( x + a ) − F ( x ) is a balanced function from V ( p ) n to V ( p ) k . Extended-affine equiv alence is a fundamental notion in the theory of (vectorial) func- tions, as it characterizes transformations that preserve imp ortant cryptographic prop er- ties, including the extended W alsh sp ectrum and, in particular, b entness. Tw o functions F, G : V ( p ) n → V ( p ) k are said to be extende d-affine e quivalent (EA- e quivalent) if G ( x ) = L 1  F ( L 2 ( x ) + a )  + L 3 ( x ) + b, where L 1 : V ( p ) k → V ( p ) k and L 2 : V ( p ) n → V ( p ) n are linear permutations, L 3 : V ( p ) n → V ( p ) k is a linear map, and a ∈ V ( p ) n , b ∈ V ( p ) k . In the special case k = 1, EA-equiv alence reduces to the follo wing relation b et w een functions f , g : V ( p ) n → F p : g ( x ) = f ( L ( x ) + a ) + ⟨ c, x ⟩ n + b, where L is a linear p ermutation on V ( p ) n , c ∈ V ( p ) n , and b ∈ F p . A class C of p -ary , respectively Boolean, b en t functions is called c omplete d if it is in v ariant under EA-. In other words, C contains a function f if and only if it con tains ev ery function EA-equiv alen t to f . The c ompletion of C , denoted b y C # , is the smallest completed class con taining C . Ben t functions can be obtained via t w o main t yp es of constructions: primary and se c- ondary . Primary constructions are direct algebraic metho ds that pro duce b ent functions from scratch, without relying on previously kno wn examples. In contrast, secondary con- structions generate new b ent functions from one or more known b en t (or related) functions b y applying transformations or com binations that preserve b en tness. While many sec- ondary constructions of b en t functions are kno wn (see [11, Section 6.1.16]), there are only t w o classical primary constructions, namely the Maiorana–McF arland construction [22] and the partial spread (Dillon) construction [14]. It is well kno wn that all b ent functions in dimensions up to six b elong to the completed Maiorana–McF arland class. The first examples of b ent functions outside the M M # class w ere constructed b y Dillon [14], who gav e explicit examples in eight v ariables. It w as 3 later shown in [20], using computer searc h, that the M M # class constitutes only a very small fraction of all b en t functions. More precisely , the n um b er of b en t functions in the M M # class is at most 2 72 , whereas there are approximately 2 106 b en t functions in dimension eigh t. This observ ation initiated an active line of researc h fo cused on the construction of b en t functions lying outside the M M # class; see, for instance, the recen t pap ers [21, 26, 27, 28], as well as the surv ey pap er [24] and the references therein. While exclusion from the completed Desarguesian partial spread P S # ap class—a dis- tinguished sub class of the partial spread construction—can b e detected using in v ariants suc h as the algebraic degree or the 2-rank (see [6, 31]), membership in the M M # class is c haracterized b y a criterion due to Dillon [14], formulated in terms of second-order deriv atives. In this pap er, although we presen t certain results in the more general setting of an arbitrary prime p , our primary focus is on the construction and classification of (vectorial) Bo olean functions. The pap er is organized as follo ws. In Section 2, w e briefly review t w o classical con- structions: Maiorana–McF arland and Desarguesian partial spread b ent functions, and their generalizations for arbitrary c haracteristic p . In Section 3, we present a construc- tion of Bo olean generalized Maiorana–McF arland functions that do not b elong to the M M # class. In Section 4, w e study the decomp osition of Bo olean generalized P S ap func- tions and show that, in general, they do not arise from secondary constructions of b en t or semib ent functions via concatenation when their degree is sufficiently small relative to the size of the finite field on whic h they are defined. Finally , in Section 5, w e pro vide a secondary construction of Bo olean b ent functions motiv ated by the concatenation of the comp onen ts of v ectorial P S ap functions. 2 Generalized Maiorana–McF arland and generalized P S ap functions In this section, we briefly recall t w o fundamen tal primary constructions of p -ary (v ecto- rial) b en t functions: the Maiorana–McF arland class, introduced indep enden tly b y Maio- rana (unpublished) and McF arland [22], and the Desarguesian partial spread class P S ap , in tro duced b y Dillon in his Ph.D. thesis [14], together with their generalizations. Throughout, w e use the biv ariate represen tation V ( p ) n = F p m × F p m , n = 2 m , and presen t the corresp onding definitions in this setting. Maiorana–McF arland ( M M ) class. A function f : F p m × F p m → F p is said to b elong to the Maiorana–McF arland class if it is of the form f ( x, y ) = T r m 1  x π ( y )  + g ( y ) , where π is a p erm utation of F p m and g : F p m → F p is an arbitrary function. 4 In the vectorial case, a function F : F p m × F p m → F p m is called Maiorana–McF arland if it can b e written as F ( x, y ) = x π ( y ) + G ( y ) , where G : F p m → F p m is an arbitrary function. It is well kno wn that f , and respectively F , is b en t if and only if π is a p erm utation of F p m . The following characterization of Bo olean b en t functions, stated in a general form for the completed M M class, w as given b y Dillon. Lemma 1. [14, Dil lon ’s criterion] L et n = 2 m . A Bo ole an b ent function f : V (2) n → F 2 b elongs to the M M # class if and only if ther e exists an m -dimensional ve ctor subsp ac e U ⊆ V (2) n such that, for al l a, b ∈ U , the se c ond-or der derivatives D a D b f ( x ) = f ( x ) + f ( x + a ) + f ( x + b ) + f ( x + a + b ) (1) vanish identic al ly. A subspace U ⊆ V (2) n for which D a D b f ( x ) = 0 for all a, b ∈ U and all x ∈ V (2) n is called an M -subsp ac e of f (see [28]). W e note that b oth the n um ber of M -subspaces of f and its maximal dimension, called the line arity index ind( f ) of f , are in v arian t under EA-equiv alence. F or a bent function f : V (2) n → F 2 , the linearit y index satisfies 1 ≤ ind( f ) ≤ n/ 2; see [24, Prop osition 5.1]. Consequently , a b ent function f on V (2) n b elongs to the M M # class if and only if ind( f ) = n/ 2. Generalized Maiorana–McF arland ( GM M ) class. Let m and k b e integers with 1 ≤ k ≤ m . F or each z ∈ V ( p ) k , let f z : V ( p ) m → F p b e a k -plateaued p -ary function, and define its W alsh supp ort b y supp( W f z ) = { b ∈ V ( p ) m : W f z ( b )  = 0 } . Assume that the W alsh supp orts are pairwise disjoint, that is, supp( W f z ) ∩ supp( W f y ) = ∅ for all z , y ∈ V ( p ) k , z  = y . Then the function f : V ( p ) m × V ( p ) k → F p , f ( x, z ) = f z ( x ) , (2) is a p -ary b ent function (see [13] for details). Observ e that, for fixed y , the Maiorana–McF arland function f ( x, y ) = T r m 1  x π ( y )  + g ( y ) is affine in x , and hence corresp onds to an m -plateaued p -ary function that is affine on each coset of F p n × { 0 } . Note also that when k = m , the construction (2) reduces to the Maiorana–McF arland class. F or this reason, functions defined by (2) are called gen- er alize d Maior ana–McF arland functions if they are affine on the cosets of a k -dimensional subspace. 5 A commonly used finite field represen tation of the GM M construction is the following. Consider the function f : F p n × F p k × F p k → F p defined b y f ( x, y , z ) = f ( z ) ( x ) + T r k 1 ( y z ) , (3) where, for eac h z ∈ F p k , the function f ( z ) : F p n → F p is b en t. F or fixed z , the function f ( · , · , z ) is a k -plateaued p -ary function on F p n × F p k , and f is affine on the cosets of { 0 } × { 0 } × F p k . Note that the corresp onding W alsh supp orts of f ( z ) are pairwise disjoin t, in accordance with the GM M construction. In [1, Theorem 3], the W alsh transform of f in (3) and the explicit form of its dual f ∗ are deriv ed. In particular, the dual function f ∗ is again a GM M function, giv en by f ∗ ( x, y , z ) =  f ( y )  ∗ ( x ) − T r k 1 ( y z ) , and satisfies f ∗∗ = ( f ∗ ) ∗ = f . Remark 1. A related notion of the generalized Maiorana–McF arland class was in tro- duced in [9, Section 4.3] as follows. Let 0 ≤ k ≤ n 2 − 1, and set r = n 2 − k and s = n 2 + k . A Bo olean function f φ,h : V (2) r × V (2) s → F 2 of the form f φ,h ( x, y ) = ⟨ x, π ( y ) ⟩ r + h ( y ) , x ∈ V (2) r , y ∈ V (2) s , (4) is said to b e a generalized Maiorana–McF arland function, where π : V (2) s → V (2) r and h : V (2) s → F 2 are arbitrary functions; see also [12]. Observ e that, for fixed x ∈ F p n and z ∈ F p k , the function defined in (3) is affine in the v ariable y . Consequently , f is equiv alent to a function on V (2) k × V (2) n + k of the form f ( x, y ) = ⟨ x, π ( y ) ⟩ k + h ( y ), where π : V (2) n + k → V (2) k and h : V (2) n + k → F 2 . Observ e also that, when k = 0, the ab ov e construction reduces to the classical Maiorana–McF arland class of b ent functions, provided that π is a p erm utation, whereas for k = n 2 − 1 the completed class defined by (4) con tains all Bo olean functions on V (2) n . There is an extensiv e researc h dev oted to the classification of functions of the form (4), in particular to the c haracterization of those functions that are b en t; see the survey pap er [24] and the references therein. Desarguesian partial spread ( P S ap ) class. Let n = 2 m . A (c omplete) spr e ad of V ( p ) n is a collection of m -dimensional F p -subspaces U 0 , U 1 , . . . , U p m ⊆ V ( p ) n suc h that U i ∩ U j = { 0 } for all 0 ≤ i < j ≤ p m . Equiv alen tly , every nonzero elemen t of V ( p ) n b elongs to exactly one of the sets U ∗ j = U j \ { 0 } . It is w ell kno wn that ev ery function f : V ( p ) n → F p constructed as follo ws is a bent function. F or every c ∈ F p , the function f maps the elements of exactly p m − 1 of the sets U ∗ j , 1 ≤ j ≤ p m , to the v alue c , and f is constant on the subspace U 0 . Similarly , one obtains v ectorial partial spread bent functions F : V ( p ) n → V ( p ) m . F or ev ery c ∈ V ( p ) m , the function F maps the elements of exactly one of the sets U ∗ j , 1 ≤ j ≤ p m , to the v alue c , and F is constant on the subspace U 0 . 6 A w ell-known example of a spread is the Desar guesian spr e ad . In the biv ariate repre- sen tation V ( p ) n ∼ = F p m × F p m , it is given by U = { (0 , y ) : y ∈ F p m } , U s = { ( x, sx ) : x ∈ F p m } , s ∈ F p m . Ben t functions arising from the Desarguesian s pread admit the explicit representation f ( x, y ) = P  y x p m − 2  , (5) where P : F p m → F p is a balanced function. Analogously , v ectorial b ent functions F : F p m × F p m → F p m arising from the Desar- guesian spread are represented as F ( x, y ) = P  y x p m − 2  , where P : F p m → F p m is a p erm utation. F unctions obtained from the Desarguesian partial spread construction are called P S ap functions. It has b een sho wn that ev ery (v ectorial) b en t function arising from a spread of V ( p ) n has algebraic degree ( p − 1) n/ 2; see [14] for the case p = 2 and [8] for o dd primes p . Generalized P S ap class. The construction of b en t functions from spreads can b e generalized via the notion of (normal) b ent p artitions [8], defined as follows: A partition of V ( p ) n in to an ( n/ 2)-dimensional subspace U and sets A 1 , A 2 , . . . , A K is called a normal b ent p artition of depth K if ev ery function f : V ( p ) n → F p satisfying the follo wing conditions is b en t: (i) F or eac h c ∈ F p , exactly K /p of the sets A j are contained in the preimage f − 1 ( c ) = { x ∈ V ( p ) n : f ( x ) = c } . (ii) The function f is constant on the subspace U . In particular, if U 0 , U 1 , . . . , U p m is a spread of V ( p ) n , then the sets U 0 , U ∗ 1 , . . . , U ∗ p m , where U ∗ i = U i \ { 0 } , form a normal b en t partition of depth p m . The first class of b en t partitions that are not equiv alen t to those arising from spreads w as introduced in [8] for arbitrary prime p , and in [23] for p = 2. Since then, sev eral other primary and secondary constructions of b ent partitions ha ve b een dev elop ed; see, for instance, [1, 4, 7, 30], as well as the survey pap er [3] and the references therein. Among these, the construction presented in [8] (and [23]) admits an explicit represen- tation and can b e describ ed as follo ws. Let m , k , and e b e in tegers suc h that k | m , e ≡ p ℓ mo d ( p k − 1), and gcd( p m − 1 , e ) = 1. Let η denote the multiplicativ e in v erse of e mo dulo ( p m − 1), that is, η e ≡ 1 mo d ( p m − 1). F or s ∈ F p m , define the sets U = { (0 , y ) : y ∈ F p m } , U s = { ( x, sx e ) : x ∈ F p m } , U ∗ s = U s \ { (0 , 0) } . (6) 7 Similarly , define V = { ( x, 0) : x ∈ F p m } , V s = { ( sx η , x ) : x ∈ F p m } , V ∗ s = V s \ { (0 , 0) } . (7) F or γ ∈ F p k , define A ( γ ) = [ s ∈ F p m T r m k ( s )= γ U ∗ s , B ( γ ) = [ s ∈ F p m T r m k ( s )= γ V ∗ s . (8) Then the collections Ω 1 = { U, A ( γ ) : γ ∈ F p k } , Ω 2 = { V , B ( γ ) : γ ∈ F p k } , (9) are b en t partitions of F p m × F p m of depth p k . Moreov er, similarly to the case of P S ap b en t functions, the b en t functions f , g : F p m × F p m → F p obtained from Ω 1 and Ω 2 , resp ectiv ely , admit explicit representations given b y f ( x, y ) = P  T r m k  y x − e  + c 0  1 − x p m − 1  , and g ( x, y ) = P  T r m k  xy − η  + c 0  1 − y p m − 1  , where P : F p k → F p is a balanced function and c 0 ∈ F p is a constant. Similarly , the v ectorial version of the generalized P S ap functions from F p m × F p m to F p k is defined using a p erm utation P : F p k → F p k and c 0 ∈ F p k . In the case k = m , the partitions Ω 1 and Ω 2 are equiv alent to the Desarguesian spread. Accordingly , they are called gener alize d Desar guesian spr e ads , and the b ent functions obtained from them are referred to as gener alize d P S ap b ent functions . F or details, w e refer to [3, 8]. In general, a generalized P S ap b en t function do es not b elong to the completed Maiorana– McF arland class. Moreo v er, there exist generalized P S ap b en t functions that lie neither in the partial spread class nor in the completed Maiorana–McF arland class. F or further details, see [6, Examples 5.1 and 5.2]. W e remark that generalized P S ap b en t functions w ere originally presented without the additional terms c 0  1 − x p m − 1  and c 0  1 − y p m − 1  , resp ectiv ely , as in the classical P S ap construction. While an y P S ap b en t function is EA-equiv alen t to one of the form giv en in Equation (5), these additional terms are nev ertheless necessary in order to obtain a complete set of inequiv alen t generalized P S ap b en t functions; see [3, Example 1]. 3 Construction of GM M functions outside the M M # class W e b egin b y examining the generalized Maiorana–McF arland construction given in [13] o v er finite fields. Sp ecifically , we consider functions f : F p n × F p k × F p k → F p defined b y f ( x, y , z ) = f ( z ) ( x ) + T r k 1 ( y z ) , (10) 8 where, for eac h z ∈ F p k , the function f ( z ) : F p n → F p is b en t. T o c haracterize functions lying outside the completed Maiorana–McF arland class, we mak e use of Dillon’s criterion; see Lemma 1. In this con text, it is essential to understand the behavior of second-order deriv ativ es under EA-equiv alence. It is sho wn in [2] that, for a Bo olean function f : V (2) n → F 2 , the prop ert y of ha ving constant second-order deriv atives is in v ariant under EA-equiv alence. F or the con v enience of the reader, w e include the pro of here. Lemma 2. L et f : V (2) n → F 2 b e a Bo ole an function, and let a, b ∈ V (2) n b e line arly indep endent. L et L : V (2) n → V (2) n b e a line ar p ermutation, and let c, d ∈ V (2) n and e ∈ F 2 . Then D a D b f ( x ) is identic al ly e qual to 0 (r esp e ctively, identic al ly e qual to 1 ) for al l x ∈ V (2) n if and only if D L − 1 ( a ) D L − 1 ( b )  f  L ( x ) + c  + ⟨ d, x ⟩ n + e  (11) is also identic al ly 0 (r esp e ctively, identic al ly 1 ) for al l x ∈ V (2) n . Pr o of. Since any affine term ⟨ d, x ⟩ n + e has v anishing second-order deriv ativ es, we may assume without loss of generalit y that g ( x ) = f ( L ( x ) + c ). By the linearit y of L , it follo ws that D L − 1 ( a ) D L − 1 ( b ) g ( x ) = g  x + L − 1 ( a ) + L − 1 ( b )  + g  x + L − 1 ( a )  + g  x + L − 1 ( b )  + g ( x ) = f  L ( x + L − 1 ( a ) + L − 1 ( b )) + c  + f  L ( x + L − 1 ( a )) + c  + f  L ( x + L − 1 ( b )) + c  + f  L ( x ) + c  = f  L ( x ) + a + b + c  + f  L ( x ) + a + c  + f  L ( x ) + b + c  + f  L ( x ) + c  . Setting y = L ( x ) + c , we obtain D L − 1 ( a ) D L − 1 ( b ) g ( x ) = f ( y + a + b ) + f ( y + a ) + f ( y + b ) + f ( y ) = D a D b f ( y ) . Since the map x 7→ L ( x ) + c is a p ermutation of V (2) n , the assertion follows. Remark 2. Let f : V (2) n → F 2 b e a Bo olean function, and let L : V (2) n → V (2) n b e a linear p erm utation. Then, b y Lemma 2, W ⊆ V (2) n is an M -subspace of f if and only if L − 1 ( W ) is an M -subspace of f ◦ L . W e now analyze the second-order deriv ativ e of the function f : F 2 n × F 2 k × F 2 k → F 2 defined in Equation (10). Recall that f ( x, y , z ) = f ( z ) ( x ) + T r k 1 ( y z ) . Let ν 1 = ( u 1 , v 1 , w 1 ) and ν 2 = ( u 2 , v 2 , w 2 ) b e t w o distinct nonzero elements of F 2 n × F 2 k × F 2 k . Then D ν 1 D ν 2 f ( x, y , z ) = f ( x, y , z ) + f ( x + u 1 , y + v 1 , z + w 1 ) + f ( x + u 2 , y + v 2 , z + w 2 ) (12) + f ( x + u 1 + u 2 , y + v 1 + v 2 , z + w 1 + w 2 ) = f ( z ) ( x ) + f ( z + w 1 ) ( x + u 1 ) + f ( z + w 2 ) ( x + u 2 ) + f ( z + w 1 + w 2 ) ( x + u 1 + u 2 ) + T r k 1  w 1 v 2 + w 2 v 1  . 9 Remark 3. In the case w 1 = w 2 = 0, Equation (12) yields D ν 1 D ν 2 f ( x, y , z ) = f ( z ) ( x ) + f ( z ) ( x + u 1 ) + f ( z ) ( x + u 2 ) + f ( z ) ( x + u 1 + u 2 ) . (13) In particular, D ν 1 D ν 2 f ( x, y , z ) = D u 1 D u 2 f ( z ) ( x ), and hence D ν 1 D ν 2 f ( x, y , z ) is indep en- den t of y . Consequen tly , if the functions f ( z ) are M M functions sharing a common M -subspace of dimension n/ 2, then, by Dillon’s criterion, the function f ( x, y , z ) is itself M M . Indeed, it suffices to note that f ( x, y , z ) admits an M -subspace of dimension k + n/ 2. Let V denote the common M -subspace of the functions f ( z ) , with dim V = n/ 2. Then, b y Equation (13), the set V × F 2 k × { 0 } forms an M -subspace of f of dimension k + n/ 2, whic h establishes the claim. Remark 3 indicates that, in order to construct a GM M function from M M functions that do es not b elong to the completed Maiorana–McF arland class, it is necessary to com bine M M functions that do not share a common M -subspace of dimension n/ 2. Motiv ated by this observ ation, we consider the construction of GM M functions from M M functions that share only trivial M -subspaces, i.e., subspaces of dimension at most one. Our approac h begins with an M M function on V (2) n admitting a unique M -subspace W of dimension n/ 2, with the additional property that ev ery nontrivial M -subspace is con tained in W . By applying a suitably chosen linear p erm utation to this function and em b edding it in to a higher-dimensional GM M framew ork, w e obtain a GM M function that admits only M -subspaces of dimension strictly smaller than those allow ed for M M functions. The follo wing prop osition formalizes this construction. Prop osition 3. L et h b e an M M function on V (2) n having a unique M -subsp ac e W of dimension n/ 2 , such that if U is a nontrivial M -subsp ac e of h , then U ⊆ W . L et f W b e a c omplementary subsp ac e of V (2) n satisfying V (2) n = W ⊕ f W (i.e., W ∩ f W = { 0 } and V (2) n = W + f W ), and let L b e a line ar p ermutation of V (2) n with L ( f W ) = W . F or e ach z ∈ F 2 k , define f ( z ) ( x ) =    h ( x ) , if z ∈ V , ( h ◦ L )( x ) , if z ∈ V , (14) wher e V =  z ∈ F 2 k : T r k 1 ( z ) = 0  . With this choic e of f ( z ) in (14) , if n > 2 k + 4 , then the function f : V (2) n × F 2 k × F 2 k → F 2 , define d by f ( x, y , z ) = f ( z ) ( x ) + T r k 1  y z  , is not (e quivalent to) an M M function. Equivalently, if U is an M -subsp ac e of f , then dim( U ) < n/ 2 + k . Pr o of. W e recall that, b y Remark 2, the subspace L − 1 ( W ) = f W is an M -subspace of h ◦ L of dimension n/ 2. Moreov er, by assumption, f W is the unique M -subspace with the prop ert y that every nontrivial M -subspace of h ◦ L is contained in f W . Equiv alen tly , for 10 distinct nonzero a, b ∈ V (2) n , the equalit y D a D b h ( L ( x )) = 0 holds for all x ∈ V (2) n if and only if a, b ∈ f W . Let U b e an M -subspace of f . W e deriv e an upp er b ound on the dimension dim( U ) of U in tw o steps. Step 1. Define U V =  ( u, v , w ) ∈ U : T r k 1 ( w ) = 0  . (15) Let ν 1 = ( u 1 , v 1 , w 1 ) and ν 2 = ( u 2 , v 2 , w 2 ) b e distinct nonzero elements of U V . Since V is a subspace, w 1 + w 2 ∈ V . Assume first that T r k 1 ( w 1 v 2 + w 2 v 1 ) = 0. Then for any z ∈ V , Equation (12) yields D ν 1 D ν 2 f ( x, y , z ) = h ( x ) + h ( x + u 1 ) + h ( x + u 2 ) + h ( x + u 1 + u 2 ) = D u 1 D u 2 h ( x ) . Hence, D ν 1 D ν 2 f ( x, y , z ) = 0 for all x ∈ V (2) n if and only if either u 1 , u 2 are distinct nonzero elemen ts of the M -subspace of h , or ( u 1 , u 2 ) ∈ { (0 , u ) , ( u, 0) , ( u, u ) : u ∈ V (2) n } . Similarly , for z / ∈ V , D ν 1 D ν 2 f ( x, y , z ) = D u 1 D u 2 h ◦ L ( x ), and v anishing o ccurs if and only if either u 1 , u 2 b elong to the M -subspace of h ◦ L , or ( u 1 , u 2 ) ∈ { (0 , u ) , ( u, 0) , ( u, u ) : u ∈ V (2) n } . Since W ∩ f W = { 0 } , w e must ha ve ( u 1 , u 2 ) ∈ { (0 , u ) , ( u, 0) , ( u, u ) : u ∈ V (2) n } . W e define S ⊆ U V to be a subset suc h that, for any ( u i , v i , w i ) , ( u j , v j , w j ) ∈ S , one has T r k 1 ( w i v j + w j v i ) = 0. In other words, S =  ν = ( u, v , w ) ∈ U V : T r k 1 ( w v i + w i v ) = 0 for all ( u i , v i , w i ) ∈ S  . Cho ose S of maximal dimension. If S con tains tw o elements with nonzero first comp o- nen ts, then these comp onents must b e the same. Hence, for a fixed nonzero u ∈ V (2) n , S is con tained in { 0 , u } ×  ( v j , w j ) ∈ F 2 2 k : T r k 1 ( w j ) = 0 and T r k 1 ( w j v i + w i v j ) = 0 for all i  , whic h is an M -subspace of f . Case (i): If S = { (0 , 0 , 0) } , then w e claim that dim( U V ) ≤ 2. Suppose, for the sake of contradiction, that U V con tains three linearly independent v ectors ν i = ( u i , v i , w i ), i = 1 , 2 , 3. Since S = { (0 , 0 , 0) } , w e ha ve T r k 1  w i v j + w j v i  = 1 for all i  = j . By linear indep endence, the v ectors ( u 1 + u 2 , v 1 + v 2 , w 1 + w 2 ) and ( u 3 , v 3 , w 3 ) are distinct nonzero elements of U V . Moreo v er, we hav e T r k 1  ( w 1 + w 2 ) v 3 + w 3 ( v 1 + v 2 )  = 0. Hence, ( u 1 + u 2 , v 1 + v 2 , w 1 + w 2 ) and ( u 3 , v 3 , w 3 ) b oth b elong to S , con tradicting the assumption that S = { (0 , 0 , 0) } . Case (ii): Supp ose that S  = { (0 , 0 , 0) } . If S = U V , then dim( S ) = dim( U V ) ≤ 2 k . In the case S ⊆ U V , choose ν 1 = ( u 1 , v 1 , w 1 ) ∈ U V \ S . By maximality , there exists ν 2 = ( u 2 , v 2 , w 2 ) ∈ S with T r k 1 ( w 1 v 2 + w 2 v 1 ) = 1, whic h forces u 2 = u and u 1 / ∈ { 0 , u } . Otherwise, h ( x ) + h ( x + u 1 ) + h ( x + u 2 ) + h  x + u 1 + u 2  = 0 , 11 whic h w ould imply that, for an y z ∈ V , D ν 1 D ν 2 f ( x, y , z ) = 1. Con v ersely , for an y ν i = ( u i , v i , w i ) ∈ S with u i = u , w e ha ve T r k 1  w 1 v i + w i v 1  = 1. Indeed, if this w ere not the case, then { u, u 1 } ⊆ W ∩ f W , whic h con tradicts our assumption that W ∩ f W = { 0 } . W e claim that any M -subspace of f con tains at most tw o vectors whose last comp o- nen ts hav e trace zero and whose first comp onents are linearly indep enden t. This implies that U V ⊆ ⟨ u 1 , u 2 ⟩ × n ( v , w ) ∈ F 2 k × F 2 k : T r k 1 ( w ) = 0 o , whic h is a subspace of dimension 2 k + 1. Supp ose, for the sake of contradiction, that there exist three v ectors ν i = ( u i , v i , w i ) ∈ U V , i = 1 , 2 , 3 , with linearly indep enden t first comp onen ts satisfying T r k 1  w i v j + w j v i  = 1 for all i  = j . Since { u 1 , u 2 , u 3 } is linearly indep enden t, the v ectors ( u 1 + u 2 , v 1 + v 2 , w 1 + w 2 ) and ( u 3 , v 3 , w 3 ) are distinct nonzero elemen ts of U V . Moreo v er, w e ha v e T r k 1  ( w 1 + w 2 ) v 3 + w 3 ( v 1 + v 2 )  = 0. By the argument ab o v e, this implies that ( u 1 + u 2 , u 3 ) ∈ { (0 , u ) , ( u, 0) , ( u, u ) : u ∈ V (2) n } . Since u 1  = u 2 and u 3  = 0, w e must hav e u 1 + u 2 = u 3 , whic h con tradicts the linear indep endence of { u 1 , u 2 , u 3 } . Th us, any M -subspace of f consisting of ( u, v , w ) with w ∈ V has dimension at most 2 k + 1. In particular, the dimension of U V is at most 2 k + 1. Step 2: W e now consider the set  ( u, v , w ) ∈ U : T r k 1 ( w ) = 1  . Equiv alently , for a fixed ( u 1 , v 1 , w 1 ) ∈ U with T r k 1 ( w 1 ) = 1, we can consider  ( u 1 , v 1 , w 1 ) , ( u 1 + u, v 1 + v , w 1 + w ) : ( u, v , w ) ∈ U , T r k 1 ( w ) = 1  , as both sets span the same subspace. Since T r k 1 ( w 1 ) = T r k 1 ( w ) = 1, w e ha v e T r k 1 ( w 1 + w ) = 0. In particular, ( u 1 + u, v 1 + v , w 1 + w ) ∈ U V , where U V is defined in Equation (15). Hence, U V and ( u 1 , v 1 , w 1 ) generate the whole space U , whic h implies that dim( U ) ≤ 2 k + 2. Then our assumption n > 2 k + 4 implies that dim( U ) < n 2 + k , whic h yields the desired conclusion. In Prop osition 3, in order to construct functions in the GM M class that do not b elong to the completed Maiorana–McF arland class, we start from Maiorana–McF arland functions. Sp ecifically , for V (2) n ∼ = F 2 m × F 2 m with n = 2 m , w e consider the function h : F 2 m × F 2 m → F 2 defined by h ( x 1 , x 2 ) = T r m 1  x 1 π ( x 2 )  , where π is a p erm utation of 12 F 2 m . F or a = ( a 1 , a 2 ) , b = ( b 1 , b 2 ) ∈ F 2 m × F 2 m , D a D b h ( x 1 , x 2 ) = T r m 1  x 1 π ( x 2 ) + ( x 1 + a 1 ) π ( x 2 + a 2 ) + ( x 1 + b 1 ) π ( x 2 + b 2 ) + ( x 1 + a 1 + b 1 ) π ( x 2 + a 2 + b 2 )  = T r m 1  x 1  π ( x 2 ) + π ( x 2 + a 2 ) + π ( x 2 + b 2 ) + π ( x 2 + a 2 + b 2 )   + T r m 1  a 1 π ( x 2 + a 2 ) + b 1 π ( x 2 + b 2 ) + ( a 1 + b 1 ) π ( x 2 + a 2 + b 2 )  . (16) Recall that W = F 2 m × { 0 } is an M -subspace of h of dimension m = n/ 2, called the canonical M -subspace. Our goal is to determine examples of π ensuring that h has this unique M -subspace W of dimension n/ 2 suc h that, if U is a nontrivial M -subspace of h , then necessarily U ⊆ W . Equiv alently , for distinct nonzero a, b ∈ F 2 m × F 2 m , D a D b h ( x 1 , x 2 ) = 0 for all ( x 1 , x 2 ) ∈ F 2 m × F 2 m ⇐ ⇒ a, b ∈ W . F rom Equation (16), w e see that D a D b h ( x 1 , x 2 ) = 0 for all ( x 1 , x 2 ) ∈ F 2 m × F 2 m if and only if, for every x 2 ∈ F 2 m , π ( x 2 ) + π ( x 2 + a 2 ) + π ( x 2 + b 2 ) + π ( x 2 + a 2 + b 2 ) = 0 , T r m 1  a 1 π ( x 2 + a 2 ) + b 1 π ( x 2 + b 2 ) + ( a 1 + b 1 ) π ( x 2 + a 2 + b 2 )  = 0 . Hence, w e obtain the follo wing result. Corollary 4. L et h : F 2 m × F 2 m → F 2 b e an M M function define d by h ( x 1 , x 2 ) = T r m 1  x 1 π ( x 2 )  , wher e π is a p ermutation of F 2 m . Then W = F 2 m × { 0 } is the unique M -subsp ac e of h of dimension m that c ontains al l nontrivial M -subsp ac es of h if and only if the p ermutation π satisfies the fol lowing pr op erty. Pr op erty (P). F or al l x ∈ F 2 m , π ( x ) + π ( x + a 2 ) + π ( x + b 2 ) + π ( x + a 2 + b 2 ) = 0 , (17) T r m 1  a 1 π ( x + a 2 ) + b 1 π ( x + b 2 ) + ( a 1 + b 1 ) π ( x + a 2 + b 2 )  = 0 , (18) if and only if one of the fol lowing holds: ( a 1 , a 2 ) = (0 , 0) , ( b 1 , b 2 ) = (0 , 0) , ( a 1 , a 2 ) = ( b 1 , b 2 ) , or a 2 = b 2 = 0 . Remark 4. A p erm utation π of F 2 m is said to satisfy pr op erty (P1) (see [10, 25]) if D a D b π  = 0 for all linearly indep enden t elements a, b ∈ F 2 m . Suc h p erm utations are used to construct M M functions f ( x, y ) = T r m 1  x π ( y )  + h ( y ) on F 2 m × F 2 m that admit no b en t 4-decomp osition (see Section 4 for the definition) or p ossess a unique m -dimensional M -subspace, namely the canonical one F 2 m × { 0 } . 13 Theorem 1. L et m, k b e p ositive inte gers, and let π b e a p ermutation of F 2 m satisfying Pr op erty (P) of Cor ol lary 4. F or e ach z ∈ F 2 k , define the Bo ole an function f ( z ) : F 2 m × F 2 m → F 2 by f ( z ) ( x 1 , x 2 ) =    T r m 1  x 1 π ( x 2 )  , if T r k 1 ( z ) = 0 , T r m 1  x 2 π ( x 1 )  , if T r k 1 ( z ) = 1 . If m > k + 2 , then the function f : F 2 m × F 2 m × F 2 k × F 2 k → F 2 , f ( x 1 , x 2 , y , z ) = f ( z ) ( x 1 , x 2 ) + T r k 1 ( y z ) , is a GM M function that do es not b elong to the M M # class. Pr o of. By Corollary 4, the function h ( x 1 , x 2 ) = T r m 1  x 1 π ( x 2 )  admits a unique M - subspace con taining all nontrivial M -subspaces of h , namely W = F 2 m × { 0 } . Set f W = { 0 } × F 2 m , and let L : F 2 m × F 2 m → F 2 m × F 2 m b e the linear permutation de- fined by L ( x 1 , x 2 ) = ( x 2 , x 1 ). Note that F 2 m × F 2 m = W ⊕ f W and L ( f W ) = W . Then the conclusion follo ws directly from Prop osition 3. W e begin with an auxiliary result required to exhibit a p erm utation π satisfying Prop ert y (P) of Corollary 4. Lemma 5. L et m ≥ 4 and let c, d b e nonzer o elements of F 2 m . Then T r m 1  d  1 x + 1 x + c  do es not vanish identic al ly on F 2 m \ { 0 , c } . Pr o of. W e sho w that T r m 1  d  1 x + 1 x + c  is not a constant function, i.e., there exist x 1 , x 2 ∈ F 2 m \ { 0 , c } such that T r m 1  d  1 x 1 + 1 x 1 + c  = 0 and T r m 1  d  1 x 2 + 1 x 2 + c  = 1. This will establish the claim. Observ e that T r m 1  d  1 x + 1 x + c  equals 0 (respectively 1) for some x ∈ F 2 m \ { 0 , c } if and only if the curve X η defined b y X η : Z 2 + Z = d  1 X + 1 X + c  + η has an affine F 2 m -rational point for some η ∈ F 2 m with T r m 1 ( η ) = 0 (resp ectiv ely T r m 1 ( η ) = 1). Let F η denote the function field of X η , namely F η = F 2 m ( x, z ) , z 2 + z = d  1 x + 1 x + c  + η . Then F η is an Artin–Schreier extension of F 2 m ( x ) of degree 2; see [29, Prop osition 3.7.8]. The only ramified places of F 2 m ( x ) in F η are ( x = 0) and ( x = c ) (i.e., the zero of x and x + c ), each with differen t exp onent 2. Consequently , F 2 m is the full constan t field of F η . By the Hurwitz genus form ula ([29, Theorem 3.4.13]), we hav e g ( F η ) = 1. Hence, by 14 the Hasse–W eil b ound ([29, Theorem 5.2.3]), the n um b er N ( F η ) of rational places of F η satisfies N ( F η ) ≥ 2 m + 1 − 2 ( m +2) / 2 . (19) Recall that an Artin–Schreier curve of the form Z 2 + Z = f ( X ) /g ( X ) has no affine singular p oin ts whenev er gcd( f ( X ) , g ( X )) = 1. In particular, the curve X η has no affine singularities. Since the highest-degree term in its defining equation is Z 2 X 2 , there are exactly tw o points at infinity , namely (0 : 1 : 0) and (1 : 0 : 0). These corresp ond to places lying ab ov e ( x = 0), ( x = c ), and ( x = ∞ ), which is the p ole of x . Because the places ( x = 0) and ( x = c ) are ramified in F η , there is a unique rational place lying ab o v e each of them. Moreov er, dep ending on whether T r m 1 ( η ) = 0 or T r m 1 ( η ) = 1, the place ( x = ∞ ) either splits into tw o rational places or remains a single place of degree 2. Hence, there are at most four rational places corresp onding to p oin ts at infinit y . Since X η has no affine singular p oints, eac h affine F 2 m -rational p oin t corresp onds to a unique rational place. Therefore, b y (19), the n um b er N of affine rational p oints of X η satisfies N ≥ 2 m − 2 ( m +2) / 2 − 3 . In particular, for m ≥ 4, there exist affine F 2 m -rational p oints on X η for b oth T r m 1 ( η ) = 0 and T r m 1 ( η ) = 1, which yields the desired conclusion. Lemma 6. L et m ≥ 4 , and let π b e the p ermutation of F 2 m define d by π ( x ) = x 2 m − 2 . Then π satisfies Pr op erty (P) of Cor ol lary 4. That is, π ( x ) satisfies Equations (17) and (18) for al l x ∈ F 2 m if and only if one of the fol lowing c onditions holds: ( a 1 , a 2 ) = (0 , 0) , ( b 1 , b 2 ) = (0 , 0) , ( a 1 , a 2 ) = ( b 1 , b 2 ) , or a 2 = b 2 = 0 . Pr o of. The sufficiency of the conditions ( a 1 , a 2 ) = (0 , 0), ( b 1 , b 2 ) = (0 , 0), ( a 1 , a 2 ) = ( b 1 , b 2 ), or a 2 = b 2 = 0 is straigh tforw ard. In eac h case, π ( x ) satisfies Equations (17) and (18) for all x ∈ F 2 m . Therefore, it remains to establish necessit y . F or an y nonzero x ∈ F 2 m , w e can write π ( x ) = 1 /x . Hence, for x ∈ F 2 m \ { 0 , a 2 , b 2 , a 2 + b 2 } , Equation (17) can therefore b e written as π ( x ) + π ( x + a 2 ) + π ( x + b 2 ) + π ( x + a 2 + b 2 ) = 1 x + 1 x + a 2 + 1 x + b 2 + 1 x + a 2 + b 2 = a 2 b 2 ( a 2 + b 2 ) x ( x + a 2 )( x + b 2 )( x + a 2 + b 2 ) . Consequen tly , if Equation (17) holds for all x ∈ F 2 m , then necessarily a 2 = 0 or b 2 = 0, or a 2 = b 2 . 15 W e next consider the case a 2 = b 2 = c for some nonzero c ∈ F 2 m . F or x ∈ F 2 m \ { 0 , c } , Equation (18) b ecomes T r m 1  a 1 π ( x + a 2 ) + b 1 π ( x + b 2 ) + ( a 1 + b 1 ) π ( x + a 2 + b 2 )  = T r m 1  ( a 1 + b 1 )  π ( x ) + π ( x + c )   = T r m 1  ( a 1 + b 1 )  1 x + 1 x + c  . (20) By Lemma 5, the ab ov e trace v anishes on F 2 m \{ 0 , c } only if a 1 = b 1 , i.e., ( a 1 , a 2 ) = ( b 1 , b 2 ). Finally , supp ose that exactly one of a 2 or b 2 is zero. A computation analogous to (20) sho ws that (18) cannot hold for all x ∈ F 2 m unless, resp ectiv ely , a 1 = 0 or b 1 = 0. This completes the pro of. As a direct consequence of Theorem 1 and Lemma 6, we obtain a class of functions that b elong to the GM M class but not to the M M # class. Corollary 7. L et m, k b e p ositive inte gers with m ≥ 4 and m > k + 2 . F or z ∈ F 2 k , define the Bo ole an function f ( z ) : F 2 m × F 2 m → F 2 by f ( z ) ( x 1 , x 2 ) =    T r m 1  x 1 x 2 m − 2 2  , if T r k 1 ( z ) = 0 , T r m 1  x 2 x 2 m − 2 1  , if T r k 1 ( z ) = 1 . Then the function f : F 2 m × F 2 m × F 2 k × F 2 k → F 2 , f ( x 1 , x 2 , y , z ) = f ( z ) ( x 1 , x 2 ) + T r k 1 ( y z ) , is a GM M function which is not in the M M # class. Similarly , w e consider the Gold functions π ( x ) = x 2 k +1 in tro duced in [15] o ver the finite field F 2 m . It is w ell known that π is a p ermutation of F 2 m if and only if gcd(2 k +1 , 2 m − 1) = 1. Using the iden tity gcd(2 k + 1 , 2 k − 1) = 1, we obtain gcd(2 k + 1 , 2 m − 1) = gcd(2 2 k − 1 , 2 m − 1) gcd(2 k − 1 , 2 m − 1) = 2 gcd(2 k,m ) − 1 2 gcd( k,m ) − 1 . Hence, gcd(2 k + 1 , 2 m − 1) = 1 if and only if gcd( k , m ) = 1 and m is o dd. Lemma 8. L et m and k b e p ositive inte gers such that gcd( k , m ) = 1 and m is o dd. Then π ( x ) = x 2 k +1 satisfies Pr op erty (P) of Cor ol lary 4. That is, π ( x ) satisfies Equa- tions (17) and (18) for al l x ∈ F 2 m if and only if one of the fol lowing c onditions holds: ( a 1 , a 2 ) = (0 , 0) , ( b 1 , b 2 ) = (0 , 0) , ( a 1 , a 2 ) = ( b 1 , b 2 ) , or a 2 = b 2 = 0 . 16 Pr o of. Since the sufficiency of the stated conditions is immediate, w e only pro v e necessit y . A direct computation shows that π ( x ) + π ( x + a 2 ) + π ( x + b 2 ) + π ( x + a 2 + b 2 ) = a 2 b 2  a 2 k − 1 2 + b 2 k − 1 2  . Since gcd(2 k − 1 , 2 m − 1) = 1, Equation (17) holds for all x ∈ F 2 m if and only if a 2 = 0 or b 2 = 0, or a 2 = b 2 . Next, a straigh tforw ard computation yields T r m 1  a 1 π ( x + a 2 ) + b 1 π ( x + b 2 ) + ( a 1 + b 1 ) π ( x + a 2 + b 2 )  = T r m 1  a 1 a 2 k +1 2 + b 1 b 2 k +1 2 + ( a 1 + b 1 )( a 2 + b 2 ) 2 k +1 + ( a 1 b 2 + b 1 a 2 ) x 2 k + ( a 1 b 2 k 2 + b 1 a 2 k 2 ) x  = T r m 1  a 1 a 2 k +1 2 + b 1 b 2 k +1 2 + ( a 1 + b 1 )( a 2 + b 2 ) 2 k +1 +  ( a 1 b 2 + b 1 a 2 ) + ( a 1 b 2 k 2 + b 1 a 2 k 2 ) 2 k  x 2 k  . (21) Therefore, Equation (18) holds for all x ∈ F 2 m only if the co efficien t of x 2 k v anishes, namely , ( a 1 b 2 + b 1 a 2 ) + ( a 1 b 2 k 2 + b 1 a 2 k 2 ) 2 k = 0 . (22) W e now analyze the p ossible cases. Case (i): a 2 = 0. If b 2 = 0 or a 1 = 0, then Equation (21) v anishes trivially . Assume b 2 a 1  = 0. Then (22 ) reduces to a 1 b 2 + a 2 k 1 b 2 2 k 2 = 0, which is equiv alen t to a 2 k − 1 1 = b − (2 2 k − 1) 2 . Since gcd(2 k − 1 , 2 m − 1) = 1, this holds if and only if a 1 = b − (2 k +1) 2 . Since m is o dd, substituting in to (21), w e obtain T r m 1  a 1 b 2 k +1 2  = T r m 1 (1) = 1 , a con tradiction. Hence, either a 2 = b 2 = 0 or ( a 1 , a 2 ) = (0 , 0). Case (ii): b 2 = 0. This case is symmetric to Case (i) and implies either a 2 = b 2 = 0 or ( b 1 , b 2 ) = (0 , 0). Case (iii): a 2 = b 2 = c for some nonzero c ∈ F 2 m . Assume that a 1  = b 1 . Then (22) b ecomes c ( a 1 + b 1 ) + c 2 2 k ( a 1 + b 1 ) 2 k = 0, which is equiv alen t to  c 2 k +1 ( a 1 + b 1 )  2 k − 1 = 1. Since gcd(2 k − 1 , 2 m − 1) = 1, this holds if and only if c 2 k +1 ( a 1 + b 1 ) = 1. Substituting in to (21), w e obtain T r m 1  c 2 k +1 ( a 1 + b 1 )  = T r m 1 (1) = 1 , a con tradiction. Therefore, a 1 = b 1 , and hence ( a 1 , a 2 ) = ( b 1 , b 2 ). Com bining all cases completes the pro of. Similarly , Theorem 1 together with Lemma 8 yields another class of functions that b elong to the GM M class but not to the completed M M class. 17 Corollary 9. L et m and k b e p ositive inte gers such that gcd( k, m ) = 1 and m is o dd with m > k + 2 . F or z ∈ F 2 k , define the Bo ole an function f ( z ) : F 2 m × F 2 m → F 2 by f ( z ) ( x 1 , x 2 ) =    T r m 1  x 1 x 2 k +1 2  , if T r k 1 ( z ) = 0 , T r m 1  x 2 x 2 k +1 1  , if T r k 1 ( z ) = 1 . Then the function f : F 2 m × F 2 m × F 2 k × F 2 k → F 2 , f ( x 1 , x 2 , y , z ) = f ( z ) ( x 1 , x 2 ) + T r k 1 ( y z ) , is a GM M function that is not in the M M # class. Remark 5. W e recall that an y Boolean b en t function on V (2) n b elonging to the completed P S ap class has algebraic degree n/ 2. Consequently , the b en t functions in Corollaries 7 and 9 lie neither in the M M # class nor in the P S # ap class. 4 Decomp osition of the generalized P S ap functions Let S ⊆ V (2) n b e a subspace of dimension n − 2, and let W ⊆ V (2) n b e a complemen tary subspace suc h that V (2) n = S ⊕ W . Let f : V (2) n → F 2 b e a Bo olean function. F or eac h w i ∈ W , define f i ( x ) = f ( x + w i ) for x ∈ S . Equiv alently , f i is the restriction of f to the coset w i + S , viewed as a Bo olean function on S . The 4-decomp osition of f with resp ect to S is then defined as the sequence ( f 1 , f 2 , f 3 , f 4 ), where { w 1 , w 2 , w 3 , w 4 } = W . Throughout the pap er, we simply refer to this as a decom- p osition (with respect to S ). If all the functions f i are b en t (resp ectiv ely , semib ent), then f is said to admit a b ent (resp ectiv ely , semib ent) decomp osition with resp ect to S . W e remark that, in the decomp osition of a function f , the functions f i are pairwise EA-equiv alent and hence hav e the same extended W alsh sp ectrum. In particular, they are b ent or semib en t simultaneously . Therefore, it suffices to consider the restriction of f to the subspace S . Moreo v er, since S has co dimension 2, there exist linearly indep endent v ectors u, v ∈ V (2) n suc h that S = ⟨ u, v ⟩ ⊥ . In other w ords, S = { x ∈ V (2) n : ⟨ u, x ⟩ n = ⟨ v , x ⟩ n = 0 } . An equiv alen t criterion for a b en t function f to admit a b ent (resp ectiv ely , semibent) decomp osition, expressed in terms of the second-order deriv ativ e of its dual function, is giv en in [10, Theorem 7] as follo ws. Lemma 10. L et f : V (2) n → F 2 b e a b ent function with dual f ∗ . L et u, v ∈ V (2) n b e line arly indep endent, and let S = ⟨ u, v ⟩ ⊥ b e the ortho gonal c omplement of ⟨ u, v ⟩ . Then the fol lowing hold: (i) f admits a b ent de c omp osition with r esp e ct to S if and only if D u D v f ∗ = 1 . (ii) f admits a semib ent de c omp osition with r esp e ct to S if and only if D u D v f ∗ = 0 . 18 F rom Lemma 10, it follows that f admits a semib ent decomp osition if and only if its dual f ∗ p ossesses a non trivial M -subspace; equiv alen tly , its linearity index satisfies ind( f ∗ ) > 1. In particular, any M M or GM M b ent function admits a semib en t decom- p osition. In contrast to M M and GM M b en t functions, it app ears that the ma jority of the P S ap b en t functions admit neither a b en t nor a semib ent decomp osition; for details, w e refer to [2]. In this section, w e inv estigate the decomp osition of b ent functions in the generalized P S ap class. As in [2], our approach relies on curv es defined o ver finite fields and on estimates for their num ber of affine rational p oin ts. How ever, due to the definition of generalized P S ap functions, the analysis requires more inv olv ed tec hnical computations. Let m , k , and e b e integers suc h that k | m , e ≡ 2 ℓ mo d (2 k − 1), and gcd(2 m − 1 , e ) = 1. Let η denote the multiplicativ e in v erse of e mo dulo (2 m − 1), that is, η e ≡ 1 mo d (2 m − 1). In particular, we consider the generalized P S ap b en t functions defined by g ( x, y ) = P  T r m k  x y − η  , (23) where P : F 2 k → F 2 is a balanced Bo olean function. T o apply Lemma 10, we first determine the dual of the function g defined in (23). Lemma 11. L et g : F 2 m × F 2 m → F 2 b e the gener alize d P S ap b ent function define d in (23) . Then the dual b ent function g ∗ is given by g ∗ ( x, y ) = P  T r m k  y x − e  2 m − ℓ  = P  T r m k  ˜ y ˜ x − e  , wher e ˜ y = y 2 m − ℓ and ˜ x = x 2 m − ℓ . Pr o of. F or ( u, v ) ∈  F 2 m × F 2 m  \ { (0 , 0) } , let χ u,v denote the c haracter of F 2 m × F 2 m defined by χ u,v ( x, y ) = ( − 1) T r m 1 ( ux + v y ) . Let Γ 2 = { V , B ( γ ) : γ ∈ F 2 k } b e the generalized Desarguesian spread defined in (9), where V and B ( γ ) are as in (7) and (8), resp ectively . W e note that g ( x, y ) = P ( γ ) for all ( x, y ) ∈ B ( γ ) and g ( x, y ) = P (0) for all ( x, y ) ∈ V , i.e., g is a b en t function arising from the bent partition Γ 2 . By [5, Prop osition 10], for γ ∈ F 2 k w e ha ve χ u,v  B ( γ )  = X ( x,y ) ∈ B ( γ ) ( − 1) T r m 1 ( ux + v y ) = ( 2 m − 2 m − k , if u  = 0 and γ 2 ℓ = T r m k ( v u − e ) , − 2 m − k , otherwise , and χ u,v ( V ) = X ( x,y ) ∈ V ( − 1) T r m 1 ( ux + v y ) = ( 0 , if u  = 0 , 2 m , otherwise . 19 W e now compute the W alsh transform of g : W g ( u, v ) = X ( x,y ) ∈ F 2 m × F 2 m ( − 1) g ( x,y )+T r m 1 ( ux + v y ) = X ( x,y ) ∈ F 2 m × F 2 m ( − 1) P ( T r m k ( x y − η )) +T r m 1 ( ux + v y ) = X γ ∈ F 2 k X ( x,y ) ∈ B ( γ ) ( − 1) P ( γ )+T r m 1 ( ux + v y ) + X ( x,y ) ∈ V ( − 1) P (0)+T r m 1 ( ux + v y ) = X γ ∈ F 2 k ( − 1) P ( γ ) χ u,v  B ( γ )  + ( − 1) P (0) χ u,v ( V ) . Using the c haracter v alues abov e together with the balancedness of P : F 2 k → F 2 , we obtain W g ( u, v ) = ( ( − 1) P (0) 2 m , if u = 0 , ( − 1) P ( γ ) 2 m , if u  = 0, where γ 2 ℓ = T r m k ( v u − e ) . This yields the claimed expression for the dual function, together with the identit y γ = γ 2 m = T r m k ( v u − e ) 2 m − ℓ = T r m k  v 2 m − ℓ u − e 2 m − ℓ  . Note that the trace map T r m k : F 2 m → F 2 k and the function P : F 2 k → F 2 are b oth balanced. Consequen tly , the comp osition P (T r m k ( z )) defines a balanced Bo olean function from F 2 m to F 2 . Hence, there exists a p erm utation Q of F 2 m suc h that P (T r m k ( z )) = T r m 1 ( Q ( z )). Therefore, in the remainder of this section, we consider the generalized P S ap b en t functions from F 2 m × F 2 m to F 2 of the form f ( x, y ) = T r m 1  Q  x y − η  . (24) W e remark that, in this case, the dual function of f is giv en by f ∗ ( x, y ) = T r m 1  Q  ˜ y ˜ x − e  , (25) where ˜ y = y 2 m − ℓ and ˜ x = x 2 m − ℓ . Moreo ver, b y Lemma 2, w e can without loss of generalit y supp ose that Q (0) = 0. Prop osition 12. L et m > 4 , and let Q b e a p ermutation of F 2 m of o dd p olynomial de gr e e less than or e qual to 2 m/ 4 − 1 3 e . L et u = ( a, b ) and v = ( c, d ) b e two line arly indep endent ve ctors in F 2 m × F 2 m , and set S = ⟨ u, v ⟩ ⊥ . L et f : F 2 m × F 2 m → F 2 b e the gener alize d P S ap b ent function define d in (24) . If ad + bc  = 0 , then f admits neither a b ent nor a semib ent de c omp osition on S . Remark 6. Let f : F 2 m × F 2 m → F 2 b e the function defined in (24), and supp ose that it satisfies the assumptions of Prop osition 12. Then, equiv alen tly , we hav e: (i) The dual function f ∗ of f do es not admit any tw o-dimensional M -subspace ⟨ ( a, b ) , ( c, d ) ⟩ with ad + bc  = 0. 20 (ii) If f admits a b ent or semib ent decomp osition on ⟨ ( a, b ) , ( c, d ) ⟩ ⊥ , then necessarily ad + bc = 0. Equiv alen tly , ( c, d ) = λ ( a, b ) for some λ ∈ F 2 m \ F 2 . Before pro ving Prop osition 12, we require some preliminary results. The first lemma is analogous to the one app earing in [2]; how ever, we include its pro of for completeness. Lemma 13. L et S b e the ( n − 2) -dimensional subsp ac e of F 2 m × F 2 m define d by T r m 1 ( ax + by ) = 0 and T r m 1 ( cx + dy ) = 0 . Assume that ad + bc  = 0 . Then the function f ( x, y ) = T r m 1  Q ( xy − η )  admits a b ent de- c omp osition (r esp e ctively, a semib ent de c omp osition) on S if and only if D (1 , 0) D (0 , 1) b f ( x, y ) is identic al ly e qual to 1 (r esp e ctively, identic al ly e qual to 0 ) for al l ( x, y ) ∈ F 2 m × F 2 m , wher e b f ( x, y ) = T r m 1  Q   b 2 m − ℓ x + d 2 m − ℓ y  a 2 m − ℓ x + c 2 m − ℓ y  − e  . Pr o of. By Lemma 10, the function f admits a b en t decomp osition (resp ectiv ely , a semib en t decomp osition) on S if and only if the second-order deriv ative D u D v f ∗ ( x ) of the dual function f ∗ ( x, y ) given in (25) is constantly equal to 1 (resp ectively , constantly equal to 0), where u = ( a, b ) and v = ( c, d ). By Lemma 2, this condition is equiv alen t to requiring that D (1 , 0) D (0 , 1) f ∗  L ( x, y )  is constan tly equal to 1 (resp ectiv ely , 0), where L is a linear p ermutation of F 2 m × F 2 m satisfying L (1 , 0) = ( a, b ) and L (0 , 1) = ( c, d ). Define the linear map L by L ( x, y ) = ( ax 2 ℓ + cy 2 ℓ , bx 2 ℓ + dy 2 ℓ ). The map L p ermutes F 2 m × F 2 m if and only if ad + bc  = 0. Under this assumption, w e indeed ha ve L (1 , 0) = ( a, b ) and L (0 , 1) = ( c, d ). Moreov er, f ∗  L ( x, y )  = T r m 1  Q  ( bx 2 ℓ + dy 2 ℓ ) 2 m − ℓ ( ax 2 ℓ + cy 2 ℓ ) − 2 m − ℓ e  = T r m 1  Q   b 2 m − ℓ x + d 2 m − ℓ y  a 2 m − ℓ x + c 2 m − ℓ y  − e  = b f ( x, y ) . This yields the desired conclusion. By Lemma 13, in order to pro ve Proposition 12, it suffices to show that, for u = (1 , 0) and v = (0 , 1), the second-order deriv ative D u D v b f ( x, y ) is nonconstant. Since a 2 m − ℓ d 2 m − ℓ + b 2 m − ℓ c 2 m − ℓ = ( ad + bc ) 2 m − ℓ , the condition ad + bc  = 0 holds if and only if a 2 m − ℓ d 2 m − ℓ + b 2 m − ℓ c 2 m − ℓ  = 0. Hence, without loss of generalit y , w e ma y assume ad + bc  = 0 and consider e f ( x, y ) = T r m 1  Q  ( bx + dy )( ax + cy ) − e  . 21 More precisely , we compute D u D v e f ( x, y ) = e f ( x + 1 , y + 1) + e f ( x + 1 , y ) + e f ( x, y + 1) + e f ( x, y ) (26) = T r m 1  Q  ( b ( x + 1) + d ( y + 1))( a ( x + 1) + c ( y + 1)) − e  + Q  ( b ( x + 1) + dy )( a ( x + 1) + cy ) − e  + Q  ( bx + d ( y + 1))( ax + c ( y + 1)) − e  + Q  ( bx + dy )( ax + cy ) − e   . The follo wing lemma is frequen tly used to c haracterize the circumstances under whic h the second-order deriv ativ e of a function cannot b e constant. Lemma 14. F or a p ositive inte ger ℓ ≥ 1 , let X ϑ b e the curve over F 2 m define d by Z 2 + Z = P  α 1 X + β 1 κ 1 X + υ 1  + · · · + P  α ℓ X + β ℓ κ ℓ X + υ ℓ  + ϑ, (27) wher e P is a p olynomial of o dd de gr e e t . Assume that α i υ i + β i κ i  = 0 for i = 1 , . . . , ℓ, and ∃ j s.t. υ j κ j  = υ i κ i for al l i  = j. Then the numb er N ϑ of affine F 2 m -r ational p oints ( x, z ) ∈ X ϑ with x  = υ i /κ i , for i = 1 , . . . , ℓ , of X ϑ satisfies N ϑ ≥ 2 m − 1 − ( ℓt + 1) ℓt 2 m 2 − ℓ ( ℓt + 2) . In p articular, if m ≥ 4 and ℓt ≤ 2 m/ 4 − 1 , then N ϑ > 0 . Pr o of. Let F ϑ = F 2 m ( x, z ) b e the function field of X ϑ . W e may regard F ϑ as an extension of the rational function field F 2 m ( x ) defined b y the equation z 2 + z = g ϑ ( x ), where g ϑ ( x ) = P  α 1 x + β 1 κ 1 x + υ 1  + · · · + P  α ℓ x + β ℓ κ ℓ x + υ ℓ  + ϑ. Note that F ϑ / F 2 m ( x ) is an Artin–Sc hreier extension of degree 2; see [29, Proposition 3.7.8]. A place R of F 2 m ( x ) ramifies in F ϑ only if the v aluation v R ( g ϑ ( x )) is negative. This can o ccur only when R = ( x = υ i /κ i ) for i = 1 , . . . , ℓ . Moreov er, by the strict triangle inequalit y (see [29, Lemma 1.1.11]), for the place R j corresp onding to x = υ j /κ j , where υ j κ j  = υ i κ i for all i  = j , we obtain v R j  g ϑ ( x )  = − deg( P ) = − t. Since t is o dd, the place R j is totally ramified in the extension F ϑ / F 2 m ( x ). Consequen tly , F ϑ is a function field with full constant field F 2 m . It follo ws that X ϑ is an absolutely irreducible curve defined ov er F 2 m ; see [29, Corollary 3.6.8]. F urthermore, X ϑ has degree 22 deg( X ϑ ) ≤ ℓt + 2. Applying the Hasse-W eil b ound (see [16, Theorem 9.57]), w e obtain the follo wing estimate for the n um b er N ( X ϑ ) of F 2 m -rational p oin ts of X ϑ in the pro jectiv e plane: N ( X ϑ ) ≥ 2 m + 1 − (deg ( X ϑ ) − 1)(deg ( X ϑ ) − 2) 2 m 2 ≥ 2 m + 1 − ( ℓt + 1) ℓt 2 m 2 . (28) As the highest-degree term in the defining equation of X ϑ is X s Z 2 with s ≤ ℓt , the curv e X ϑ has at most tw o F 2 m -rational points at infinity , namely (0 : 1 : 0) and (1 : 0 : 0). In order to ensure that the denominators in (27) do not v anish, w e m ust also exclude all p oin ts lying on the ℓ lines defined by κ i X + υ i = 0. By B ´ ezout’s theorem, a line intersects X ϑ in at most deg( X ϑ ) p oints. Hence, b y subtracting 2 + ℓ deg ( X ϑ ) from (28), w e obtain the desired b ound on the n um b er N ϑ of affine F 2 m -rational points. Moreo v er, whenever m ≥ 4 and ℓt ≤ 2 m/ 4 − 1, w e obtain ( ℓt + 1) ℓt 2 m 2 + ℓ ( ℓt + 2) + 1 < 2 m , and hence N ϑ > 0. Pr o of of Pr op osition 12. The pro of is carried out b y a case-b y-case analysis. Case (i): b = d . Note that the condition b = d implies b  = 0 and a  = c , since ad + bc  = 0. In this case, w e set y = x . Then, by (26), for x  = a a + c , c a + c , w e obtain D u D v e f ( x, x ) = T r m 1  Q  b (( a + c ) x + a ) − e  + Q  d (( a + c ) x + c ) − e  = T r m 1  Q  b (( a + c ) x + a ) e  + Q  b (( a + c ) x + c ) e  . Since gcd( e, 2 m − 1) = 1, the map x 7→ x e p erm utes F 2 m . Th us, there exists ˜ b ∈ F 2 m suc h that b = ˜ b e . Setting P ( X ) = Q ( X e ), w e see that there exists x ∈ F 2 m \  a a + c , c a + c  suc h that D u D v ˜ f ( x, x ) = 0, resp ectively D u D v ˜ f ( x, x ) = 1, if and only if the curve Z 2 + Z = P ˜ b ( a + c ) X + a ! + P ˜ b ( a + c ) X + c ! + ϑ has an affine rational p oin t ( x, z ) with x / ∈  a a + c , c a + c  , where ϑ ∈ F 2 m satisfies T r m 1 ( ϑ ) = 0, resp ectiv ely T r m 1 ( ϑ ) = 1. By setting in (27) ℓ = 2, ( α 1 , β 1 , κ 1 , υ 1 ) = (0 , ˜ b, a + c, a ) , and ( α 2 , β 2 , κ 2 , υ 2 ) = (0 , ˜ b, a + c, c ) , this condition is equiv alen t to the curv e X ϑ ha ving an affine F 2 m -rational p oin t ( x, z ) with x / ∈  a a + c , c a + c  . Note that α 1 υ 1 + β 1 κ 1  = 0 and α 2 υ 2 + β 2 κ 2  = 0 since b  = 0 and 23 a  = c . Moreo ver, υ 1 κ 1  = υ 2 κ 2 , as a  = c . Therefore, the existence of suc h a p oin t follows from Lemma 14. Case (ii): bd  = 0 and b  = d . Set y = b d x and δ = a + cb d . By the assumption ad + bc  = 0, we ha v e δ  = 0. Then, for x  = a + c δ , a δ , c δ , w e obtain D u D v e f  x, b d x  = T r m 1  Q  b + d ( δ x + ( a + c )) e  + Q  b ( δ x + a ) e  + Q  d ( δ x + c ) e  . (29) As the map x 7→ x e p erm utes F 2 m , there exist elements ˜ b, ˜ d, ] b + d ∈ F 2 m suc h that b = ˜ b e , d = ˜ d e , b + d =  ] b + d  e . Defining P ( X ) = Q ( X e ), Equation (29) can b e rewritten as D u D v e f  x, b d x  = T r m 1 P ] b + d δ x + ( a + c ) ! + P ˜ b δ x + a ! + P ˜ d δ x + c !! . Consequen tly , there exists x ∈ F 2 m \  a + c δ , a δ , c δ  suc h that D u D v ˜ f ( x, b d x ) = 0, resp ectively D u D v ˜ f ( x, b d x ) = 1, if and only if the curv e defined by Z 2 + Z = P ] b + d δ X + ( a + c ) ! + P ˜ b δ X + a ! + P ˜ d δ X + c ! + ϑ has an affine F 2 m -rational point ( x, z ) with x / ∈  a + c δ , a δ , c δ  , where ϑ ∈ F 2 m satisfies T r m 1 ( ϑ ) = 0, resp ectiv ely T r m 1 ( ϑ ) = 1. W e now verify that the assumptions of (14) are satisfied. T o this end, in (27) we set ℓ = 3 and ( α 1 , β 1 , κ 1 , υ 1 ) = (0 , ] b + d, δ, a + c ) , ( α 2 , β 2 , κ 2 , υ 2 ) = (0 , ˜ b, δ, a ) , ( α 3 , β 3 , κ 3 , υ 3 ) = (0 , ˜ d, δ, c ) . Note that for every i w e hav e α i υ i + β i κ i = β i δ , whic h is nonzero since δ  = 0, bd  = 0, and b  = d . Moreo v er, as the κ i are iden tical, the second assumption reduces to showing that at least one of the υ i differs from the other tw o. The equalities a + c = a = c w ould hold only if a = 0 and c = 0, whic h is imp ossible b ecause ad + bc  = 0. Therefore, b y (14), suc h a rational p oin t alw a ys exists. Case (iii): b = 0 or d = 0. Note that b and d cannot v anish sim ultaneously since ad + bc  = 0. As the case d = 0 is analogous to the case b = 0, w e may assume without loss of generality that b = 0. Then ad  = 0, and hence D u D v e f ( x, y ) = T r m 1  Q  d ( y + 1)( a ( x + 1) + c ( y + 1)) − e  + Q  dy ( a ( x + 1) + cy ) − e  + Q  d ( y + 1)( ax + c ( y + 1)) − e  + Q  dy ( ax + cy ) − e   . 24 W e now set y = 0. Then, for x  = c a , 1 + c a , w e obtain D u D v e f ( x, 0) = T r m 1  Q  d ( ax + ( a + c )) e  + Q  d ( ax + c ) e  . Consequen tly , there exists x ∈ F 2 m \ { c a , 1 + c a } suc h that D u D v ˜ f ( x, 0) = 0, resp ectively D u D v ˜ f ( x, 0) = 1, if and only if the curve defined b y Z 2 + Z = P ˜ d aX + ( a + c ) ! + P ˜ d aX + c ! + ϑ has an affine F 2 m -rational p oin t ( x, z ) with x / ∈ { c a , 1 + c a } , where P ( X ) = Q ( X e ), d = ˜ d e , and ϑ ∈ F 2 m satisfies T r m 1 ( ϑ ) = 0, resp ectiv ely T r m 1 ( ϑ ) = 1. By setting in (27) ℓ = 2, ( α 1 , β 1 , κ 1 , υ 1 ) = (0 , ˜ d, a, a + c ) , and ( α 2 , β 2 , κ 2 , υ 2 ) = (0 , ˜ d, a, c ) , this condition is equiv alen t to the curv e X ϑ ha ving an affine F 2 m -rational p oin t ( x, z ) with x / ∈ { c a , 1 + c a } . Note that α i υ i + β i κ i  = 0 for i = 1 , 2 since ad  = 0. More- o v er, υ 1 /κ 1  = υ 2 /κ 2 since a  = 0. Therefore, the existence of suc h a p oin t follows from Lemma 14. □ W e now give a complete analysis of the generalized P S ap b en t function in the case Q ( x ) = x . That is, we consider the b en t function f : F 2 m × F 2 m → F 2 , f ( x, y ) = T r m 1  x y − η  . Corollary 15. L et m , k , and e b e p ositive inte gers such that k | m, e ≡ 2 ℓ mo d (2 k − 1) , gcd(2 m − 1 , e ) = 1 . L et η denote the multiplic ative inverse of e mo dulo 2 m − 1 . If k ≤ m 4 − 3 , then the gener alize d P S ap b ent function f ( x, y ) = T r m 1 ( x y − η ) on F 2 m × F 2 m satisfies the fol lowing pr op erties: (i) The function f admits a semib ent de c omp osition on S = ⟨ u, v ⟩ ⊥ , for line arly inde- p endent u, v ∈ F 2 m × F 2 m , if and only if u, v ∈ { 0 } × F 2 m . (ii) The function f do es not admit any b ent de c omp osition. Pr o of. Let u = ( a, b ) and v = ( c, d ) b e tw o linearly indep enden t v ectors in F 2 m × F 2 m . By Prop osition 12, the function f do es not admit any semib en t or b en t decomp osition on S whenev er ad + bc  = 0. Therefore, we restrict our attention to the case ad + bc = 0. Recall that, since u = ( a, b ) and v = ( c, d ) are linearly indep enden t vectors in F 2 m × F 2 m , the condition ad + bc = 0 is equiv alen t to ( c, d ) = λ ( a, b ) for some λ ∈ F 2 m \ F 2 . 25 By Lemma 10, the function f admits a semib en t (resp ectiv ely , b en t) decomp osition on S if and only if its dual f ∗ ( x, y ) = T r m 1  y 2 ℓ x − 2 ℓ e  satisfies D u D v f ∗ = 0 (resp ectiv ely , D u D v f ∗ = 1). Note that for the linear map L ( x, y ) = ( x 2 ℓ , y 2 ℓ ) we ha v e f ∗ ( x, y ) = e f ( L ( x, y )), where e f ( x, y ) = T r m 1 ( y x − e ). Hence, b y Lemma 2, it suffices to consider the second-order deriv a- tiv e of e f . By replacing c = λa and d = λb , the second-order deriv ative D u D v e f ( x, y ) can b e expressed as D u D v e f ( x, y ) = T r m 1  ( y + b (1 + λ )) ( x + a (1 + λ )) − e + ( y + b ) ( x + a ) − e + ( y + λb ) ( x + λa ) − e + y x − e  . W e now pro ceed b y a case-b y-case analysis. Case (i): a = 0, and hence b  = 0. In this case, we obtain D u D v e f = 0 for all suc h c hoices of u and v . Therefore, e f admits a semib en t decomp osition on ⟨ u, v ⟩ ⊥ . In particular, for an y linearly indep enden t u, v ∈ { 0 } × F 2 m , the function e f admits a semib en t decomp osition on ⟨ u, v ⟩ ⊥ . Since L − 1 ( { 0 } × F 2 m ) = { 0 } × F 2 m , it follows from Lemma 2 that f ∗ ( x, y ) admits a semib en t decomp osition on ⟨ u, v ⟩ ⊥ for all linearly indep enden t u, v ∈ { 0 } × F 2 m . Case (ii): b = 0, and hence a  = 0. Then applying the changes of v ariables x 7→ ax and y 7→ a e y and then setting y = 1, w e obtain g ( x ) = T r m 1  ( x + 1 + λ ) − e + ( x + 1) − e + ( x + λ ) − e + x − e  . (30) Hence, if D u D v f ∗ ( x, y ) is a constant function, then the function g ( x ) in (30) must also be constan t, which is imp ossible b y Lemma 14. Then, by Lemma 2, the function f ∗ do es not admit neither a b en t or a semib en t decomp osition on ⟨ u, v ⟩ ⊥ for an y linearly indep enden t u, v ∈ F 2 m × { 0 } . Case (iii): ab  = 0. Pro ceeding as b efore, w e apply the change of v ariables x 7→ ax and y 7→ by , and then set α = ba − e and y = 0. This yields g ( x ) = T r m 1  α  (1 + λ ) ( x + 1 + λ ) − e + ( x + 1) − e + λ ( x + λ ) − e   . (31) As in the previous cases, if D u D v e f ( x, y ) w ere a constan t function, then g ( x ) in (31) w ould also hav e to b e constan t. This, how ever, is imp ossible by Lemma 14. Then we similarly 26 conclude that f ∗ admits neither a b ent nor a semib en t decomp osition on ⟨ u, v ⟩ ⊥ for an y suc h linearly indep enden t elemen ts u, v . This completes the pro of. Remark 7. Note that the generalized P S ap b en t function f in Corollary 15, and hence its dual f ∗ , is also an M M function. Consequently , f and f ∗ admit m -dimensional M - subspaces, namely the canonical ones F 2 m × { 0 } and { 0 } × F 2 m , resp ectiv ely . It follows from Corollary 15 that { 0 } × F 2 m is the unique m -dimensional M -subspace of f ∗ . W e can similarly generalize Corollary 15 to arbitrary p erm utation p olynomials Q ( x ) whose degree is sufficien tly small compared to the cardinalit y of the underlying finite field. In particular, combined with Prop osition 12, we obtain the following sufficien t conditions under whic h the generalized P S ap b en t function admits neither a semib en t nor a b en t decomp osition. The proof pro ceeds analogously to that of Corollary 15; therefore, w e only giv e a sk etc h. Theorem 2. L et m , k , and e b e p ositive inte gers such that k | m, e ≡ 2 ℓ mo d (2 k − 1) , gcd(2 m − 1 , e ) = 1 . L et η denote the multiplic ative inverse of e mo dulo 2 m − 1 . L et Q b e a p ermutation p olynomial of F 2 m of o dd de gr e e at most 2 m/ 4 − 1 4 e satisfying Q (0) = 0 . Define the gener alize d P S ap b ent function f : F 2 m × F 2 m → F 2 by f ( x, y ) = T r m 1 ( Q ( x y − η )) . Then f admits neither a semib ent nor a b ent de c omp osition pr ovide d that, for every λ ∈ F 2 m \ F 2 , the Bo ole an function g λ ( x ) = T r m 1  Q  (1 + λ ) x − e  + Q  x − e  + Q  λx − e  (32) on F 2 m is nonc onstant. Sketch of the pr o of. Similarly , by Prop osition 12, it suffices to consider bent or semib en t decompositions on S = ⟨ u, v ⟩ ⊥ , where u = ( a, b ) and v = ( c, d ) are linearly indep enden t vectors in F 2 m × F 2 m satisfying ad + bc = 0. This condition implies that ( c, d ) = λ ( a, b ) for some λ ∈ F 2 m \ F 2 . Moreo v er, b y Lemma 2, it suffices to consider e f ( x, y ) = T r m 1 ( Q ( y x − e )). Case a = 0 (and henc e b  = 0 ). Setting y = 0 and applying the change of v ariables x 7→ ˜ bx , where ˜ b e = b , to D u D v e f ( x, y ), w e obtain the function g λ ( x ) in (32). Hence, if g λ ( x ) is not constant, then D u D v f ∗ ( x, y ) cannot b e constant. Case b = 0 (and henc e a  = 0 ). Applying the c hanges of v ariables x 7→ ax and y 7→ a e y to D u D v e f ( x, y ) and then setting y = 1, we obtain T r m 1  Q  ( x + 1 + λ ) − e  + Q  ( x + 1) − e  + Q  ( x + λ ) − e  + Q  x − e   , whic h cannot b e constant by Lemma 14. Consequen tly , D u D v f ∗ ( x, y ) cannot b e constan t. 27 Case ab  = 0 . W e apply the changes of v ariables x 7→ ax and y 7→ by , and then set α = ba − e and y = 0. This yields T r m 1  Q  α (1 + λ )( x + 1 + λ ) − e  + Q  α ( x + 1) − e  + Q  αλ ( x + λ ) − e  . Again, b y Lemma 14, this expression cannot be constan t, and hence D u D v f ∗ ( x, y ) cannot b e constant. The result no w follo ws from Lemma 10, whic h completes the pro of. □ Example 4.1. L et m b e an o dd inte ger and let k b e a p ositive inte ger with gcd( k , m ) = 1 . L et Q ( x ) = x 2 k +1 b e the Gold function. We r e c al l that Q is a p ermutation of F 2 m if and only if m is o dd and gcd( k , m ) = 1 . F or λ ∈ F 2 m \ F 2 , the function g λ ( x ) define d in (32) satisfies g λ ( x ) = T r m 1  (1 + λ ) 2 k +1 x − e (2 k +1) + x − e (2 k +1) + λ 2 k +1 x − e (2 k +1)  = T r m 1   (1 + λ ) 2 k +1 + 1 + λ 2 k +1  x − e (2 k +1)  = T r m 1  λ 2 k x − e (2 k +1)  . Sinc e gcd( e (2 k + 1) , 2 m − 1) = 1 , the function g λ is b alanc e d for every nonzer o λ ∈ F 2 m . Ther efor e, by The or em 2, the gener alize d P S ap b ent function f : F 2 m × F 2 m → F 2 , f ( x, y ) = T r m 1  x 2 k +1 y − η (2 k +1)  , wher e η is the unique inte ger satisfying η e ≡ 1 mo d (2 m − 1) , do es not admit any semib ent or b ent de c omp osition for al l sufficiently smal l inte gers k r elative to 2 m . 5 Decomp osition and concatenation of b en t functions Let u, v b e t wo linearly indep enden t elements of V (2) n , and let S = ⟨ u, v ⟩ ⊥ . Since ⟨· , ·⟩ n is a nondegenerate inner pro duct on V (2) n , w e can write V (2) n = S ⊕ ⟨ u, v ⟩ . Let f : V (2) n → F 2 b e a Bo olean function, and let ( f 1 , f 2 , f 3 , f 4 ) b e its decomp osition with resp ect to S . After a suitable c hange of co ordinates as described abov e, we ma y iden tify S = V (2) n − 2 and u = (1 , 0), v = (0 , 1). Th us, we ma y regard f as a function on V (2) n − 2 × F 2 × F 2 obtained by concatenation of f 1 , f 2 , f 3 , f 4 : V (2) n − 2 → F 2 . That is, in this represen tation, the decomp osition ( f 1 , f 2 , f 3 , f 4 ) satisfies f ( x, y , z ) =          f 1 ( x ) , if ( y , z ) = (0 , 0) , f 2 ( x ) , if ( y , z ) = (0 , 1) , f 3 ( x ) , if ( y , z ) = (1 , 0) , f 4 ( x ) , if ( y , z ) = (1 , 1) , 28 where x ∈ V (2) n − 2 and y , z ∈ F 2 . Equiv alently , f ( x, y , z ) = f 1 ( x ) + y z  f 1 + f 2 + f 3 + f 4  ( x ) + y  f 1 + f 3  ( x ) + z  f 1 + f 2  ( x ) . In other w ords, f is obtained b y concatenation of the Bo olean functions f 1 , f 2 , f 3 , f 4 . W e define the 4-concatenation (or simply , concatenation) of arbitrary Bo olean func- tions f 1 , f 2 , f 3 , f 4 : V (2) n → F 2 as the Bo olean function f = f 1 ∥ f 2 ∥ f 3 ∥ f 4 from V (2) n × F 2 2 to F 2 giv en b y f ( x, y , z ) = f 1 ( x ) + y z  f 1 + f 2 + f 3 + f 4  ( x ) + y  f 1 + f 3  ( x ) + z  f 1 + f 2  ( x ) , where x ∈ V (2) n and y , z ∈ F 2 . W e remark that the concatenation metho d has b een used efficiently for the secondary construction of Bo olean bent functions outside the M M # class. F or instance, see [18, 25, 27] as w ell as the survey paper [24] and the references therein. The necessary and sufficien t condition for the concatenation of b en t functions to b e b ent is given as follows. Lemma 16 ([17, Theorem I I I.1]) . L et f 1 , f 2 , f 3 , f 4 b e four Bo ole an b ent functions. Then the c onc atenation f = f 1 ∥ f 2 ∥ f 3 ∥ f 4 is b ent if and only if f ∗ 1 + f ∗ 2 + f ∗ 3 + f ∗ 4 = 1 . W e recall that a GM M function f : F 2 n × F 2 2 k → F 2 is defined b y f ( x, y , z ) = f ( z ) ( x ) + T r k 1 ( y z ), where, for each z ∈ F 2 k , the function f ( z ) : F 2 n → F 2 is b ent. In the sp ecial case k = 1, that is, for f : F 2 n × F 2 × F 2 → F 2 giv en b y f ( x, y , z ) = f ( z ) ( x ) + y z , w e observe that f ( x, y , z ) = f (0) ( x ) whenev er z = 0, whereas for z = 1 we hav e f ( x, y , z ) = f (1) ( x ) if y = 0 and f ( x, y , z ) = f (1) ( x ) + 1 if y = 1. Consequen tly , as noted in [2], f can b e expressed as the concatenation f = f (0) ∥ f (1) ∥ f (0) ∥  f (1) + 1  . In this case, the necessary and sufficien t condition on the dual functions in Lemma 16 is trivially satisfied. More generally , as in the case k = 1, GM M functions may b e view ed as concatenations of b en t functions o v er F 2 n . In general, concatenations of b ent functions (particularly the ones of the form f = f (0) ∥ f (1) ∥ f (0) ∥  f (1) + 1  ) hav e b een studied for constructing b en t functions outside the completed M M class. In the recen t pap er [2], the concatenation of vectorial P S ap b en t functions is in v estigated. The motiv ation stems from the fact that the comp onen ts of a vectorial P S ap b en t function F : V (2) n → V (2) m satisfy the follo wing property: for an y nonzero α, β ∈ V (2) m with α  = β , w e ha ve ( F α ) ∗ + ( F β ) ∗ = ( F α + β ) ∗ . (33) The prop ert y in (33) holds for almost all b en t functions arising from bent parti- tions; in particular, it applies to vectorial generalized P S ap b en t functions F : V (2) n → V (2) m . Owing to their explicit representation, analogous to that of P S ap b en t func- tions, one can construct b ent functions by concatenating the comp onen ts of F as f = F α ∥ F β ∥ F γ ∥ F α + β + γ +1, where α, β , γ ∈ V (2) m are nonzero elements such that α + β + γ  = 0, as stated b elo w. The pro of pro ceeds by analogous argumen ts and is therefore omitted. 29 Theorem 3. L et m , k b e inte gers such that k divides m and gcd(2 m − 1 , 2 k + 1) = 1 . Set e = 2 k + 1 . L et P b e any p ermutation of F 2 k , and let α, β , γ b e any nonzer o elements of F 2 k such that α + β + γ  = 0 . Then f ( x, y , z 1 , z 2 ) = T r k 1  (1 + z 1 + z 2 ) α + z 2 β + z 1 γ  P  T r m k  y x − e  + z 1 z 2 . (34) is a b ent function fr om F 2 m × F 2 m × F 2 × F 2 to F 2 . W e conclude this section with a more general construction of b ent functions arising from the structure of the preimage distribution of f giv en in (34). Without loss of generalit y , we ma y assume that P is a permutation of F 2 m satisfying P (0) = 0. Under this assumption, the preimage distribution of f in (34) is given b elo w. Note that the collection of subsets  U × { ( z 1 , z 2 ) }  ∪  A ( γ ) × { ( z 1 , z 2 ) } : γ ∈ F 2 k  forms a disjoin t cov er of F 2 m × F 2 m × F 2 × F 2 , where U and A ( γ ) are defined in (6) and (8), resp ectiv ely . The subsets U × { (0 , 0) } , U × { (0 , 1) } , and U × { (1 , 0) } are mapp ed to 0, whereas U × { (1 , 1) } is mapp ed to 1. Moreo ver, exactly half of the sets {A ( γ ) × { ( y , z ) } : γ ∈ F 2 k } are mapp ed to 0; consequently , the remaining 2 k − 1 are mapp ed to 1 whenever ( y , z ) ∈ { (0 , 0) , (1 , 0) , (0 , 1) } . Finally , exactly half of the sets {A ( γ ) × { (1 , 1) } : γ ∈ F 2 k } are mapp ed to 0 in suc h a wa y that, for eac h γ ∈ F 2 k , if A ( γ ) × { (0 , 0) } , A ( γ ) × { (0 , 1) } , and A ( γ ) × { (1 , 0) } are mapp ed to a 0 , a 1 , and a 2 , resp ectiv ely , then A ( γ ) × { (1 , 1) } is mapp ed to a 0 + a 1 + a 2 + 1. Motiv ated b y the preimage distribution of f in (34), w e present the following con- struction of b en t functions, analogous to that obtained from the Desarguesian spread in [2]. Although the arguments are more in v olv ed, they are similar to those in [2]; hence, w e omit the pro of. Theorem 4. L et Ω 1 = { U, A ( γ ) : γ ∈ F 2 k } b e the gener alize d Desar guesian spr e ad of F 2 m × F 2 m given in (9) . This induc es a p artition of F 2 m × F 2 m × F 2 × F 2 into the sets U × { ( y , z ) } and A ( γ ) × { ( y , z ) } , wher e γ ∈ F 2 k and ( y , z ) ∈ F 2 2 . Assign to e ach quadruple ( a 0 , a 1 , a 2 , a 3 ) ∈ F 4 2 with an o dd numb er of 1 ’s exactly 2 k − 3 sets A ( γ ) . Define f : F 2 m × F 2 m × F 2 × F 2 → F 2 to b e c onstant on e ach p art of this p artition as fol lows: f = 0 on U × { ( y , z ) } for ( y , z ) ∈ { (0 , 0) , (0 , 1) , (1 , 0) } , and f = 1 on U × { (1 , 1) } . If A ( γ ) is assigne d to ( a 0 , a 1 , a 2 , a 3 ) , then f ( x, y , z 1 , z 2 ) = a i whenever ( x, y ) ∈ A ( γ ) and ( z 1 , z 2 ) = (0 , 0) , (0 , 1) , (1 , 0) , (1 , 1) for i = 0 , 1 , 2 , 3 , r esp e ctively. Then f is a b ent function. Ac kno wledgemen ts The initial w ork on this pro ject began during the “W omen in Num b ers Europe 5 (WINE- 5)” w orkshop, held at the Univ ersit y of Split in August 2025. The authors are grateful to the Universit y of Split and the supp orting institutions for making this conference and 30 the resulting collab oration p ossible. They w ould esp ecially lik e to thank the organizers of WINE-5 —Marcela Hanzer, Bork a Jadrijevi ´ c, Pınar Kılı¸ cer, and Lejla Sma jlo vi´ c—for their dedication and hard w ork, whic h made the meeting b oth exceptionally fruitful and enjo y able. This study w as supp orted b y the Scientific and T ec hnological Research Council of T urkey (T ¨ UB ˙ IT AK) under Gran t Number 125F396. S. A. and N. A. thank T ¨ UB ˙ IT AK for its supp ort. T. K. is supp orted by the FWF Pro ject P 35138. B. T. is supp orted b y the Swiss National F oundation through gran t no. 212865. References [1] S. Alk an, N. An bar, T. Kalaycı, W. Meidl, Bent partition, v ectorial dual-b ent func- tion, and LP-Pac king constructions. IEEE T rans. Inform. Theory 71 (2025), 752–767. [2] N. Anbar, T. Kalaycı, S. Kudin, W. Meidl, E. P asalic, A. 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