Nonvanishing $k$-flats of Boolean and vectorial functions

Non v anishing k -flats of Bo olean and v ectorial functions Christian Kasp ers Otto v on Guerick e Univ ersit y Magdeburg, F aculty of Mathematics, Institute for Algebra and Geometry , 39106 Magdeburg, German y ∗ Abstract k th-order sum-free functions are a natural generalization of APN functions using the concept of (non)v anishing flats. In this pap er, w e in tro duce a new com binatorial tec hnique to study the non v anishing flats of Boolean functions. This approach allo ws us to determine the n umber of non v anishing flats for an infinite family of Bo olean functions. W e moreo ver use it to sho w that any k th-order sum-free ( n, n )-function of algebraic degree k giv es rise to an ( n − k )th-order sum-free ( n, n )- function of algebraic degree n − k . This implies the existence of millions of ( n − 2)th-order sum-free functions. Key W ords: Bo olean function, vectorial function, almost perfect nonlinear, k th-order sum-free, v anishing flats. 1 In tro duction A k -dimensional affine subspace, in brief a k -flat, A of F n 2 is said to b e vanishing with respect to an ( n, m )-function F if P x ∈ A F ( x ) = 0. If this sum is nonzero, we call A nonvanishing with resp ect to F . V anishing k -flats w ere introduced by [ LMP + 20 ] for k = 2. In this case, they are closely related to almost p erfect nonlinear (APN) functions: An ( n, n )-function is APN if and only if it has no v anishing 2-flats [ Hou06 , BL08 ]. APN functions ha ve optimal differen tial prop erties. They hav e b een intensiv ely studied and, b y now, several infinite families and millions of sp oradic examples of these functions are known [LK24, BLL + 25]. Because of the connection mentioned ab o ve, the num ber of v anishing 2-flats is a go o d measure of how close a function is to an APN function. W e ∗ c hristian.kasp ers@ovgu.de This is the revised extended abstract accepted for presen tation at WCC 2026. 1 refer to [ LMP + 20 ] for more results on the n umber of v anishing 2-flats of ( n, n )-functions. F or any ( n, m )-function, this num b er is also directly related to the W alsh spectrum of the function [ Ars18 ]. F urthermore, v anishing 2-flats can b e used to construct t -designs from certain ( n, m )-functions [MPP23]. So while there are multiple results ab out v anishing 2-flats, the situation is m uch less clear if k ≥ 3. Ho wev er, suc h v anishing k -flats recen tly came in to fo cus with the in tro duction of k th-order sum-free functions: An ( n, n )- function is called k th-or der sum-fr e e if it has no v anishing k -flats [ Car25b ]. By now, these functions hav e b een studied in several pap ers, many fo cusing on the question for which v alues of n and k the inv erse function x 2 n − 2 is k th-order sum-free [ Car25b , EHRZ24 , CH24 , HZ25b , HZ25a ]. W e refer to [ Car25c ] for the cryptographic background, an ov erview and more prop erties of k th-order sum-free functions. Clearly , APN functions are precisely second- order sum-free functions. F or k ≥ 3, ho wev er, k th-order sum-free functions seem to b e rare and so far only one infinite family of these functions is known: the ( n, n )-function x 1+2 j + ··· +2 j ( k − 1) is k th-order sum-free if gcd ( j, n ) = 1 [ Car25c , Car25a ]. Also regarding the n um b er of non v anishing k -flats of ( n, m )-functions, so far no results for k ≥ 3 are kno wn. In this pap er, we study the nonv anishing k -flats of Bo olean and v ectorial functions. In Section 3, w e giv e an ov erview of the case k = 2. In Section 4, w e translate the problem of finding the nonv anishing k -flats of a Bo olean function in to finding matrices with prescrib ed sets of linearly indep enden t columns. W e b elieve that this new combinatorial technique, whic h do es not use finite fields, is p ow erful and may lead to new theoretical and computational results. In Section 5, w e apply our new approach to Bo olean and vectorial functions. W e show that each k th-order sum-free function of algebraic degree k giv es rise to an ( n − k )th-order sum-free function of algebraic degree n − k (Theorem 5.12). W e also determine the n umber of nonv anishing k -flats of an infinite family of Bo olean functions whose terms share a constant set of v ariables (Theorem 5.4). Because of page limitations w e omit several computational results in this pap er. W e will include them in an extended v ersion later. 2 Preliminaries Let k ∈ { 0 , . . . , n } . W e denote the set of all k -dimensional linear subspaces of F n 2 , or k -subsp ac es in short, b y U n,k . The num b er of k -subspaces of F n 2 is |U n,k | = " n k # 2 = (2 n − 1)(2 n − 1 − 1) · · · (2 n − k +1 − 1) (2 k − 1)(2 k − 1 − 1) · · · (2 − 1) , (1) where   q denotes the q -binomial co efficien t. Let U ∈ U n,k . W e write U = ⟨ u 1 , . . . , u k ⟩ if u 1 , . . . , u k ∈ F n 2 span U . F or a ∈ F n 2 , we call U + a := { x + a : x ∈ U } a k -dimensional affine subsp ac e , 2 or in brief a k -flat , of F n 2 . Denote C U := { U + a : a ∈ F n 2 } and note that | C U | = 2 n − k . W e denote the set of all k -flats, of F n 2 b y A n,k . Clearly , |A n,k | = 2 n − k |U n,k | . (2) It is well kno wn that A n, 2 is the set of 4-subsets of F n 2 whose elements sum to 0. Note that this do es not hold if k > 2. F or k ≥ 3, the elements of a k -flat still sum to 0, but not ev ery 2 k -set whose elemen ts sum to 0 is a k -flat. In the following, we denote [ n ] := { 1 , . . . , n } . An n -variable Bo ole an function is a function f : F n 2 → F 2 whic h we usually consider by its algebraic normal form (ANF). If I ⊆ [ n ], we write x I for the monomial Q i ∈ I x i and define V ar ( x I ) := I . A ve ctorial function is a function F : F n 2 → F m 2 , in brief an ( n, m ) -function . Note that an ( n, 1)-function is a Boolean function. W e define an ( n, m )-function usually by its ANF, i. e. the ANF s of its n - v ariable Bo olean co ordinate f unctions f 1 , . . . , f m . F or ( n, n )-functions, w e sometimes also use their univ ariate representation on the finite field F 2 n . The algebr aic de gr e e deg alg ( F ) of an ( n, m )-function F is the maxim um degree of its co ordinate functions. If the ANF of an ( n, m )-function con tains only terms of degree k , w e say that F is k -homo gene ous . Denote by N k ( F ) the set of nonv anishing k -subspaces and b y N A ,k ( F ) the set of nonv anishing k -flats of F , resp ectiv ely , so w e ha ve N k ( F ) = { U ∈ U n,k : P x ∈ U F ( x )  = 0 } and N A ,k ( F ) = { A ∈ A n,k : P x ∈ A F ( x )  = 0 } . In the following Prop osition 2.1 and Corollary 2.2, w e sho w that if F has algebraic degree k , then N A ,k ( F ) can b e easily derived from N k ( F ). Let F b e an ( n, m )-function of algebraic degree k . W e define the derivative D a F of F in direction a ∈ F n 2 as the ( n, m )-function D a F with D a F ( x ) = F ( x ) + F ( x + a ) + F ( a ) + F (0), and the ℓ th-order deriv ativ e D a 1 · · · D a ℓ F of F in direction ( a 1 , . . . , a ℓ ) ∈ ( F n 2 ) ℓ b y the comp osition D a 1 · · · D a ℓ F ( x ) = ( D a ℓ F ◦ · · · ◦ D a 1 F )( x ). It is easy to see that D a 1 · · · D a ℓ F has algebraic degree at most k − ℓ and no constan t term [ Car20 , Prop osition 5 and Corollary 1]. In particular, the k th-order deriv ativ e of F is alw ays 0. Prop osition 2.1. L et k ∈ [ n ] , U ∈ U n,k , and let F b e an ( n, m ) -function. If the algebr aic de gr e e of F is k , then P x ∈ U F ( x ) = P x ∈ A F ( x ) for al l A ∈ C U . Pr o of. Let F b e an ( n, n ) function of algebraic degree k . F or c  = a , we ha ve D a F ( c ) = P x ∈⟨ a,c ⟩ F ( x ) and, analogously , for c / ∈ ⟨ a 1 , . . . , a ℓ ⟩ , w e ha ve D a 1 · · · D a ℓ F ( c ) = P x ∈⟨ c,a 1 ,...,a ℓ ⟩ F ( x ). Let x 1 , . . . , x k form a basis of U , and let c ∈ F n 2 \ U . Then the k th-order deriv ativ e D x 1 · · · D x k F ( c ) = P x ∈⟨ c,x 1 ,...,x k ⟩ F ( x ) = P x ∈ U F ( x ) + P x ∈ U + c F ( x ), which equals 0 according to ab o ve. Hence, P x ∈ U F ( x ) = P x ∈ U + c F ( x ). Prop osition 2.1 immediately implies Corollary 2.2. Corollary 2.2. L et F b e an ( n, m ) -function. If F has algebr aic de gr e e k , then | N A ,k ( F ) | = 2 n − k | N k ( F ) | . In p articular, in this c ase, F is k th-or der sum-fr e e if and only if | N k ( F ) | = 0 . 3 Corollary 2.3 follo ws from the fact men tioned ab ov e that every k -th order deriv ativ e of a function of algebraic degree k is 0. Corollary 2.3 ([ Car25c ]) . L et F b e an ( n, m ) -function, and let k ∈ [ n ] . I f deg alg ( F ) < k , then F has no nonvanishing k -flats, i. e. N A ,k ( F ) = ∅ . This implies that there are no k th-order sum-free functions of algebraic degree less than k , and considering Corollary 2.2, it seems to b e easier to find k th-order sum-free functions of algebraic degree k than of an y algebraic degree r > k . The k th-order sum-free functions from [ Car25c , Car25a ] all ha ve algebraic degree k . So far, for k > 2, the only known example with r > k is the inv erse function x 2 n − 2 of algebraic degree n − 1: it is ( n − 2)th-order sum-free if n is o dd [Car25b]. Corollary 2.3 also leads to Corollary 2.4. Corollary 2.4. L et k ∈ [ n ] , and let F and G b e ( n, m ) -functions, such that deg alg ( G ) < k . Then N A ,k ( F + G ) = N A ,k ( F ) . In p articular, if F is k th-or der sum-fr e e, so is F + G . W e con tinue b y studying the behavior of | N A ,k ( F ) | under equiv alence, see [ Car20 ] for details on EA- and CCZ-equiv alence. In general, | N A ,k ( F ) | is an EA-in v arian t but no CCZ-inv ariant [ Car25c ]. In the APN case, i. e. k = 2, the n um b er of non v anishing k -flats is preserv ed under CCZ-equiv alence [ LMP + 20 ]. F or k ≥ 3, this do es not hold in general: On F 2 5 , the p erm uta- tion x 7 is third-order sum-free. Its inv erse x 9 , which is CCZ-equiv alen t to x 7 , is quadratic and thus, b y Corollary 2.3, not third-order sum-free [Car25c]. Corollary 2.4 motiv ates the following definition that in tro duces a general- ization of EA-equiv alence preserving the num b er of v anishing k -flats. Definition 1. Let r ∈ { 0 , . . . , n } . T wo ( n, m )-functions F , G are said to b e de gr e e- r e quivalent if there exists an affine ( n, n )-p erm utation L , an affine ( m, m )-p erm utation M , and an ( n, m )-function R with deg alg ( R ) = r suc h that G = M ◦ F ◦ L + R . Note that degree-1 equiv alence is precisely EA-equiv alence. Corollary 2.5 is a direct consequence of Corollary 2.4 and | N A ,k ( F ) | b eing EA-in v arian t. Corollary 2.5. L et F b e an ( n, m ) -function. The numb er of nonvanishing k -flats | N A ,k ( F ) | is invariant under de gr e e- ( k − 1) -e quivalenc e. In p articular, the k th-or der sum-fr e e dom of F is a de gr e e- ( k − 1) -invariant. The W alsh tr ansform of an n -v ariable Bo olean function f is the function W f : F n 2 → Z defined by W f ( a ) = P x ∈ F n 2 ( − 1) f ( x )+ ⟨ x,a ⟩ , where ⟨ , ⟩ denotes a scalar pro duct on F n 2 . The v alues of W f are called the W alsh c o efficients of F , and the m ultiset W f = { W f ( a ) : a ∈ F n 2 } is the W alsh sp e ctrum of f . The W alsh sp ectrum of an ( n, m )-function F is the union W f = S b ∈ F n 2 ,b  =0 W f b , where f b = ⟨ b, F ( x 1 , . . . , x m ) ⟩ , of the W alsh sp ectra of the component func- tions of F . The nonline arity nl( F ) of F is 2 n − 1 − 1 2 max W ∈W f | W | . 4 An n -v ariable Bo olean function with maxim um nonlinearity 2 n − 1 − 2 n 2 − 1 is a b ent function . Ben t functions ha ve W alsh co efficients ± 2 n 2 and exist only if n is even. An o dd-dimension analogue are semi-b ent functions with W alsh co efficien ts 0 , ± 2 n +1 2 . Ho wev er, for n ≥ 9 there exist functions with higher nonlinearit y than semi-b en t functions [KMSY06, KY10]. In Section 4, we use a one-to-one corresp ondence b et ween the k -subspaces of F n 2 and the k × n -matrices o ver F 2 of full rank k in reduced ro w echelon form. A matrix ov er F 2 is said to b e in r e duc e d r ow e chelon form (RREF) if it is in ec helon form and in ev ery column with a leading entry all other en tries are 0. Using elemen tary row op erations, any matrix can b e transformed in to a unique RREF. So if we consider U ∈ U n,k as the row span of a k × n matrix of rank k and transform this matrix in to RREF, w e obtain a unique RREF matrix of rank k , whose rows form a basis of U . Vice v ersa, every full rank k × n matrix in RREF giv es rise to a unique k -subspace of F n 2 . F rom no w on, for any U ∈ U n,k , we denote the corresponding k × n matrix of rank k in RREF b y G U and call it the RREF-gener ator matrix of U . W e will often consider matrices formed by some columns of G U : If I ⊆ [ n ], w e denote the matrix formed b y the columns i 1 , . . . , i | I | ∈ I of G U b y G U [ I ]. 3 Non v anishing 2 -flats and the W alsh sp ectrum A ccording to [ Ars18 , Theorem 2.5], w e can calculate the num b er of v anishing 2-flats of an ( n, m )-function F from its W alsh sp ectrum: this n umber is 1 24  1 2 n + m  2 4 n + ω  − 3 · 2 2 n + 2 n +1  , where ω = P W ∈W F W 4 is the sum of the fourth p ow ers of the W alsh co efficients of F . W e determine the num b er of non v anishing 2-flats | N A , 2 | of F b y subtracting |A n,k | , see (2), from the expression ab o ve: Theorem 3.1. L et F b e an ( n, m ) -function. The function F has | N A , 2 ( F ) | = 2 4 n (2 m − 1) − ω 3 · 2 n + m +3 nonvanishing 2 -flats. If m = 1 , then | N A , 2 ( f ) | = 2 4 n − ω 3 · 2 n +4 . The following result was also observ ed b y [ Ars18 ]. W e add a short pro of. Theorem 3.2. L et f b e an n -variable Bo ole an function. W e have | N A , 2 ( f ) | ≤ 2 2 n − 4 (2 n − 1) 3 and N A , 2 ( f ) |A n, 2 | ≤ 2 n − 2 2 n − 1 − 1 . Equality holds if and only if f is b ent. Pr o of. T o maximize | N A , 2 ( f ) | , according to Theorem 3.1, w e need to mini- mize ω . Recall that the W alsh co efficien ts need to satisfy P arsev al’s relation P W ∈W ( f ) W 2 = 2 2 n . Th us, ω = P W ∈W ( f ) W 4 is minimal if | W | = 2 n 2 for all W ∈ W f whic h is precisely the d efinition of a b en t function and is only p ossible if n is ev en. The second relation follo ws from simplifying the fraction, where | N A , 2 ( f ) | is as in the first relation and |A 2 ,k | is as in (2). Th us, b en t functions are the functions with the most nonv anishing 2-flats and the situation is clear for even n . How ev er, the question which functions ha ve the most non v anishing 2-flats if n is o dd is still widely op en. 5 Prop osition 3.3. L et n b e o dd, and let f b e an n -variable Bo ole an function. If f is semi-b ent, then | N A , 2 ( f ) | = 2 2 n − 3 (2 n − 1 − 1) 3 and N A , 2 ( f ) |A n, 2 | = 2 n − 1 2 n − 1 . Pr o of. The nonzero W alsh co efficien ts of f are ± 2 n +1 2 with a combined m ultiplicity of 2 n − 1 . The result then follo ws from Theorem 3.1. Con trary to n ev en, semi-bent functions do not maximize the num ber of non v anishing 2-flats if n is o dd. This w as kno wn for n = 9, where [ Ars18 ] sho wed that the functions from [ KMSY06 ] with higher nonlinearity (241) than semi-b en t functions (240) also hav e more non v anishing 2-flats. W e confirmed computationally that the functions from [ KY10 ] of nonlinearit y 242 ha ve even more nonv anishing 2-flats. F urthermore, we verified that also functions with the same nonlinearit y as semi-bent functions can hav e more non v anishing 2-flats: This is the case for the functions with fiv e-v alued W alsh sp ectrum from [MS02, Theorems 7 and 9] in dimension 5, 7 and 9. 4 A new technique to determine the non v anishing k -flats of Bo olean functions In this section, w e presen t and apply our new technique to determine the non v anishing k -flats of a Bo olean function. W e start with an easy lemma. Lemma 4.1. L et v 1 , . . . , v k ∈ F k 2 . Then for any v ∈ F k 2 , the e quation α 1 v 1 + · · · + α k v k = v with α 1 , . . . , α k ∈ F 2 has an o dd numb er of solutions if and only if v 1 , . . . , v k ar e line arly indep endent. Pr o of. The solution set of α 1 v 1 + · · · + α k v k = v is empty or a flat of F k 2 . The only flats containing an o dd n umber of elemen ts are those of dimension 0 whic h implies that v 1 , . . . , v k are linearly indep endent. In the pro of of the following theorem we use the n -ary symmetric dif- ference. Recall that for a family of sets A = { A 1 , . . . , A n } , the symmetric difference △ A := A 1 △ · · · △ A n is defined as the set of elements o ccurring in an o dd num b er of the sets A 1 , . . . , A n . The cardinalit y of △ A is well known, w e hav e | △ A | = n X ℓ =1 ( − 2) ℓ − 1 X { i 1 ,...,i ℓ }∈ ( n ℓ )       ℓ \ j =1 A i j       , (3) where  n ℓ  denotes the set of all ℓ -subsets of [ n ]. Theorem 4.2. L et f = m 1 + · · · + m t b e a k -homo gene ous n -variable Bo ole an function. W e have | N k ( f ) | = P t ℓ =1 ( − 2) ℓ − 1 P { i 1 ,...,i ℓ }∈ ( n ℓ )    T ℓ j =1 N k ( m i j )    . Pr o of. Let U ∈ U n,k . Then U ∈ N k ( f ) if and only if U ∈ N k ( m i ) for an o dd n umber of i ∈ [ t ]. Consequently , N k ( f ) is precisely the symmetric difference N k ( f ) = N k ( m 1 ) △ · · · △ N k ( m t ), and the result follo ws from (3). 6 W e next sho w ho w to determine T t i =1 N k ( m i ). W e first presen t an approac h for the nonv anishing k -flats of a single monomial m and then extend it to the intersection of the sets of non v anishing k -flats of t monomials: Theorem 4.3. L et m b e an n -variable monomial of de gr e e k . W e have N k ( m ) = { U ∈ U n,k : rank( G U [V ar( m )]) = k } . Pr o of. Let x = ( x 1 , . . . , x n ). Recall that U ∈ N k ( m ) if P x ∈ U m ( x )  = 0 whic h is equiv alent to m ( x ) = 1 for an o dd n umber of x ∈ U . Clearly , m ( x ) = 1 if and only if x i = 1 for all i ∈ V ar ( m ). Th us, U ∈ N k ( m ) if and only if the n umber of x ∈ U with x i = 1 for all i ∈ V ar ( m ) is o dd. According to Lemma 4.1 this is the case if and only if the rows of G U [ V ar ( m )] are linearly indep enden t, since then m ( x ) = 1 for a unique x ∈ U . Since G U [ V ar ( m )] is a quadratic k × k matrix, U ∈ N k ( m ) if and only if rank ( G U [ V ar ( m )]) = k . Theorem 4.3 extends naturally to the intersection of sev eral monomials: Corollary 4.4. L et m 1 , . . . , m t b e n -variable monomials of de gr e e k . Then T t i =1 N k ( m i ) = { U ∈ U n,k : rank( G U [V ar( m i )]) = k for i ∈ [ t ] } . Com bining Theorem 4.2 and Corollary 4.4, we obtain a new technique to determine the num b er of nonv anishing k -flats of a Bo olean function. 5 Applications to Bo olean and vectorial functions In this section, w e use the new approach presented in Section 4 to determine the n umber of nonv anishing k -flats for an infinite family of Bo olean functions and to study the nonv anishing k -flats of the so-called complement of an ( n, m )-function. Note that we restrict ourselv es mostly to k -homogeneous functions in this section because for them it is enough to study their v anishing k -subspaces according to Corollary 2.2. F or arbitrary functions of degree k , w e ma y simply omit the terms of degree less than k , according to Corollary 2.4, and use the technique for the resulting k -homogeneous function. W e first observ e that the set of nonv anishing k -flats of the monomial x [ k ] with x [ k ] = x 1 · · · x k has an easy description: Prop osition 5.1. F or the n -variable monomial x [ k ] , we have N k ( x [ k ] ) = { U ∈ U n,k : G U [[ k ]] = I k } and | N k ( x [ k ] ) | = 2 k ( n − k ) , wher e I k denotes the k × k identity matrix. Pr o of. The first iden tit y follows from Theorem 4.3. W e then obtain | N k ( x [ k ] ) | b y observing that the matrix G U has 2 k ( n − k ) en tries that can b e arbitrarily c hosen from F 2 to obtain a unique subspace U ∈ N k ( x [ k ] ). So for x [ k ] precisely those subspaces U ∈ U n,k are nonv anishing for which G U is of the shap e ( I k | A n − k ) , where A n − k is an arbitrary k × ( n − k ) matrix. Since we can obtain x [ k ] from any other monomial of degree k b y permuting the v ariables the following Corollary 5.2 holds. 7 Corollary 5.2. L et m b e an n -variable monomial of de gr e e k . W e have | N k ( m ) | = 2 k ( n − k ) . Analogously , b y p ermuting the v ariables we can represen t any Bo olean function f of degree k in the form f = x [ k ] + g for some function g . This allo ws us to fo cus on matrices of shap e ( I k | R n − k ) . In Theorem 5.4, we precisely determine the num b er of nonv anishing k -flats for the following family of Bo olean functions. Definition 2. Let 0 ≤ d ≤ n − 1, and let f = m 1 + · · · + m t b e an n - v ariable Bo olean function. W e say that f is d -interse cting if there exists a subset D ⊆ [ n ] with | D | = d suc h that V ar ( m i ) ∩ V ar ( m j ) = D for all i, j ∈ [ t ] with i  = j . Note that clearly 0 ≤ d ≤ deg( f ) − 1 and V ar( m 1 ) ∩ · · · ∩ V ar( m t ) = D . Lemma 5.3. L et f = m 1 + · · · + m t b e a d -interse cting, k -homo gene ous n -variable Bo ole an function. Then      t \ i =1 N k ( m i )      = 2 k ( n − tk +( t − 1) d ) k − d − 1 Y i =0 (2 k − 2 i + d ) t − 1 . (4) Pr o of. Using Corollary 4.4, we coun t all U ∈ U n,k with rank ( G U [ V ar ( m i )]) = k for all i ∈ [ t ] . Without loss of generality supp ose m 1 = x [ k ] and D = [ d ] if d > 0. Then G U has shap e ( I k | B ( k − d ) 1 | · · · | B ( k − d ) t − 1 | A n − tk +( t − 1) d ), where B ( k − d ) i is a k × ( k − d ) matrix such that rank ( e 1 , . . . , e d | B ( k − d ) i ) = k for all i ∈ [ t − 1], and the en tries of A n − tk +( t − 1) d can b e arbitrarily c hosen. It follo ws that we hav e 2 k ( n − tk +( t − 1) d ) distinct c hoices for A n − tk +( t − 1) d that w e can combine with Q k − 1 j = d (2 k − 2 j ) choices for B ( k − d ) i for each i ∈ [ t − 1]. Lemma 5.3 allows us to pro ve our first main theorem. Theorem 5.4. L et f b e a d -interse cting, k -homo gene ous n -variable Bo ole an function. W e have | N k ( f ) | = t X ℓ =1 ( − 2) ℓ − 1 t ℓ ! 2 k ( n − ℓk +( ℓ − 1) d ) k − d − 1 Y i =0 (2 k − 2 i + d ) ℓ − 1 . (5) Pr o of. Let f = m 1 + · · · + m t . W e use Theorem 4.2 to calculate | N k ( f ) | with    T t i =1 N k ( m i )    as in (4), and take into account that for an y ℓ ∈ [ t ] the in tersection size    T t i =1 N k ( m i )    is constant for all { i 1 , . . . , i ℓ } ⊆  t ℓ  . In the case d = 0, we can simplify Theorem 5.4: Prop osition 5.5. L et f b e a 0 -interse cting, k -homo gene ous n -variable Bo ole an function. A nd define G = Q k − 1 i =0 (2 k − 2 i ) . W e have | N k ( f ) | = 2 kn − 1 G 1 −  1 − G 2 k 2 − 1  t ! . 8 Pr o of. W e consider (5) with d = 0. F actoring out 2 kn − 2 G , we obtain | N k ( f ) | = − 2 kn − 1 G P t ℓ =1  t ℓ   − G 2 k 2 − 1  ℓ . A dding and subtracting 1 and using the binomial theorem, the right-hand side b ecomes − 2 kn − 1 G  − 1 +  1 − G 2 k 2 − 1  t  . Note that w e can use Prop osition 5.5 to determine the num b er of non- v anishing 2-flats of the b en t function x 1 x 2 + · · · + x n − 1 x n if n is even and of the semi-b en t function x 1 x 2 + · · · + x n − 2 x n − 1 if n is o dd as these are 0-in tersecting functions, see Theorem 3.2 and Proposition 3.3. As another application of our n ew technique, we in tro duce the complement of a Bo olean function and present a bijection b etw een the non v anishing k -flats of a function and those of its complement. Definition 3. Let m b e an n -v ariable monomial. W e call the n -v ariable monomial m ′ with V ar ( m ′ ) = [ n ] \ V ar ( m ) the c omplement of m and denote it by m . Analogously , for a Bo olean function f defined by f = m 1 + · · · + m t w e define its c omplement by m 1 + · · · + m t and denote it by f . Clearly , the follo wing three prop erties hold: f = f ; if f has degree r , then f has degree n − r ; and if f is homogeneous so is f . Moreov er, equiv alence is preserved by taking the complemen t of t wo homogeneous functions: Prop osition 5.6 ([ Hou96 , LL08 ]) . L et f , g b e r -homo gene ous n -variable Bo ole an functions. Then f and g ar e de gr e e- ( r − 1) e quivalent if and only if f and g ar e de gr e e- ( n − r − 1) e quivalent. More precisely , if g = f ◦ L + h for some L ∈ GL ( n, 2) and a Bo olean function h of algebraic degree at most r − 1, then g = f ◦ ( L − 1 ) T + h ′ for a Bo olean function h ′ of algebraic degree at most n − r − 1. T o establish the aforementioned bijection, we recall the follo wing defini- tion: If U is a k -subspace of an n -dimensional vector space V , we call the set U ⊥ := { v ∈ V |⟨ v , u ⟩ = 0 for all u ∈ U } the ortho gonal c omplement of U . It is well kno wn that U ⊥ is an ( n − k )-subspace of V . Note that U ⊥ = k er G U . Lemma 5.7. L et U ∈ U n,k , and let I b e a k -subset of [ n ] . W e have rank( G U [ I ]) = k if and only if rank( G U ⊥ [[ n ] \ I ]) = n − k . Pr o of. W e consider U as a linear [ n, k ] co de with generator matrix G U whose columns i 1 , . . . , i k ∈ I are linearly independent. Clearly , U is equiv alen t to a co de U ′ with generator matrix G U ′ = G U σ , where σ ∈ S n , σ = (1 i i ) · · · ( k i k ), p erm utes the columns of G U suc h that i 1 , . . . , i k are the first columns of G U ′ . W e transform G U ′ in to standard form ( I k | P ). Then the parity chec k matrix of U ′ is H U ′ = ( P T | I n − k ), and H U = H U ′ σ is a parity c heck matrix of U . Clearly , the columns j 1 , . . . , j n − k ∈ [ n ] \ I of H U are linearly indep endent, and we can transform H U in to G U ⊥ b y elementary ro w op erations which preserv e this prop ert y . 9 Theorem 5.8. L et f b e a k -homo gene ous n -variable Bo ole an function, and let U ∈ U n,k . Then U ∈ N k ( f ) if and only if U ⊥ ∈ N k ( f ) . Pr o of. The result follo ws from com bining Theorem 4.3 with Lemma 5.7. W e remark that Theorem 5.8 do es not hold if f is not k -homogeneous: Supp ose m is a term of f of degree ℓ < k . A ccording to Corollary 2.3, then m sums to zero ov er every k -flat. How ev er, in this case, m has degree n − ℓ > n − k and, thus, do es not necessarily sum to 0 o ver every ( n − k )- flat. So in this case, U ∈ N k ( f ) do es not imply U ⊥ ∈ N k ( f ). W e obtain Theorem 5.9 as an immediate consequence of Theorem 5.8 and Theorem 3.2. Theorem 5.9. L et n b e even, and let f b e an n -variable Bo ole an function of de gr e e n − 2 . Then | N A ,n − 2 ( f ) | ≤ 2 2 n − 4 (2 n − 1) 3 and N A ,n − 2 ( f ) |A n, 2 | ≤ 2 n − 2 2 n − 1 − 1 . Equality holds if and only if f ′ is EA-e quivalent to a b ent function, wher e f ′ is an ( n − 2) -homo gene ous function that is de gr e e- ( n − 3) e quivalent to f . Ev entually , we extend Definition 3 and Theorem 5.8 to vectorial functions. Definition 4. Let F b e an ( n, m )-function defined by Bo olean co ordinate functions f 1 , . . . , f m . W e call the ( n, m )-function F defined b y the comple- men ts f 1 , . . . , f m of f 1 , . . . , f m the c omplement of F . Note that Prop osition 5.6 transfers immediately to vectorial f unctions: Corollary 5.10. L et F and G b e r -homo gene ous ( n, m ) -functions. Then F and G ar e de gr e e- ( r − 1) e quivalent if and only if F and G ar e de gr e e- ( n − r − 1) e quivalent. More precisely , if G = M ◦ F ◦ L + H for L ∈ GL ( n, 2), M ∈ GL ( m, 2) and some ( n, m )-function H of algebraic degree at most r − 1, then G = M ◦ F ◦ ( L − 1 ) T + H ′ for some ( n, m )-function H ′ of algebraic degree at most n − r − 1. The follo wing result is a direct consequence of Theorem 5.8: Corollary 5.11. L et F b e a k -homo gene ous ( n, m ) -function, and let U ∈ U k . Then U ∈ N k ( F ) if and only if U ⊥ ∈ N k ( F ) . Corollary 5.11 in particular implies our second main theorem. Theorem 5.12. L et F b e a k -homo gene ous ( n, m ) -function. Then F is k -th or der sum-fr e e if and only if F is ( n − k ) th-or der sum-fr e e. Thanks to Corollary 5.10, degree-( k − 1) inequiv alen t k -th order sum-free functions yield degree-( n − k − 1) inequiv alen t ( n − k )th-order sum-free functions. Therefore, Theorem 5.12 implies that every quadratic APN ( n, n )- function F giv es rise to an ( n − 2)th-order sum-free ( n, n )-function: W e omit the affine terms in the ANF of F to obtain F ′ , which preserv es the APN prop ert y , and then F ′ is ( n − 2)th-order sum-free. 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