On Third-Order Nonlinearity of Biquadratic Monomial Boolean Functions
The th-order nonlinearity of Boolean function plays a central role against several known attacks on stream and block ciphers. Because of the fact that its maximum equals the covering radius of the th-order Reed-Muller code, it also plays an important role in coding theory. The computation of exact value or high lower bound on the th-order nonlinearity of a Boolean function is very complicated problem, especially when . This paper is concerned with the computation of the lower bounds for third-order nonlinearities of two classes of Boolean functions of the form for all , , where , where , , and are integers such that and , and , where is a positive integer such that and .
Boolean functions are the building blocks for the design and the security of symmetric cryptographic systems and for the definition of some kinds of error correcting codes, sequences, and designs. The th-order nonlinearity, , of a Boolean function is defined by the minimum Hamming distance of to -Reed-Muller code of length and order . The nonlinearity of is given by and is related to the immunity of against best affine approximation attacks  and fast correlation attacks , when is used as a combiner function or a filter function in a stream cipher. The th-order nonlinearity is an important parameter, which measures the resistance of the function against various low-order approximation attacks [1, 3, 4]. In cryptographic framework, within a trade-off with the other important criteria, the th-order nonlinearity must be as large as possible; see [5–9]. Since, the maximal th-order nonlinearity of all Boolean functions equals the covering radius of , it also has an application in coding theory. Besides these applications, an interesting connection between the th-order nonlinearity and the fast algebraic attacks has been introduced, recently in , which claims that a cryptographic Boolean function should have high th-order nonlinearity to resist the fast algebraic attack.
Unlike nonlinearity there is no efficient algorithm to compute second-order nonlinearities for . The most efficient algorithm is introduced by Fourquet and Tavernier  which works for and up to for some special functions. Thus, to identify a class of Boolean function with high th-order nonlinearity, even for , is a very relevant area of research. In 2008, Carlet has devolved a technique to compute th-order nonlinearity recursively in , and using this technique he has obtained the lower bounds of nonlinearity profiles for functions belonging to several classes of functions: Kasami functions, Welch functions, inverse functions, and so forth. Based on this technique, the lower bound for th-order nonlinearity, for , is obtained for some specific classes of Boolean functions, in many articles; see, for example, [11–14] and the references therein. The best known asymptotic upper bound for given by Carlet and Mesnager  is as follows: The classes of Boolean functions for which the lower bound of third nonlinearity is known are inverse functions , Dillon functions , and Kasami functions, . In this paper, we deduce the theoretical lower bounds on third-order nonlinearities of two classes of biquadratic monomial Boolean functions for all , where and (a) , where , , and are integers such that and , and (b) , where is a positive integer such that and .
Remainder of the paper is organized as follows. In Section 2 some basic definitions and notations required for the subsequent sections are reviewed. The main results on lower bounds of third-order nonlinearities are presented in Section 3. The numerical compression of our bounds with the previous known results is provided in Section 4. Section 5 is conclusion.
Let be the finite field consisting of elements. The group of units of , denoted by , is a cyclic group consisting of elements. An element is said to be a primitive element if it is a generator of the multiplicative group . A function from to is said to be a Boolean function on variables; the set of such functions is denoted by . Let and , where is a positive integer, denote the ring of integers and integers modulo , respectively. A cyclotomic coset modulo of is defined as where is the smallest positive integer such that [17, page 104]. It is a convention to choose the subscript to be the smallest integer in and refer to it as the coset leader of and denotes the size of . The trace function is defined by for all . The trace representation  of a function is where is the set of all coset leaders modulo and , , for all . A Boolean function is said to be a monomial trace function if its trace representation consists of single trace term. The binary representation of an integer is where . The Hamming weight of is , where the sum is over . The algebraic degree, denoted by , of , as represented in (3), is the largest positive integer for which and . The support of is . The weight of is , where is the cardinality of any set . The Hamming distance between two functions , is defined by .
The Walsh-Hadamard transform (WHT) of a Boolean function at is defined by . The nonlinearity of in terms of its Walsh-Hadamard spectrum (WHS) is given by The set is referred to as the WHS of which satisfies the Parseval’s identity: which implies that , and so . The function achieving maximum possible nonlinearity are said to be bent functions (exists only for even ), were introduced by Rothaus .
The derivative of with respect to is defined by for all . The second-order derivatives of with respect to is the Boolean function which is defined by , where is two-dimensional subspace of generated by and ; for details on higher derivatives, see [5, 11]. The th-order nonlinearity of is defined as The sequence is called the nonlinearity profile of . Also, because . The notion of th-order bent functions was introduced by Iwata and Kurosawa . A function is said to be th-order bent (for ) if and only if , for even , and , for odd .
Carlet’s  recursive lower bounds for third-order nonlinearities which we use to compute our bounds, are given below in Propositions 1 and 2.
Proposition 1 (see [11, Proposition 2]). Let ; then .
Proposition 2 (see [11, Equation ]). Let . Then
Proposition 3 (see [17, Chapter 15, Corollary 13] (McEliece’s theorem)). The th-order nonlinearities of a Boolean function with algebraic degree are divisible by , where denotes the ceiling of (the smallest integer greater than or equal to ).
Proposition 4 (see [20, Corollary 1]). Let be a linearized polynomial over , where , are positive integers such that . Then zeroes of the linearized polynomial in are at most .
The result in Proposition 4 above was introduced by Bracken et al. . The bilinear form  associated with a quadratic Boolean function is defined by and the kernel, of is the subspace of defined by An element is called a linear structure of . Next, if is a vector space over a field of characteristic and a quadratic form, then and have the same parity . The distribution of the WHT values of a quadratic Boolean function is given in the following theorem which claims that the weight distribution of the values in the WHS of depends only on the dimension of .
Theorem 5 (see [17, 21]). Let be a quadratic Boolean function and , where is defined in (8); then the weight distribution of the WHT values of is given by
3. Main Results
In this section, using Carlet’s recursive technique , the theoretical lower bounds for third-order nonlinearities of two general classes of monomial Boolean functions of degree 4 are obtained.
Theorem 6. Let , for all , where and , , and are integers such that and . Then In particular, if , then
Proof. Derivative of with respect to is
where is quadratic. The second derivative with respect to , where , is
where is an affine function. If is quadratic, then the WHS of is equivalent to the WHS of the function obtained by removing from :
Further, for all , where is the bilinear form associated with . Now, using , , and , for all , we compute as follows
Let . Using , , , , and , for all , we have
The coefficient of in is zero if and only if ; that is, which implies that . Therefore, for every , such that , the degree of linearized polynomial, , in is at most ; this implies that the dimension of the kernel associated with is if is even; otherwise . The WHT of at is
Using Proposition 1, we have
In particular, if , we have if is even; otherwise for all such that and . Therefore, (20) holds for all such that and .
Using Proposition 2, we have the following. (i)When , (ii)When ,
Theorem 7. Let , for all and , where is a positive integer such that and . Then
Proof. The proof is similar to that of Theorem 6 up to (18). Here the kernel of associated with is , where is obtained by replacing , , and in (18) by , , and , respectively:
The coefficient of in is zero if and only if ; that is, . Moreover, and so, by Proposition 4, . The polynomial as represented in (25) is of the form and so, again by Proposition 4, the equation has at most roots for all such that and . This implies that if is even; otherwise . The WHT of at is
Using Proposition 1, we have
Using Proposition 2, we have the following. (i)When , (ii)When ,
Remark 8. Let be a biquadratic Boolean function. If there exists at least elements such that is quadratic, then . This result follows from Proposition 1 and the fact that the nonlinearity of any quadratic function in is at least [11, 22].
The theoretical lower bounds for third-order nonlinearities obtained by using Theorem 6 for and , are taken in such a way that and reported in Tables 1 and 2. The bounds are compared with the general bounds for third-order nonlinearity: , for any biquadratic Boolean function. It is evident that the bounds for are efficiently large and decrease with increasing the value of . It is to be noted that Class is the more general class of biquadratic monomial Boolean functions containing several classes of highly nonlinear Boolean functions. In particular, for , , and Class coincides with Kasami functions of algebraic degree .
The theoretical bounds for third-order nonlinearities obtained by using Theorem 7 and Proposition 3 are compared with known classes of functions [4, 11, 12] and reported in Tables 3 and 4. It is to be noted that the lower bounds for third-order nonlinearities of the inverse functions are larger than that of the Dillon functions for all . Thus, it is demonstrated that the lower bound obtained by Theorem 7 is better than the bounds obtained by Gode and Gangopadhyay  for Kasami functions: , Iwata and Kurosawa’s general bound  for all . Also these bounds are improved upon Carlet’s  bound for inverse function when is odd, or , and equal for the rest of values of even .
In this paper, using recursive approach introduced in , we have computed the lower bounds of third-order nonlinearities of two general classes of biquadratic monomial Boolean functions. It is demonstrated that in some cases our bounds are better than the bounds obtained previously.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author would like to thank the anonymous referees for their time, effort, and extensive comments on the revision of the paper which improve the quality of the presentation of the paper. The work is supported by Council of Scientific and Industrial Research, New Delhi, India.
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