On Boolean Functions with Low Polynomial Degree and Higher Order Sensitivity

07/23/2021 ∙ by Subhamoy Maitra, et al. ∙ Naval Postgraduate School 0

Boolean functions are important primitives in different domains of cryptology, complexity and coding theory. In this paper, we connect the tools from cryptology and complexity theory in the domain of Boolean functions with low polynomial degree and high sensitivity. It is well known that the polynomial degree of of a Boolean function and its resiliency are directly connected. Using this connection we analyze the polynomial degree-sensitivity values through the lens of resiliency, demonstrating existence and non-existence results of functions with low polynomial degree and high sensitivity on small number of variables (upto 10). In this process, borrowing an idea from complexity theory, we show that one can implement resilient Boolean functions on a large number of variables with linear size and logarithmic depth. Finally, we extend the notion of sensitivity to higher order and note that the existing construction idea of Nisan and Szegedy (1994) can provide only constant higher order sensitivity when aiming for polynomial degree of n-ω(1). In this direction, we present a construction with low (n-ω(1)) polynomial degree and super-constant ω(1) order sensitivity exploiting Maiorana-McFarland constructions, that we borrow from construction of resilient functions. The questions we raise identify novel combinatorial problems in the domain of Boolean functions.

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1 Introduction

Real polynomial degree () and sensitivity () are two central properties of Boolean functions in complexity theory, and have been studied extensively over the past three decades. These notions have important implications in the domain of query complexity [3] (and not only), where finding functions with lower polynomial degree than sensitivity generates more candidates for obtaining super-linear separation between two of the query complexity models, the classical deterministic and exact quantum models. A detailed study of these properties and the relations can be found in [3].

Determining the maximum possible separation between sensitivity and polynomial degree is an open problem that has been studied widely. Implicitly, the problem reduces to finding the separation between the number of variables and the real polynomial degree in a fully sensitive function (a function on variables with ). Informally, the sufficient and necessary condition for obtaining full sensitivity in a function on variables is to have an input point such that , where is obtained by altering the value of the -th bit of . Thus, we fix the output corresponding to some input points, of the total input points. Interestingly, this greatly restricts the real polynomial degree of the function. Without any restriction, a function that depends on all of its variables can have as low as . However, one of the seminal papers [11] in the study of Boolean functions dictate that . Furthermore, apart from the famous recursive amplification method (which is also known as the function composition), there does not exist any known method to obtain functions with non-constant separation between and . In fact, the maximum separation known between and is achieved by finding a variable function with and (due to Kushilevitz [10]) and then recursively amplifying it. This results in a function on variables with full sensitivity () and polynomial degree of so that  [11]. On the other hand, the real polynomial degree is intrinsically connected to the cryptographically important property of resiliency. Specifically if a function on variables has then the function is -resilient, where is the all variable (symmetric) linear function. In this paper we refer to and as each other’s dual. In this regard, we define the dual sensitivity () property, where a function has full dual sensitivity if and only if there exists a point such that . This results in a one-to-one connection between the - relationship and resiliency order- relationship, where is the dual of . We should remark here that the resilient Boolean functions have received a lot of interest in construction of symmetric ciphers as evident from [6, 8, 14].

In this paper, we show that the techniques from complexity theory and cryptology can supplement each other with respect to combinatorial aspects of Boolean functions which are important in their own interests, and extend the notion of sensitivity to higher order towards a better understanding how fixing of outputs with respect to flipping of more than one input bits can effect the lower bound on the polynomial degree of a function. That is, we use different properties of resiliency and polynomial degree to obtain results and constructions that apply to both paradigms of cryptology and complexity theory.

1.1 Contribution and Organization

Our paper is centered around two related concepts. In the first part we exploit the well known connection between resiliency and polynomial degree.

We observe an one to one connection between “low polynomial degree-high sensitivity” and “high resiliency-dual sensitivity” of Boolean functions, via their Fourier spectrum based definitions. We use this connection to search for separation between and for functions on all small number of variables through the resiliency approach. We find new classes of functions with maximum separation and obtain super-linear separation between and for second and third order sensitive functions. Specifically we find the following which were not known earlier.

  • There exists second order six variable functions with . That is the maximum known separation for and is same for first and second order sensitivity.

  • There does not exist any variable fully sensitive function with .

Further, We analyze the famous recursive amplification (RA) method. We show that its generalization allow us to obtain additional classes of functions with super-linear separation between and . Next, using the RA method we design efficient circuits (linear size and logarithmic depth (in ) for highly resilient functions , improving upon best known results. We further obtain bounds on cryptographic properties of these functions, such as nonlinearity and describe different trade-offs.

We use our resiliency based search method to obtain second order sensitive variable functions with . Coupled with the modified recursive amplification method that we propose in Section 3 (Further elaborated in Theorem B in Appendix B ), this gives us second order functions with variables and , matching the best known bound between and for first order sensitivity. This raises the question of whether asymptotic separation between and is a strictly decreasing function when plotted against sensitivity order.

In the second part, we take a deeper look on higher (-th) order sensitivity. Sensitivity of a function , at a point , is the number of bits of so that, upon flipping any one of them, the output of the function also flips. Extending this, a function is called -th order sensitive, if there is a point such that changing any of the input bits changes the output of the function. This is a much stronger notion than simple sensitivity, where a fully sensitive function of present literature () is a first order one. We are interested in understanding the level of restriction higher order sensitivity (dual sensitivity) has on polynomial degree (resiliency).

  • In this direction we observe for constant sensitivity order, we can design functions with sub-linear polynomial degree using the recursive amplification method. However, even for functions with , we can only have constant sensitivity order through this method.

  • Finally, we show that the Maiorana-McFarland (MM) construction can be used to show super-constant separation between and for functions with super-constant () sensitivity order. We first design a first order sensitive function with , which is worse than the recursive amplification method as mentioned above. However, by concatenating non-linear symmetric function in one of the half spaces of the MM function, we obtain the desirable trade-off. We obtain -th order sensitive functions, with . This gives us functions with and . For example, if we set , we get -order sensitive functions with polynomial degree of .

The paper is organized as follows. In Section 1.2 we recall the definitions of sensitivity and polynomial degree and the connection of polynomial degree with resiliency. Then we briefly describe the concept dual sensitivity, followed by that of higher order sensitivity and higher order dual sensitivity with the fundamental equivalences. In Section 2 we obtain the search based results on sensitivity and higher order sensitivity vs. polynomial degree. In Section 3 we study the recursive amplification method and obtain the related results. Section 4 is dedicated towards finding functions with higher order sensitivity and lower than () polynomial degree. We conclude the paper in Section 5.

1.2 Preliminaries

The definitions of resiliency and polynomial degree are based on the Fourier spectrum of a Boolean function [12]. A function has polynomial degree iff its Fourier spectrum values are for all with hamming weight of being greater than . A function is -resilient if and only if its Fourier spectrum values are for all . We call a function to be -th order resilient when it is -resilient but not -resilient. Given a function , where is the linear function on variables with obtained by flipping each bit of . These structural arguments gave rise to the famous result connecting the resiliency order and polynomial degree of Boolean functions. [[12], page 150] If a function is -th order resilient then the function will have a polynomial degree equal to , where .

Next, we study the notion of sensitivity and also, that of dual sensitivity, which we define to better understand the connection between resiliency and polynomial degree.

1.3 Sensitivity

Sensitivity is one of the most studied properties of Boolean function. For any , we let to be with the -th bit of flipped (complemented). The sensitivity of a Boolean function at a point can be defined as , and the sensitivity of a function is It is natural to consider the situation where we want the function to have the same value even if multiple input bits of are flipped regardless of their position. In this direction, we define the -th order sensitivity of a Boolean function.

-th order sensitivity: For any set and the input point we define as the input point obtained by flipping the -th bit of for all . Sensitivity is defined around the notion of flipping any single component corresponding to a given input where the output of the function remains unchanged. In this regard we define -th order sensitivity of a function in the following manner.

We call a function -th order sensitive if there exists such that

That is, is -th order sensitive if there exists an input so that flipping any of the component bits of the input, changes the function’s output. Thus a first order sensitive function is simply a function with . The main implication of -th order sensitivity is that it indeed further restricts how low the degree of the real polynomial corresponding to the function can be. Without any restrictions we know can be as low as for functions that depend on variables. If we fix then the polynomial degree is . The paper is centered around obtaining functions with -order sensitivity and -polynomial degree.

1.4 Dual sensitivity

Given a function on variables with polynomial degree , we call the function , the dual of , which has resiliency. The dual sensitivity of a function at a point is defined as . The dual sensitivity of is This notion can be extended to -th order dual sensitivity in the following manner.

We say a function is -th order dual sensitive if there exists , such that, for all we have:

  • If , then .

  • If , then .

That is, a function is

-th order dual sensitive if there is an input point such that if we flip the values of any odd number

of input bits then the function’s output remains unchanged and if we flip any even number of input bits then the function’s output gets complemented.

A function on variables is -th order sensitive if and only if its dual is -th order dual sensitive.

We use the following notations:

  • An -function is a Boolean function of variables that is -th order sensitive and has real polynomial degree at most .

  • An -function is a Boolean function of variables that is -th order dual sensitive and -resilient.

Thus we have the following proposition. Thus, if is an -function then is an -function. Let us now move onto the search based results.

2 Search On Small Variables

As we shall observe in Section 3, upon some modification, the recursive amplification method can be used to obtain -th order sensitive functions on variables with polynomial degree of , starting from a function on variables and . Here is called the base function. Thus results of low of -th order sensitive functions on small variables directly generate super-linear separations between and for -th order sensitive functions.

For example, if we obtain a fully sensitive (first oder sensitive) function on variables with , or a variable first order sensitive function with polynomial degree , then it would improve upon the best known separation between and . In this direction, the functions on up to variables can be exhaustively searched to obtain all existing combinations. However, for functions on and more variables, an exhaustive search is not possible given the size of search space ( for , for and so on), and we instead use the properties of resiliency and dual sensitivity to completely exhaust the case of fully sensitive functions for and in terms of obtaining all functions and proving non existence respectively.

2.1 -functions:

It can be checked with a simple search that there does not exist any fully sensitive function on variables with . In fact, if that would have been the case then we could use the recursive amplification method to obtain a function on variables with and , which would give us quadratic separation between sensitivity and polynomial degree, demonstrating a tight lower bound. Thus we look into the functions, which are duals of functions. We have the following counts.

  • There are approximately many -functions.

  • Only of these functions are -functions and there are no -functions.

2.2 -functions:

In the case of variable functions, the possible polynomial degree is between and . As we have already observed the case of -resiliency ( polynomial degree in the dual) for and , we compute the -resilient functions in this case, and get the following counts:

Finally let us look into the case of variable functions, for which we have the best base function for first order sensitivity, which is the Kushilevitz function.

2.3 -functions:

There are total Boolean functions on variables, and checking the resiliency and sensitivity of all possible functions requires computational resources that is unattainable. We instead use properties of dual sensitivity and resiliency to obtain all possible -functions by concatenating the truth tables of two variable functions. Any variable function can be written as Where and are functions on variables. Then we have the following constraints on the properties of .

  1. If is -resilient then either both and are -resilient or both are -resilient [8].

  2. If is fully dual sensitive then at least one of and are fully dual sensitive. This is easy to see as if neither nor are dual sensitive then there is no input point for which the whole function can have full dual sensitivity.

Now we have only many -functions and many -functions, which reduces the effective search space to approximately from the naive . Using these constraints we get the full characterization of -functions, which was not previously reported.

  • We find that there are many -functions. Here it should be noted that the dual of any such function is a -function. We can use the modified recursive amplification technique of Theorem B on all such functions to obtain -functions, which gives us the best known separation between sensitivity and polynomial degree, same as the function by Kushilevitz [10].

  • We also get many -functions, and this gives us the maximum super-linear separation between number of variables and real polynomial degree in second order sensitive functions, which is , which is also the currently best known separation for first order sensitivity.

    Furthermore, there is no -functions.

2.4 Nonexistence of -functions and Searching the Rotation Symmetric Functions

The existence or non-existence of a -function is central to understanding the maximum separation between and . If there does exist a -function then we can obtain a -function using the recursive amplification method, which gives , improving on the best known result. However the total number of functions on variables is

and therefore checking all functions for this profile through brute force is not computationally possible. Against this background we use a mixed integer linear program (MILP) to investigate the existence of such a function. If

is a variable Boolean function with and

iff there is a vector

such that for every and for every with . For every . We run the MILP and it returns no solution for all choice of . This shows there are no -functions. One can refer to Appendix A.1 for a formal description of the constraints.

Even with our strategy, it not possible to search for all fully sensitive and higher order sensitive functions on more than variables because of the size of the search space. In this regard we search and variable rotation symmetric functions, which is another cryptographically important class of functions to obtain with fully sensitive (first order sensitive functions) using least possible polynomial degree (maximum resiliency in the dual function). Our findings can be found in Appendix A.2. Let us now proceed towards the recursive amplification method.

3 The Recursive Amplification Method

We have noted that fixing the value of a function corresponding to input points to make a function fully sensitive, will restrict the polynomial degree to . The best known results in this paradigm is derived through the recursive amplification method, which is also the function composition method. This is a well known technique that is used to obtain super-linear separation between and and is also used to obtain super-linear separation between and . In this section we use this technique and obtain the following results:

  1. A slight modification of the recursive amplification method to obtain super-linear separation between and by starting from any candidate base function.

  2. We build highly resilient functions with good nonlinearity, circuit size and circuit depth.

  3. We obtain super-linear separation between number of variables() and polynomial degree () for functions with constant order sensitivity.

Recursive amplification was used to obtain the largest known separation between sensitivity and polynomial degree of Boolean functions [3], as well as the first example of separation between exact quantum query complexity and deterministic query complexity [1], among other separation results. Let be a function on variables with polynomial degree . Then the recursive amplification method generates the function on variables as:

  1. .

  2. .

Then for any we have . Thus if the sensitivity also gets amplified, we could start with any variable function with and obtain with super-linear whenever . However, sensitivity is not always amplified in the similar manner, and can be arbitrarily low. To this end we propose a construction so that we can get super-linear separation between and starting from any function. Furthermore the results also follow for higher-order sensitivity. Let be obtained by concatenating copies of . Then we define the amplification method w.r.t a base function on variables as . Then is a -th order sensitive function on variables and , with -th order sensitivity achieved at the input point . One can refer to Theorem B in Appendix B for the formal representation and proof.

Now we look into the functions of variables and then discuss how the recursive amplification method can be used to obtain highly resilient functions with good nonlinearity.

3.1 Low cost resilient functions with recursive amplification

Let us consider a function on variables with , where is a constant. Then we can recursively amplify the function to obtain a function on variables with . Now if we add the all variables linear function to it we get an variable function with resiliency. However, there already exists many methods of obtaining Boolean functions with high resiliency and other cryptographically important properties such as high nonlinearity.

Here the advantage of the recursive amplification method is the circuit size for building such functions. Building efficient low depth circuits for cryptographically important functions with large number of input variables is a challenging problem. In this regard the work by Sarkar et al. [15, 2003] is important. This work shows how to start with an -resilient function on some variables and generate an -resilient function on variables that requires depth, which is effectively as is constant for any given construction. In fact, this has been the best known result in this direction for almost two decades in building efficient circuits for resilient functions on large variables starting from base functions. Improving on this, we have the following result.

Result 1.

Given a function on variables with we can obtain a function on on variables with resiliency such that there is a circuit of linear size and logarithmic depth (in ) for it. Here is the dual of where is the function on variables obtained by recursively amplifying .

The proof can be found in Theorem B in the appendix, followed by Figure 1 that gives an example of building a variable function using instances of the circuit corresponding to a -variable function . We refer to Appendix C for elaborate discussion on the nonlinearity lower bounds we have derived for these highly resilient functions, along with algebraic degree-resiliency trade-offs. Finally, we show that we can obtain super-linear separation between and for functions with any constant order sensitivity. This raises the interesting problem of understanding the nature maximum super-linear separation possible between and with increasing, constant order sensitivity . Specifically we have the following result.

Result 2.

Given any constant there exists a -th order sensitive function on variables such that , if is even and , if is odd.

One may refer to Section C.1 for the detailed formal explanation.

4 Higher Order Sensitivity

Until now we have discussed functions with a constant higher order sensitivity, and have found classes of functions defined on the number of variables for which we could obtain super-linear separation between and using the recursive amplification method. However, we cannot obtain any -order sensitive (or dual sensitive) function, where is an increasing function on using any recursive amplification process, whenever we intend the function to have less than polynomial degree.

The general recursive amplification process cannot obtain a function that has super-constant order of sensitivity where the polynomial degree of less than the number of variables, where the recursive amplification process is defined as

  • A base function on some variables.

  • where .

Proof.

Let us consider any function on variables and -order sensitivity where . Then is a function on variables and there exists such that . However it is easy to see (via induction) that any function built with a base function on variables and sensitivity order less than cannot be -th order sensitive. ∎

Thus the recursive amplification process does not help us anymore when we consider -order sensitive functions. In this regard we next explore the class of Maiorana-McFarland (MM) constructions, a heavily studied class for cryptographic and coding theoretic purposes.

Here, we use it from the perspective of polynomial degree-sensitivity to obtain results in the domain of functions with non-constant order of sensitivity. The layout of this section is as follows. We first discuss the Maiorana-McFarland construction and then obtain separation between and while making the function first order sensitive (fully sensitive). We extend this construction while discussing first order sensitivity only for ease of understanding. We analyze the polynomial structure of these functions and obtain logarithmic separation between and . Finally we show that this construction can be modified for super-constant orders of sensitivity (upto ) with only a few tweaks.

4.1 Maiorana-McFarland construction

The Maiorana-McFarland (MM) construction [4] is based on dividing the input variable space into two parts and attaching different linear functions from one subspace to each point in the other subspace, defined as follows.

A function is called an MM function if it can be expressed as , where

  • , ,

  • is a mapping of the form

  • is an arbitrary Boolean function defined on the subspace .

The Boolean functions due to Maiorana-McFarland construction can be visualized in different ways. We view them as different linear functions defined on attached to activating values in . Let there be an MM Boolean function with any arbitrary map and some Boolean function . Corresponding to any , the quantity is essentially the outcome of the linear equation . Thus equals for all in . Let us denote this function defined on as . We now describe two real polynomial structures.

  1. is defined as , so that

  2. corresponding to each . Any linear function on can be expressed as . Then the .

Then we have the following real polynomial w.r.t to any MM type function.

Given an MM function on variables with and , the corresponding real polynomial can be defined as

(1)

Let us now note down a simple result that this polynomial structure entails.

For any we have and .

The structure of the rest of this section is as follows. First we describe some sufficient condition that allows a MM type function to have . Next we obtain a MM type function with polynomial degree and then extend this technique to obtain a function with . Finally we extend this notion to higher order sensitivity by adding non-linear functions on , which is one of the main results of the paper.

We first ensure . We add two restriction to a MM type function.

  1. and .

  2. and .

Then for all such functions we have . We denote this class of MM functions as . Refer to the formal presentation in Lemma D.1 in Appendix D.1. Now we describe the construction for the first separation.

4.2 First Order Sensitive Functions With Lower Polynomial Degree

First we show a construction for getting .

Construction 1.

Let be an MM function defined on variables so that with . If is defined using , , the sensitivity of is and the polynomial degree is at most . Refer to Appendix D.2 for the proof, along with examples and a count on the number of such functions.

Let us now better understand how the polynomial structure of the MM type functions can be modified so that the modified polynomial still represents a Boolean function, but with lower polynomial degree.

4.3 Interpreting real polynomial terms via the MM construction

We saw in Proposition 4.1 that the real polynomial corresponding to any MM type function can be expressed as We know that .

If , it is easy to see Corresponding to a Boolean function defined on variables , we define three non-empty mutually disjoint sets and such that with . Let the variables indexed by elements in be denoted as and be represented as . Then the real polynomial can be represented as that is, . This implies that corresponding to an MM type function defined on variables, the polynomial can be interpreted as

We represent it as Thus, we get

(2)

These considerations imply the following result. Let be an MM function such that if then and . Then the polynomial corresponding to the Boolean function can be written as and the polynomial can be written as , where

and this represents another MM type function which differs from only in the points . Using this result we can attempt to obtain an MM type Boolean function with a pre-decided real polynomial structure. We start with the real polynomial of a particular MM type function, and then modifying its corresponding polynomial by adding keeping in mind the respective necessary constraints we have discussed in terms of and . This gives us another function whose structure and its properties can be recovered from . Using this combinatorial approach we next have the following result.

Result 3.

There exists a Boolean function with .

One can refer to Appendix D.3 for the buildup, along with the proof. Finally we extend our constructions and results for super-constant orders of sensitivity.

4.4 Extending to super-constant higher order sensitivity via the MM construction

We have so far observed the situation where we have defined a function on variables in the MM class as where and with . The simplest interpretation is choosing a linear function in (or its complement depending on ) corresponding to each point . We can extend this to nonlinear functions in being fixed with respect to the points in , with being the non-linear function on to be evaluated when . Then the real polynomial corresponding to the function can be written as where is the real polynomial corresponding to the function . Now let us discuss some sufficient conditions to obtain -th order sensitivity by choosing the proper functions.

4.5 Obtaining -th order sensitivity an reducing polynomial degree:

We start by defining a function in that is -th order sensitive itself. We define this function as . The algebraic normal form of the function contains all degree monomials. For an example . Next we observe the sensitivity order of this function.

The function is a function defined on variables. -th order sensitive around the all zero input point . The proof can be found in Section E.1. Next we define an MM type function with nonlinear functions in , which is -th order sensitive.

Construction 2.

Any function with the algebraic normal form where and for all is -th order sensitive.

Finally we extend the technique of Section D.3 to obtain non-constant separation between number of variables and real polynomial degree in functions with super-constant order of sensitivity.

Construction 3.

There exists a -th order sensitive function in with real polynomial degree.

The formal proofs of Constructions 2 and 3 can be found in Appendix E.2. To reflect on the implications of this result, we can have -order sensitive functions with . In fact as long as , we have . These results could not be achieved via the recursive amplification constructions.

5 Conclusion

In this paper we have studied the interplay of resiliency and polynomial with respect to sensitivity, and have also extend the notion of sensitivity to higher order sensitivity. In this direction based on properties of resilient functions, we have obtained new classes of -variable first order sensitive functions with , while also obtaining the same result for second order sensitivity. Which indicates that the function of minimum polynomial degree vs. sensitivity order and may not be a strictly increasing functions.

Next we have studied the recursive amplification method and have designed slight modifications that allow us to start with base function, removing the restrictions of the simple function composition method. Furthermore, we use the resiliency-polynomial degree connection to design efficient circuits with linear size and logarithmic depth to realize highly-resilient () functions. Our result improves on the best result known in this domain.

Finally we observe that for constant orders of sensitivity, we can have functions with polynomial degree using the recursive amplification method. Against this backdrop we take the MM constructions and first obtain first order sensitive function with . Then we modify the MM construction with nonlinear function concatenation, and obtain functions with polynomial degree and order sensitivity. Specifically, we show construction of -th order sensitive function with . Our results enrich the domain of cryptographically important Boolean functions as long as lay down important combinatorial problems that should further enhance our understanding of real polynomial degree of Boolean functions.

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Appendix A Notes on Search Based Results

a.1 Nonexistence of variable fully sensitive function with Polynomial Degree Value of

We know that, for any -variable Boolean function , for every with if and only if . Thus, is an -variable Boolean function with full sensitivity and polynomial degree if and only if is an -variable Boolean function with full sensitivity and polynomial degree . Therefore, if there are -variable Boolean functions with sensitivity and polynomial degree , there exist a -variable Boolean function with sensitivity , polynomial degree , and a point such that and for all where ’s are pairwise distinct vectors with Hamming weight . Assume that , where is the binary expansion of , and when