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COVERING RADIUS OF THE (

*N − *3)-RD ORDER
REED-MULLER CODE IN THE SET OF RESILIENT FUNCTIONS
Yuri BorissovInstitute of Mathematics and Informatics,Bulgarian Academy of Sciences,8 G.Bonchev, 1113 Sofia, Bulgariayborisov@moi.math.bas.bgAn Braeken, Svetla NikovaDepartment Electrical Engineering, ESAT/COSIC,Katholieke Universiteit Leuven, Kasteelpark Arenberg 10,B-3001 Heverlee-Leuven, Belgiuman.braeken,svetla.nikova@kuleuven.ac.be
In an important class of stream ciphers, called combination generators, the
key stream is produced by combining the outputs of several independent LinearFeedback Shift Register (LFSR) sequences with a nonlinear Boolean function.

Siegenthaler [12] was the first to point out that the combining function shouldpossess certain properties in order to resist divide-and-conquer attacks.A Booleanfunction to be used in the combination generator (or more general also in streamciphers) should satisfy several properties.

*Balancedness *– the Boolean functionhas to output zeros and ones with equal probabilities.

*High nonlinearity *- theBoolean function has to be at sufficiently high distance from any affine func-tion.

*Correlation-immunity *(of order

*t*) - the output of the function should bestatistically independent of the combination of any

*t *of its inputs. A balancedcorrelation-immune function is called

*resilient*.

Besides the divide-and-conquer attacks, another important class of attacks
on combination generators are the algebraic attacks [4, 5]. The central idea in thealgebraic attacks is to use a lower degree approximation of the combining Booleanfunction and then to solve an over-defined system of nonlinear multivariate equa-tions of low degree by efficient methods such as XL or simple linearization [3]. Inorder to resist these attacks, the Boolean function should have not only a a highalgebraic degree but also a high distance to lower order degree functions. Thetrade-off between resiliency and algebraic degree is well-known. To achieve the
desired trade-off designers typically fix one or two parameters and try to optimizethe others.

In this paper, we investigate the generalization of the trade-off between re-
siliency and algebraic degree. In particular, we study the relation between re-siliency and distance to lower order degree functions. In order to define a the-oretic model for combining these properties, Kurosawa

*et al. *[6] have intro-duced a new covering radius ˆ(

*t, r, n*), which measures the maximum distancebetween

*t*-resilient functions and

*r*-th degree functions or the

*r*-th order Reed-Muller code

*RM *(

*r, n*). That is ˆ(

*t, r, n*) = max

*d*(

*f *(

*x*)

*, RM*(

*r, n*)), where themaximum is taken over the set

*Rt,n *of

*t*-resilient Boolean functions of

*n *vari-ables. Note that as the covering radius of Reed-Muller codes is defined by
(

*r, n*) = max

*d*(

*f, RM*(

*r, n*)) where the maximum is taken over all Boolean func-
tions

*f *, it holds that 0

*≤ *ˆ(

*t, r, n*)

*≤ *(

*r, n*). Kurosawa

*et al. *also provide a tablewith certain lower and upper bounds for ˆ(

*t, r, n*). In [1] some exact values andnew bounds for the covering radius of the second order Reed-Muller codes in theset of resilient functions were found.

In this paper we find the exact value of the covering radius of

*RM *(

*n − *3

*, n*)
in the set of 1-resilient Boolean functions of

*n *variables, when

*n/*2 = 1mod 2.

We also improve the lower bounds for covering radius of the Reed-Muller codes

*RM*(

*r, n*) in the set of

*t*-resilient functions, where

*r/*2 = 0mod 2,

*t ≤ n − r − *2and

*n ≥ r *+ 3. We start with some background on Boolean functions.

Any Boolean function

*f *(

*x*) on F

*n *can be uniquely expressed in the algebraic

*hf *(

*a*1

*, . . . , an*)

*xa*1

*· · · xan,*
(

*a*1

*,.,an*)

*∈*F

*n*
with

*hf *a function on F

*n*, defined by

*h*
*f *(

*x*) for any

*a ∈ *F

*n*, where

*x ≤ a *means that

*xi ≤ ai *for all

*i ∈ {*1

*, . . . , n}*. The algebraic degree of

*f*, denotedby deg(

*f *) or shortly

*d*, is defined as the number of variables in the highest term

*xa*1

*· · · xan *in the ANF of

*f *for which

*h*
*f *(

*a*1

*, . . . , an*) = 0. The suport of

*f *, denoted
by

*sup*(

*f *), is the set of all vectors

*x *for which

*f *(

*x*) = 0. The Walsh transform of

*f *(

*x*) is a real-valued function over F

*n *that is defined as
(

*−*1)

*f*(

*x*)+

*x·ω,*
where

*x · w *denotes the dot product of the vectors

*x *and

*w*, i.e.,

*x · w *=

*x*1

*w*1 +

*· · · *+

*xnwn*.

Definition 1

*A function f *(

*x*)

*is called t-th order correlation-immune if its Walshtransform satisfies Wf *(

*ω*) = 0

*, for *1

*≤ wt*(

*ω*)

*≤ t, where wt*(

*x*)

*denotes theHamming weight of x. Balanced t-th order correlation-immune functions arecalled t-resilient functions, i.e. Wf *(

*ω*) = 0

*, for *0

*≤ wt*(

*ω*)

*≤ t.*
By the well-known

*Siegenthaler’s inequality *[11] the maximal possible alge-
braic degree of

*t*-resilient function

*f *of

*n *variables is equal to

*n − t − *1 when

*t < n − *1. The problem for constructing resilient functions (in particular suchof maximal possible degree) attracted the attention of many authors in the past.

Among other works we mention [11], [2] and [10]. The next theorem shows howwe can easily construct (

*t *+ 1)-resilient function on F

*n*+1 from

*t*-resilient function
Lemma 2

*[2] Let xn*+1

*be a linear variable, i.e., f*(

*x*1

*, . . . , xn, xn*+1) =

*g*(

*x*1

*, . . . , xn*)+

*xn*+1

*, where g*(

*x*1

*, . . . , xn*)

*is t-resilient. Then f*(

*x*1

*, . . . , xn, xn*+1)

*is *(

*t *+ 1)

*-resilient.*
We also make use of the following theorem:
Theorem 3

*[7] The covering radius of RM *(

*n − *3

*, n*)

*is equal to n *+ 2

*if n iseven. If n is odd, the covering radius is equal to n *+ 1

*.*
To prove the theorem, McLoughlin constructed a coset for which the minimalweight is equal to

*n *+ 2 when

*n *is even, and

*n *+ 1 when

*n *is odd. This cosetcontains

*σn−*2, the symmetric polynomial consisting of all terms of degree

*n − *2.

THE COVERING RADIUS OF (

*N − *3)-RD REED-MULLER CODES IN THESET OF 1-RESILIENT BOOLEAN FUNCTIONS
In order to prove the main theorem of this paper we will need the following
Lemma 4

*Let σi*(

*x*)

*be the symmetric polynomial of n variables containing allterms of degree i (σ*0(

*x*) = 1

*) and S*(

*x*) =
0

*, n − *1

*, n when n is even;*
*v ∈ sup*(

*S*)

*if and only if wt*(

*v*) =

*Proof. *Let

*v ∈ *F

*n *be a vector of weight

*w*. It is easy to see that the number of
terms in

*σi*(

*v*) equal to 1 is

*w *(as usual

*w *= 0, when

*w < i*). Therefore the
number of terms in

*S*(

*v*) that are equal to 1 is

*N*(

*w*) =

*N*(

*w*) mod 2. There are four cases to be considered:
1. If

*w *= 0, then

*S*(0) = 1;
2. If 0

*< w < n − *1, then

*N*(

*w*) = 2

*w *and thus

*S*(

*v*) =

*N*(

*w*) mod 2 = 0;
3. If

*w *=

*n − *1, we have

*N*(

*n − *1) =
= 2

*n−*1

*− *1 and therefore
4. If

*w *=

*n*, we have

*N*(

*n*) =

*n−*2

*n *= 2

*n − *(

*n *+ 1). Therefore
Lemma 5

*Let S*(

*x*)

*be the symmetric Boolean function of n variables, defined inLemma 4, where n is equal to *4

*k *+ 2

*or equal to *4

*k *+ 3

*. Let v be an arbitraryvector of weight *2

*k *+ 1

*or of weight *2

*k *+ 2

*. Then the Walsh transform valueWS*(

*v*) = 0

*.*
*Proof. *Let us consider the following two linear functions:

*L*1(

*x*) =

*i*. Arranging the set

*sup*(

*S*) in decreasing lexicographic order,
it is easy to see that

*Lj *= 0

*, j *= 1

*, *2 for the half of the vectors from

*sup*(

*S*).

Since the linear functions are balanced the same is true for the complement set of

*sup*(

*S*), in which

*S *takes value 0. Therefore

*L*1 and

*L*2 differ from

*S *in 2

*n−*1 pointsi.e.

*d*(

*Lj, S*) = 2

*n−*1

*, j *= 1

*, *2. By using the relation

*Wf *(

*ω*) = 2

*n − *2

*d*(

*ω, x , f*)we get

*WS*(

*v*) = 0 where

*v *is either the vector having only ones in the first 2

*k *+ 1or in the first 2

*k *+ 2 coordinates. Since

*S*(

*x*) is a symmetric function this holdsfor any vector of weight 2

*k *+ 1 or 2

*k *+ 2.

Let

*T *be a subset of F

*n*. The rank of

*T *, denoted by

*rank*(

*T *), is defined as
the maximal number of linearly independent elements from

*T *.

Lemma 6

*Let n be equal to *4

*k *+ 2

*or equal to *4

*k *+ 3

*and Z *=

*{v ∈ *F

*n *:

*wt*(

*v*) =
2

*k *+ 1

*or *2

*k *+ 2

*}. Denote by v*1

*the vector *(1

*, *1

*, *1

*, .*1

*, *0

*, *0

*, *0

*, .*0)

*of weight *2

*k *+ 1

*.*

Then the set Z +

*v*1

*has rank n.*
*Proof. *Note that the following vectors of weight 2
(1

*, *0

*, *0

*, ., *0

*, *1

*, *0

*, .*0)

*, *(0

*, *1

*, *0

*, ., *0

*, *1

*, *0

*.*0)

*, . . . , *(0

*, *0

*, *0

*, ., *1

*, *1

*, *0

*.*0)

*,*
where the second “1” is in the (2

*k *+ 2)-nd position, belong to

*Z *+

*v*1. The sameis valid for the vectors having only one “1” in positions 2

*k *+ 2 till

*n*. Obviously,these are

*n *linearly independent vectors and the proof is complete.

Theorem 7

*The covering radius of RM(n-3,n) in the set of 1-resilient Booleanfunctions of n variables is equal to:*
*Proof. *By the result of McLoughlin [7] (see Theorem 3), the Boolean function

*S*(

*x*) defined in Lemma 4, belongs to the coset of

*RM *(

*n − *3

*, n*) with a maximalpossible minimal weight. By Lemma 5 and Lemma 6 and using the procedure for“change the basis” described by Maitra and Pasalic [9] the function

*S*(

*x*) is affinereducible to 1-resilient function.

Finally, let us consider the case

*n *= 4. It is easy to see that

*σ*2 is affine
equivalent to some function in the coset of

*RM*(1

*, *4) containing the function

*f *=

*x*1

*x*2 +

*x*3

*x*4. However

*f *is a bent function and therefore the coset

*σ*2 +

*RM*(1

*, *4)contains no balanced functions. By Dickson [8] theorem the remaining two typesof cosets (which are interesting when consider 1

*−*resilient functions of 4 variables),are RM(1,4) itself and these equivalent to

*x*1

*x*2 +

*RM*(1

*, *4). In fact the function

*g *=

*x*1

*x*2 +

*x*3 +

*x*4 is 1-resilient and the minimal weight of its coset is 4. Hencethe covering radius of interest is 4 (see also numerical results in [6]).

DERIVING NEW LOWER BOUNDS ON THE COVERING RADIUS OF REED-MULLER CODE IN THE SET OF RESILIENT FUNCTIONS
By induction, using Theorem 3 and Theorem 7, we can also generalize the
lower bounds for

*RM *(

*r, n*) in the set of

*t*-resilient functions where

*r/*2 =0 mod 2,

*t ≤ n − r − *2 and

*n ≥ r *+ 3.

Theorem 8

*The covering radius of the Reed-Muller code RM *(

*r, n*)

*in the setRt,n for r/*2 = 0 mod 2

*, t ≤ n − r − *2

*and n ≥ r *+ 3

*is bounded from below by*2

*n−*3

*.*
In particular, for

*r *= 3 and

*r *= 4, this leads to the following lower bound:
Corollary 9

*The covering radius of the Reed-Muller code RM *(3

*, n*)

*in the setRt,n for t ≤ n − *5

*is bounded from below by *2

*n−*3

*, when n ≥ *6

*. The coveringradius of the Reed-Muller code RM*(4

*, n*)

*in the set Rt,n for t ≤ n − *6

*is boundedfrom below by *2

*n−*3

*, when n ≥ *7

*, i.e.*
ˆ(

*t, *3

*, n*)

*≥ *2

*n−*3
ˆ(

*t, *4

*, n*)

*≥ *2

*n−*3

*for t ≤ n − *6

*, n ≥ *7

*.*
In this paper, we continued the study of the covering radius in the set of
resilient functions, which has been defined by Kurosawa

*et al. *[6]. This newconcept is meaningful to cryptography especially in the context of the new classof algebraic attacks on stream ciphers proposed by Courtois and Meier at Euro-crypt 2003 [4] and Courtois at Crypto 2003 [5]. In order to resist such attacksthe combining Boolean function should be at high distance from lower degreefunctions.

Using a result from coding theory on the covering radius of (

*n − *3)-rd Reed-
Muller codes, we establish exact values of the the covering radius of

*RM *(

*n − *3

*, n*)in the set of 1-resilient Boolean functions of

*n *variables, when

*n/*2 = 1mod 2.

We also improve the lower bounds for covering radius of the Reed-Muller codes

*RM*(

*r, n*) in the set of

*t*-resilient functions, where

*r/*2 = 0mod 2,

*t ≤ n − r − *2and

*n ≥ r *+ 3.

In the table below we present the improved numerical values of the covering
radius for resilient functions. The entry

*α − β *means that

*α ≤ *ˆ(

*t, r, n*)

*≤ β*.

Table 1: Numerical data of the bounds on ˆ(

*t, r, n*)
[1] Y. Borissov, A. Braeken, S. Nikova, B. Preneel, On the Covering Radius of
Second Order Binary Reed-Muller Code in the Set of Resilient Boolean Func-tions, IMA International Conference on Cryptography and Coding, Springer-Verlag LNCS 2898, 2003, pp. 82-92.

[2] P. Camion, C. Carlet, P. Charpin, N. Sendrier, On Correlation Immune
Functions,

*CRYPTO’91*, LNCS 576, Springer-Verlag 1991, pp. 87-100.

[3] N. Courtois, A. Klimov, J. Patarin, A. Shamir, Efficient Algorithms for
Solving Overdefined Systems of Multivariate Polynomial Equations,

*Euro-crypt’00*, LNCS 1807, Springer-Verlag, 2000, pp. 392-407.

[4] N. Courtois, W. Meier, Algebraic Attacks on Stream Ciphers with Linear
Feedback,

*Eurocrypt’03*, LNCS 2656, Springer-Verlag 2003, pp. 345-359.

[5] N. Courtois, Fast Algebraic Attacks on Stream Ciphers with Linear Feedback

*Crypto’03*, LNCS 2729, Springer-Verlag 2003, pp. 176-194.

[6] K. Kurosawa, T. Iwata, T. Yoshiwara, New Covering Radius of Reed-Muller
Codes for

*t*-Resilient Functions,

*SAC’01*, LNCS 2259, Springer-Verlag 2001,pp. 75-86.

[7] A. McLoughlin, The Covering Radius of the (

*m − *3)

*−*rd Order Reed-Muller
Codes and a Lower Bound on the (

*m − *4)

*−*th Order Reed-Muller Codes,

*SIAM J. Appl. Mathematics*, vol. 37, No. 2, October 1979, pp. 419-422.

[8] F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-Correcting Codes,
North-Holland Publishing Company 1977.

[9] S. Maitra, E. Pasalic, Further Constructions of Resilient Boolean Functions
with Very High Nonlinearity,

*IEEE Transactions on Information Theory*,vol. 48, No.7, July 2002, pp. 1825-1834.

[10] J. Seberry, J. Zhang, Y. Zheng, On Constructions and Nonlinearity of Cor-
relation Immune Functions,

*Eurocrypt’93*, LNCS 765, Springer-Verlag 1994,pp. 181-199.

[11] T. Siegenthaler, Correlation-Immunity of Non-linear Combining Functions
for Cryptographic Applications,

*IEEE IT*, vol. 30, No. 5, 1984, pp. 776-780.

[12] T. Siegenthaler, Decrypting a Class of Stream Ciphers Using Ciphertext
Only,

*IEEE Trans. Comp.*, vol 34, No. 1, 1985, pp. 81-85.

[13] Y. Tarannikov, On Resilient Functions with Maximun Possible Nonlinearity,

*Indocrypt 2000*, LNCS 1977, pp. 19-30.

Source: https://eprint.iacr.org/2004/202.pdf

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