Jong Yoon Hyun

MacWilliams duality and a Gleason-type theorem on self-dual bent functions

We prove that the MacWilliams duality holds for bent functions. It enables us to derive the concept of formally self-dual Boolean functions with respect to their near weight enumerators. By using this concept, we prove the Gleason-type theorem on self-dual bent functions. As an application, we provide the total number of (self-dual) bent functions in two and four variables obtaining from formally self-dual Boolean functions.

Explicit Criteria for Construction of Plateaued Functions

Plateaued functions are very important cryptographic functions due to their desirable cryptographic characteristics. We find explicit criteria for the construction of p-ary r-plateaued functions with an odd prime p. We point out that 0-plateaued functions are bent functions, and so plateaued functions generalize the notion of bent functions. We first derive an explicit form for the Walsh-Hadamard transform of a p-ary r-plateaued function. We then obtain an upper bound on the degree of p-ary r-plateaued functions, and we classify p-ary (n - 1)-plateaued functions in n variables. We also obtain explicit criteria for the existence of p-ary r-plateaued functions. Accordingly, these results lead to improved bounds on the existence of p-ary bent functions.

Necessary Conditions for the Existence of Regular

We find some necessary conditions for the existence of regular p-ary bent functions (from Zⁿp to Zp), where p is a prime. In more detail, we show that there is no regular p-ary bent function f in n variables with w(M_f) larger than n/2, and for a given nonnegative integer k, there is no regular p-ary bent function f in n variables with w(M_f)=n/2-k ( n+3/2-k, respectively) for an even n ≥ N_p,k (an odd n ≥ N_p,k, respectively), where N_p,k is some positive integer, which is explicitly determined and the w(M_f) of a p-ary function f is some value related to the power of each monomial of f. For the proof of our main results, we use some properties of regular p-ary bent functions, such as the MacWilliams duality, which is proved to hold for regular p-ary bent functions in this paper.

p

In this paper, we derive the relationship between local weight enumerator of q-ary 1-perfect code in a face and that in the orthogonal face. As an application of our result, we compute the local weight enumerators of a shortened, doubly-shortened, and triply-shortened q-ary 1-perfect code.

-Ary Bent Functions

We denote Q_n the set of binary words of length n. A partition {C₁,C₂,_¿,C_r} of Q_n with quotient matrix B = (b_ij)_r×r is equitable if for all i and j, any word in d has exactly bij neighbors in C_j. The equitable partitions of Q_n can be obtained from completely regular codes. We derive a bound on equitable partitions of Q_n that does not depend on the size of the partition.

Local duality theorem for q-ary 1-perfect codes

We introduce a new class of Boolean functions for which the MacWilliams duality holds, called MacWilliams-dual functions, by considering a dual notion on Boolean functions. By using the MacWilliams duality, we prove the Gleason-type theorem on MacWilliams-dual functions. We show that a collection of MacWilliams-dual functions contains all the bent functions and all formally self-dual functions. We also obtain the Pless power moments for MacWilliams-dual functions. Furthermore, as an application, we prove the nonexistence of bent functions in 2n variables with minimum degree n−k for any nonnegative integer k and n ≥ N with some positive integer N under a certain condition.

A Bound on Equitable Partitions of the Hamming Space

The class of plateaued functions (or r-plateaued functions) are Boolean functions with many cryptographically desirable properties, and this class of plateaued functions include bent functions. In fact, bent functions are exactly 0-plateaued functions. There are some results on the nonexistence of homogeneous 0-plateaued functions in n variables by Xia et al., Meng et al., and by the authors. In this paper we present a result on the nonexistence of r-plateaued functions in n variables (0=N and r, we prove the nonexistence of r-plateaued functions with certain degrees, where N is some integer depending on r.

Boolean functions with MacWilliams duality

We prove that MDS linear poset-codes satisfy Gilbert-Varshamov bound for their Hamming weights asymptotically. We also construct MDS linear poset-codes on arbitrary poset-metric spaces by using the Dilworths chain decomposition theorem and results about the Hermite interpolation problem over a finite field. We prove that there exist linear poset-codes with large weights for both poset-metrics and Hamming metrics, as well.

Nonexistence of certain types of plateaued functions

In this paper, we determine explicitly the zeta polynomials of near-MDS codes and obtain necessary and sufficient conditions for near-MDS codes to satisfy the Riemann hypothesis analogue.