Partial difference sets and amorphic Cayley schemes in non‐abelian 2‐groups

In this paper, we consider regular automorphism groups of graphs in the RT2 family and the Davis‐Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non‐abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT2 family and Davis‐Xiang family and their amorphic abelian Cayley schemes to produce amorphic non‐abelian Cayley schemes.     

Vectorial bent functions and partial difference sets

Designs, Codes and Cryptography - The objective of this article is to broaden the understanding of the connections between bent functions and partial difference sets. Recently, the first two...     

Negative Latin square type partial difference sets and amorphic association schemes with Galois rings

A partial difference set (PDS) having parameters (n², r(n?1), n+r²?3r, r²?r) is called a Latin square type PDS, while a PDS having parameters (n², r(n+1), ?n+r²+3r, r² +r) is called a negative Latin square type PDS. There are relatively few known constructions of negative Latin square type PDSs, and nearly all of these are in elementary abelian groups. We show that there are three different groups of order 256 that have all possible negative Latin square type parameters. We then give generalized constructions of negative Latin square type PDSs in 2‐groups. We conclude by discussing how these results fit into the context of amorphic association schemes and by stating some open problems. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 266‐282, 2009     

Association schemes arising from bent functions

We give a construction of 3-class and 4-class association schemes from s-nonlinear and differentially 2 ^s -uniform functions, and a construction of p-class association schemes from weakly regular p-ary bent functions, where p is an odd prime.     

Nonlinear functions and difference sets on group actions

There are many generalizations of the classical Boolean bent functions. Let G, H be finite groups and let X be a finite G-set. G-perfect nonlinear functions from X to H have been studied in several papers. They are generalizations of perfect nonlinear functions from G itself to H. By introducing the concept of a (G, H)-related difference family of X, we obtain a characterization of G-perfect nonlinear functions on X in terms of a (G, H)-related difference family. When G is abelian, we prove that there is a normalized G-dual set \(\widehat{X}\) of X, and characterize a G-difference set of X by the Fourier transform on a normalized G-dual set \({{\widehat{X}}}\). We will also investigate the existence and constructions of G-perfect nonlinear functions and G-bent functions. Several known results (IEEE Trans Inf Theory 47(7):2934–2943, 2001; Des Codes Cryptogr 46:83–96, 2008; GESTS Int Trans Comput Sci Eng 12:1–14, 2005; Linear Algebra Appl 452:89–105, 2014) are direct consequences of our results.     

Partial difference sets

A (υ,k,α,β)-partial difference set in a finite group G of order υ is a subset D of G with k distinct elements such that expressions d_nd^?1₂ for d₁ and d₂ in D, represent each non-identity element not contained in D exactly α times and each non-identity element contained in D exactly α+β times. Such a set is closely related to association schemes of PBIB designs with two associate classes.     

Partial difference sets inp-groups

This work is partially supported by NSA grant # MDA 904-92-H-3067     

Partial spread and vectorial generalized bent functions

In 2007, Sun et al. (IEEE Trans Inf Theory 53(8):2922–2933, 2007) presented new variants of RSA, called Dual RSA, whose key generation algorithm outputs two distinct RSA moduli having the same public and private exponents, with an advantage of reducing storage requirements for keys. These variants can be used in some applications like blind signatures and authentication/secrecy. In this paper, we give an improved analysis on Dual RSA and obtain that when the private exponent is smaller than \(N^{0.368}\), the Dual RSA can be broken, where N is an integer with the same bitlength as the modulus of Dual RSA. The point of our work is based on the observation that we can split the private exponent into two much smaller unknown variables and solve a related modular equation on the two unknown variables and other auxiliary variables by making use of lattice based methods. Moreover, we extend this method to analyze the common private exponent RSA scheme, a variant of Dual RSA, and obtain a better bound than previous analyses. While our analyses cannot be proven to work in general, since we rely on some unproven assumptions, our experimental results have shown they work in practice.     

Partial geometric difference sets and partial geometric difference families

The concept of a partial geometric difference set (or $1\frac{1}{2}$-difference set) was introduced by Olmez in 2014. Recently, Nowak, Olmez and Song introduced the notion of a partial geometric difference family, which generalizes both the classical difference family and the partial geometric difference set. It was shown that partial geometric difference sets and partial difference families give rise to partial geometric designs. In this paper, a number of new infinite families of partial geometric difference sets and partial geometric difference families are constructed. From these partial geometric difference sets and difference families, we generate a list of infinite families of partial geometric designs.     

Amorphic association schemes from bent partitions

Bent partitions are partitions of an elementary abelian group, which have similarities to partitions from spreads. In fact, a spread partition is a special case of a bent partition. In particular, bent partitions give rise to a large number of (vectorial) bent functions. Examples of bent partitions, which generalize the Desarguesian spread, have been presented by Anbar, Meidl and Pirsic, 2021, 2022. Bent partitions, which generalize some other classes of (pre)semifield spreads, have been presented by Anbar, Kalaycı, Meidl 2023. In this article, it is shown that these bent partitions induce $(p^k+1)$-class amorphic associations schemes on ${\mathbb{F}}_{p^m}\times {\mathbb{F}}_{p^m}$, where k is a divisor of m with special properties. This implies a construction of amorphic association schemes from some classes of (pre)semifields.     

Fourier transforms on finite group actions and bent functions

Journal of Algebraic Combinatorics - Highly nonlinear functions (bent functions, perfect nonlinear functions, etc.) on finite fields and finite (abelian or nonabelian) groups have been studied in...     

Construction of bent functions from near-bent functions

We give a construction of bent functions in dimension 2m from near-bent functions in dimension 2m−1. In particular, we give the first ever examples of non-weakly-normal bent functions in dimensions 10 and 12, which demonstrates the significance of our construction.     

Imprimitive symmetric association schemes of classes 5 and 6 arising from ternary non-weakly regular bent functions

Journal of Algebraic Combinatorics - Let F be a ternary non-weakly regular bent function in GMMF class whose dual $$F^*$$ is bent. We prove that if F satisfies certain conditions, then collecting...     

Generalized bent functions and class group of imaginary quadratic fields

Several new results on non-existence of generalized bent functions are presented. The results are related to the class number of imaginary quadratic fields.     

Intersection theorems for group divisible difference sets

Let D be an (m,n;k;λ₁,λ₂)-group divisible difference set (GDDS) of a group G, written additively, relative to H, i.e. D is a k-element subset of G, H is a normal subgroup of G of index m and order n and for every nonzero element g of G,?{(d₁,d₂)?,d₁,d₂?D,d₁?d₂=g}? is equal to λ₁ if g is in H, and equal to λ₂ if g is not in H. Let H₁,H₂,…,H_m be distinct cosets of H in G and S_i=D∩H_i for all i=1,2,…,m. Some properties of S₁,S₂,…,S_m are studied here. Table 1 shows all possible cardinalities of S_i's when the order of G is not greater than 50 and not a prime. A matrix characterization of cyclic GDDS's with λ₁=0 implies that there exists a cyclic affine plane of even order, say n, only if n is divisible by 4 and there exists a cyclic (n?1,$\text{1}\text{2}$n?1,$\text{1}\text{4}$n?1)-difference set.     

A construction of bent functions from plateaued functions

In this presentation, a technique for constructing bent functions from plateaued functions is introduced and analyzed. This generalizes earlier techniques for constructing bent from near-bent functions. Using this construction, we obtain a big variety of inequivalent bent functions, some weakly regular and some non-weakly regular. Classes of bent functions having some additional properties that enable the construction of strongly regular graphs are formed, and explicit expressions for bent functions with maximal degree are presented.     

Perfect difference sets constructed from Sidon sets

A set of positive integers is a perfect difference set if every nonzero integer has a unique representation as the difference of two elements of . We construct dense perfect difference sets from dense Sidon sets. As a consequence of this new approach we prove that there exists a perfect difference set such that