Quantitative structure of stable sets in arbitrary finite groups

Perfect nonlinear functions between two finite abelian groups were studied by Carlet and Ding (J Complex 20:205---244, 2004) and Pott (Discret Math Appl 138:177---193, 2004), which can be regarded as a generalization of bent functions on finite abelian groups studied by Logachev et al. (Discret Math Appl 7:547---564, 1997). Poinsot (J Discret Math Sci Cryptogr 9:349---364, 2006), (Cryptogr Commun 4:1---23, 2012) extended this research to arbitrary finite groups, and characterized bent functions on finite nonabelian groups as well as perfect nonlinear functions between two arbitrary finite groups by the Fourier transforms of the related functions at irreducible unitary representations. The purpose of this paper is to study the characterizations of the bentness (perfect nonlinearity) of functions on arbitrary finite groups by the Fourier transforms of the related functions at irreducible characters. We will also give a characterization of a perfect nonlinear function by the relative pseudo-difference family.

In recent years, progress in practical applications of multi-party computation (MPC), fully homomorphic encryption (FHE), and zero-knowledge proofs (ZKP) motivates people to explore symmetric-key cryptographic algorithms, as well as corresponding cryptanalysis techniques (such as differential cryptanalysis, linear cryptanalysis), over finite Abelian groups or prime fields $${\mathbb {F}}_p$$ F p for large p . In this paper, we establish the links between linear cryptanalysis and differential cryptanalysis over general finite Abelian groups. As the first application, we revisit linear cryptanalysis and give general results of linear approximations over arbitrary finite Abelian groups. More precisely, we consider the linearity , which is the maximal non-trivial linear approximation, to characterize the resistance of a function against linear cryptanalysis. This thereby generalizes the work of Pott in 2004 and completes the generalization of Sidelnikov–Chabaud–Vaudenay’s bound from $${\mathbb {F}}_2^n$$ F 2 n to finite Abelian groups. As the second application, we give an exact expression for the correlation of differential-linear approximations over arbitrary finite Abelian groups ( $${\mathbb {F}}_p^n$$ F p n ) under the sole assumption that the two parts of the cipher are independent of each other. In particular, we completely generalize the differential-linear cryptanalysis from $${\mathbb {F}}_2^n$$ F 2 n to arbitrary finite Abelian groups ( $${\mathbb {F}}_p^n$$ F p n ).

Plateaued functions on finite fields have been studied in many papers in recent years. As a generalization of plateaued functions on finite fields, we introduce the notion of a plateaued function on a finite abelian group. We will give a characterization of a plateaued function in terms of an equation of the matrix associated to the function. Then we establish a one‐to‐one correspondence between the ‐valued plateaued functions and partial geometric difference sets (with specific parameters) in finite abelian groups. We will also discuss two general methods (extension and lifting) for the construction of new partial geometric difference sets from old ones in (abelian or nonabelian) finite groups, and construct many partial geometric difference sets and plateaued functions. A one‐to‐one correspondence between partial geometric difference sets (in arbitrary finite groups) and partial geometric designs will be proved.

1. Difference sets in finite groups have been studied at great lengtll in a paper by R. H. Bruck [I]. Marshall Hall, Jr.'s notion of a multiplier of a difference set has been extended by Bruck to difference sets in groups. The latter has also extended a theorem of Hall on multipliers [3] to abelian groups. Hall has later obtained a generalization of his theorem. The object of this paper is, in the first instance, to extend this generalized theorem of Hall on multipliers to difference sets in abelian groups. A new feature that has been introduced here is the application of the theory of characters to the discussion of the problem. By this method I have also been able to obtain a new class of different sets. The later part of the paper is concerned with the discussion of this new class. Let G be an abelian group of order v and let D be a subset of G containing k elements such that every element of G other than the identity can be expressed exactly X times in the form ab-' where a, b are elements of D. Then D is called a difference set of type (v, k, X). The numbers v, k, X satisfy the relation

We prove Bogolyubov–Ruzsa-type results for finite subsets of groups with small tripling, |A3| ≤ O(|A|), or small alternation, |AA−1A| ≤ O(|A|). As applications, we obtain a qualitative analogue of Bogolyubov’s lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularity lemma for sets of bounded VC-dimension in finite groups of bounded exponent. The latter result generalizes the abelian case, due to Alon, Fox and Zhao, and gives a quantitative version of previous work of the author, Pillay and Terry.

The -norm gives a useful measure of quasirandomness for real- or complex-valued functions defined on finite (or, more generally, locally compact) groups. A simple Fourier-analytic argument yields an inverse theorem, which shows that a bounded function with a large -norm defined on a finite Abelian group must correlate significantly with a character. In this paper we generalize this statement to functions that are defined on arbitrary finite groups and that take values in . The conclusion now is that the function correlates with a representation — though with the twist that the dimension of the representation is shown to be within a constant of rather than being exactly equal to . There are easy examples that show that this weakening of the obvious conclusion is necessary. The proof is much less straightforward than it is in the case of scalar functions on Abelian groups. As an easy corollary, we prove a stability theorem for near representations. It states that if is a finite group and is a function that is close to a representation in the sense that has a small Hilbert-Schmidt norm (also known as the Frobenius norm) for every , then there must be a representation such that has small Hilbert-Schmidt norm for every . Again, the dimension of need not be exactly , but it must be close to . We also obtain stability theorems for other Schatten -norms. Bibliography: 14 titles.

Tambara functors are an equivariant generalization of commutative rings. In previous work, Nakaoka introduced the spectrum of prime ideals of a Tambara functor and computed the spectrum of the Burnside Tambara functor, the equivariant analogue of the Zariski spectrum of the integers, over cyclic $p$-groups. Subsequently, Calle and Ginnett computed the spectrum of the Burnside Tambara functor over arbitrary finite cyclic groups using a generalization of Dress’ ghost coordinates for Burnside rings. In this paper, we compute the spectrum of prime ideals in the Burnside Tambara functor over an arbitrary finite group. Our proof uses recent advances in the commutative algebra of Tambara functors, as well as a Tambara functor analogue of ghost coordinates, which works over arbitrary finite groups and clarifies some previous computations. As examples, we explicitly compute the spectrum of the Burnside Tambara functor over all dihederal groups, the quaternion group $Q_{8}$, the alternating group $A_{4}$, and the general linear group $GL_{3}(\mathbb{F}_{2})$.

Sum-free subsets of finite abelian groups of type III

Modular invariance property of association schemes is recalled in connection with our joint work with François Jaeger. Then we survey codes over F 2 discussing how codes, through their (various kinds of) weight enumerators, are related to (various kinds of) modular forms through polynomial invariants of certain finite group actions and theta series. Recently, not only codes over an arbitrary finite field but also codes over finite rings and finite abelian groups are considered and have been studied extensively. We show how the determination of the solutions of the modular invariance property of finite abelian groups (our joint work with Jaeger) is used to define the concept of Type II codes over arbitrary finite abelian groups. As an example of the usefulness of such Type II codes, we give an application to hermitian modular forms.

A group is called weakly conjugate biprimitively finite if each its element of prime order generates a finite subgroup with any of its conjugate elements. A binary finite group is a periodic group in which any two elements generate a finite subgroup. If $\mathfrak{X}$ is some set of finite groups, then the group $G$ saturated with groups from the set $\mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $G$, isomorphic to some group from $\mathfrak{X}$. A group $G = F \leftthreetimes H$ is a Frobenius group with kernel $F$ and a complement $H$ if $H \cap H^f = 1$ for all $f \in F^{\#}$ and each element from $G \setminus F$ belongs to a one conjugated to $H$ subgroup of $G$. In the paper we prove that a saturated with finite Frobenius groups periodic weakly conjugate biprimitive finite group with a nontrivial locally finite radical is a Frobenius group. A number of properties of such groups and their quotient groups by a locally finite radical are found. A similar result was obtained for binary finite groups with the indicated conditions. Examples of periodic non locally finite groups with the properties above are given, and a number of questions on combinatorial group theory are raised.

Quantum normalizer circuits were recently introduced as generalizations of Clifford circuits: a normalizer circuit over a finite Abelian group G is composed of the quantum Fourier transform (QFT) over G, together with gates which compute quadratic functions and automorphisms. In \cite{VDNest_12_QFTs} it was shown that every normalizer circuit can be simulated efficiently classically. This result provides a nontrivial example of a family of quantum circuits that cannot yield exponential speed-ups in spite of usage of the QFT, the latter being a central quantum algorithmic primitive. Here we extend the aforementioned result in several ways. Most importantly, we show that normalizer circuits supplemented with intermediate measurements can also be simulated efficiently classically, even when the computation proceeds adaptively. This yields a generalization of the Gottesman-Knill theorem (valid for n-qubit Clifford operations) to quantum circuits described by arbitrary finite Abelian groups. Moreover, our simulations are twofold: we present efficient classical algorithms to sample the measurement probability distribution of any adaptive-normalizer computation, as well as to compute the amplitudes of the state vector in every step of it. Finally we develop a generalization of the stabilizer formalism relative to arbitrary finite Abelian groups: for example we characterize how to update stabilizers under generalized Pauli measurements and provide a normal form of the amplitudes of generalized stabilizer states using quadratic functions and subgroup cosets.

The security of many cryptographic techniques depends on the intractability of the discrete logarithm problem (DLP). As a starting point, we consider the particular case of this problem, the discrete logarithm problem in subgroups of Zopf <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> * (p prime number), which is of special interest because its presumed intractability is the basis for the security of the U.S. Government NIST Digital Signature Algorithm, among other cryptographic techniques. Our intention is to generalize the discrete logarithm problem in subgroups of Zopf <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> *, first by considering an arbitrary finite cyclic group G, instead of Zopf <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> *; and then, more generally, by considering an arbitrary finite group G instead of Zopf <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> *. Then, following the same idea, we try to generalize a problem closely related to the DLP, the Diffie-Hellman problem (DHP), which is of significance to public-key cryptography because its apparent intractability forms the basis for the security of many cryptographic schemes, including Diffie-Hellman key agreement and its derivatives, and ElGamal public-key encryption. Our paper will give the mathematical description of the general problems, using group theory, as well as provide a mathematical algorithm for solving them.

This work initiates a study of Luby-Racko. ciphers when the bitwise exclusive-or (XOR) operation in the underlying Feistel network is replaced by a binary operation in an arbitrary finite group. We obtain various interesting results in this context: - First, we analyze the security of three-round Feistel ladders over arbitrary groups. We examine various Luby-Racko. ciphers known to be insecure when XOR is used. In some cases, we can break these ciphers over arbitrary Abelian groups and in other cases, however, the security remains an open problem. - Next, we construct a four round Luby-Racko. cipher, operating over finite groups of characteristic greater than 2, that is not only completely secure against adaptive chosen plaintext and ciphertext attacks, but has better time / space complexity and uses fewer random bits than all previously considered Luby-Racko. ciphers of equivalent security in the literature. Surprisingly, when the group is of characteristic 2 (i.e., the underlying operation on strings is bitwise exclusive-or), the cipher can be completely broken in a constant number of queries. Notably, for the former set of results dealing with three rounds (where we report no difference) we need new techniques. However for the latter set of results dealing with four rounds (where we prove a new theorem) we rely on a generalization of known techniques albeit requires a new type of hash function family, called a monosymmetric hash function family, which we introduce in this work. We also discuss the existence (and construction) of this function family over various groups, and argue the necessity of this family in our construction. Moreover, these functions can be very easily and efficiently implemented on most current microprocessors thereby rendering the four round construction very practical.

Multidimensional Fourier transforms and nonlinear functions on finite groups

In the study of the problem of existence and conjugacy in an arbitrary finite group it is known the Blessenohl-Laue result that in any finite group G there exists a unique class of conjugate quasinilpotent injectors which are exactly the ℜ∗-maximal subgroups of G containing the generalised Fitting subgroup F ∗(G). In this paper, with the use of constructions of the Blessenohl–Laue and Gaschütz classes, we extend the Blessenohl-Laue result to the case of the Fitting class 𐔉 = 𐕳𐔅, where 𐕳 is a non-empty Fitting class and 𐔅 is a Blessenohl-Laue class, and thus we distinguish a new class of conjugate 𐔉-injectors in the classes 𐔈 of all finite groups and 𐔖π of all finite π-solvable groups respectively. Moreover, we prove that the 𐔉-injectors of the group G are exactly all 𐔉-maximal subgroups of G, which contain its 𐔉-radical G𐔉. Special cases of such injectors are the injectors for many known Fitting classes. In particular, such injectors in the class 𐔖 of all finite solvable groups were described by B. Hartley, B. Fischer, W. Frantz, and P. Lockett.