This article contains a Wiener Lemma for the convolution algebra
ℓ
1
(
H
,
C
)
and group
C
∗
-algebra
C
∗
(
H
)
of the discrete Heisenberg group
H
. At first, a short review of Wiener’s Lemma in its ...classical form and general results about invertibility in group algebras of nilpotent groups will be presented. The known literature on this topic suggests that invertibility investigations in the group algebras of
H
rely on the complete knowledge of
H
^
—the dual of
H
, i.e., the space of unitary equivalence classes of irreducible unitary representations. We will describe the dual of
H
explicitly and discuss its structure. Wiener’s Lemma provides a convenient condition to verify invertibility in
ℓ
1
(
H
,
C
)
and
C
∗
(
H
)
which bypasses
H
^
. The proof of Wiener’s Lemma for
H
relies on local principles and can be generalised to countable nilpotent groups. As our analysis shows, the main representation theoretical objects to study invertibility in group algebras of nilpotent groups are the corresponding primitive ideal spaces. Wiener’s Lemma for
H
has interesting applications in algebraic dynamics and time-frequency analysis which will be presented in this article as well.
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The Internet of things have been widely used to improve our daily life. Public-key cryptosystem plays a central role in the security of IoT. However, classic public-key cryptosystems such as RSA and ...ECC are not suitable for IoT because of the complicated encryption and decryption operations. NTRU is an efficient public-key cryptosystem. In this paper, we first generalize NTRU and propose a group-based NTRU-like public-key cryptosystem, namely GTRU. Then, we construct high-performance GTRU for IoT. Finally, security analysis shows that our proposed GTRU for IoT is more secure than NTRU against lattice-based attacks.
We execute the average running time of the various binary strings of different length and evaluate the other useful characteristics of a hash function which is to be Pseudo-Randomness of a hash ...function,which can assigned Uniformly over the range 0,p) to reduce the chance of occurring collisions. We also verify the security properties of the hash function using different attacks.
We prove that the irreducible representations with finite weight of discrete nilpotent groups of class 2, including the discrete Heisenberg groups, are monomial representations and obtain their ...classification for the group Heis(3, ℤ).
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We define a class of monomial not necessary unitary representations of discrete Heisenberg groups. Their moduli space is a complex-analytic manifold. There exist characters of the representations ...which are automorphic forms on the moduli space.
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