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In this paper, we consider the challenge of ensuring secure communication in block-fading wiretap channels using encoding techniques. We investigate the coding for block-fading wiretap channels using stacked lattice codes constructed over completely real number fields, which is a well-established technique. In this paper, we consider degree-four complex multiplication field ${𝒦}=\mathbb{Q}({\varsigma }_8)$ over $\mathbb{Q}$. We employ binary codes to generate a lattice over $\mathbb{Q}({\varsigma }_8)$, which is subsequently used to form an integral lattice. The resulting integral lattice can be effectively applied to enhance the security of communication within block-fading wiretap channels.

Proxy Re-Encryption (PRE) is a cryptographic primitive that allows a proxy to turn an Alice’s ciphertext into a Bob’s ciphertext on the same plaintext. All of the PRE schemes are public key encryption and semantic security. Deterministic Public Key Encryption (D-PKE) provides an alternative to randomized public key encryption in various scenarios where the latter exhibits inherent drawbacks. In this paper, we construct the first multi-use unidirectional D-PRE scheme from Lattices in the auxiliary-input setting. We also prove that it is PRIV1-INDr secure in the standard model based on the LWR. Finally, an identity-based D-PRE is obtained from the basic construction.

We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885–903]. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and, if connected, it is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to ${ℝ}_{\mathrm{\text{an}},\mathrm{\text{exp}}}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that ${\mathrm{\Sigma }}_{i=1}^m(X-X)$ is an approximate subgroup of $G$.

Reducing the size of a logic circuit through lattice identities is an important and well-studied discrete optimization problem. In this paper, we consider a related problem of integrating several circuits into a single hypercircuit using the recently developed concept of lattice hyperterms. We give a combinatorial algorithm for integrating k-out-of-n symmetrical diagrams which play an important role in reliability theory. Our results show that the integration can reduce the number of circuit gates by more than twice.

We propose in this work a Sieve algorithm that we called OrthogonalInteger sieve algorithm for some orthogonal integer lattices and particularly the case of integer lattices $\mathrm{\Lambda }\subset {\mathbb{Z}}^n$, root lattices of type $A_n$ ($n\ge 1$) and of type $D_n$ ($n\ge 2$). In these cases, we use the famous $LLL$ algorithm to find the shortest vector of these lattices. Indeed, in general, a sieve algorithm builds a list of short random vectors which are not necessarily in the lattice, and try to produce short lattice vectors by taking linear combinations of the vectors in the list. But in our case, we built a list of short vectors in the lattice. From the first column of the $LLL$-reduced basis of the considered basis, we have the list of at least $n$ and at most $2^n$ short vectors for the general case (where $n$ is the dimension of the lattice) of orthogonal integer lattices $\mathrm{\Lambda }\subset {\mathbb{Z}}^n$. For the lattices ${\mathbb{Z}}^n$, $A_n$ ($n\ge 1$) and $D_n$ ($n\ge 2$), we have, respectively, $2n$, $n(n+1)$ and $2n(n-1)$ short vectors. The proposed sieve algorithm for integer lattice ${\mathbb{Z}}^n$ runs in space $O(2n)$ and the OrthogonalInteger sieve algorithm performs $O(n{2}^n)$ arithmetic operations and is polynomial in space.

Let ℓ > 0 be a square free integer and the ring of integers of the imaginary quadratic field . Codes C over K determine lattices Λ_ℓ(C) over rings . The theta functions θ_Λℓ(C) of such lattices are known to determine the symmetrized weight enumerator swe(C) for small primes p = 2, 3; see [1, 10].

In this paper we explore such constructions for any p. If p ∤ ℓ then the ring is isomorphic to 𝔽_p² or 𝔽_p × 𝔽_p. Given a code C over we define new theta functions on the corresponding lattices. We prove that the theta series θ_Λℓ(C) can be written in terms of the complete weight enumerator of C and that θ_Λℓ(C) is the same for almost all ℓ. Furthermore, for large enough ℓ, there is a unique complete weight enumerator polynomial which corresponds to θ_Λℓ(C).