In recent work Cheon, Han, Lee, Ryu, and Stehlé presented an attack on the multilinear map of Coron, Lepoint, and Tibouchi (CLT). They show that given many low-level encodings of zero, the CLT multilinear map can be completely broken, recovering the secret factorization of the CLT modulus. The attack is a generalization of the “zeroizing” attack of Garg, Gentry, and Halevi. We first strengthen ...

Let Mn(R) be the algebra of all n×n matrices over a unital commutative ring R with 6 invertible. We say that A ∈ Mn(R) is a Jordan product determined point if for every R-module X and every symmetric R-bilinear map {·, ·} : Mn(R)×Mn(R) → X the following two conditions are equivalent: (i) there exists a fixed element w ∈ X such that {x, y} = w whenever x ◦ y = A, x, y ∈ Mn(R); (ii) there exists ...

In this paper, we describe linear maps between complex Banach algebras that preserve products equal to fixed elements. This generalizes some important special cases where the elements are zero or identity element. First show if such map preserves a finite-rank operator, then it must also product. several instances, is enough product preserving be scalar multiple of an algebra homomorphism. Seco...

In 1996, Harris and Kadison posed the following problem: show that a linear bijection between C∗-algebras that preserves the identity and the set of invertible elements is a Jordan isomorphism. In this paper, we show that if A and B are semisimple Banach algebras andΦ : A→ B is a linear map onto B that preserves the spectrum of elements, thenΦ is a Jordan isomorphism if either A or B is a C∗-al...

This paper investigates the effect of structure-preserving perturbations on the eigenvalues of linearly and nonlinearly structured eigenvalue problems. Particular attention is paid to structures that form Jordan algebras, Lie algebras, and automorphism groups of a scalar product. Bounds and computable expressions for structured eigenvalue condition numbers are derived for these classes of matri...

Let Φ be a trace-preserving, positivity-preserving linear map on the algebra of complex 2 × 2 matrices, and let Ω be any finite-dimensional completely positive map. For p = 2 and p ≥ 4, we prove that the maximal p-norm of the product map Φ⊗Ω is the product of the maximal p-norms of Φ and Ω. Restricting Φ to the class of completely positive maps, this settles the multiplicativity question for al...

Let A and B be Banach algebras with bounded approximate identities let Φ:A→B a surjective continuous linear map which preserves two-sided zero products (i.e., Φ(a)Φ(b)=Φ(b)Φ(a)=0 whenever ab=ba=0). We show that Φ is weighted Jordan homomorphism provided product determined weakly amenable. These conditions are in particular fulfilled when the group algebra L1(G) G any locally compact group. also...

Let X,Y be locally compact Hausdorff spaces and M,N be Banach algebras. Let θ : C0(X,M) → C0(Y,N ) be a zero-product preserving bounded linear map with dense range. We show that θ is given by a continuous field of algebra homomorphisms from M into N if N is irreducible. As corollaries, such a surjective θ arises from an algebra homomorphism, provided thatM is a W*-algebra and N is a semi-simple...