encryption – Wrap key operation in Azure Key Vault – symmetric keys

Could anyone explain why the bolded part of the wrap key description?

Wraps a symmetric key using a specified key. The WRAP operation
supports encryption of a symmetric key using a key encryption key that
has previously been stored in an Azure Key Vault. The WRAP operation
is only strictly necessary for symmetric keys stored in Azure Key
Vault since protection with an asymmetric key can be performed using
the public portion of the key.
This operation is supported for
asymmetric keys as a convenience for callers that have a key-reference
but do not have access to the public key material. This operation
requires the keys/wrapKey permission.

AFAIK, all the keys in Azure Key Vault are stored at rest in HSM modules. Why is key wrapping necessary for symmetric keys? What does ‘protection’ mean in this case? Using a public key to encrypt data?

If HSM are securing all the keys in Key Vault (using its built-in symmetric key), then why would encrypting a symmetric key be necessary as quoted?

encryption – Parameters for HSM based symmetric Key Derivation Function (KDF)

I have a quick question regarding parameters for HSM based symmetric Key Derivation.

My situation is that I have to implement HSM based symmetric key derivation for encryption of sensitive data to be stored inside DB. Each data entry should have distinct AES-256 key used only for that records encryption. There are two cases:

  1. Users ID has to be encrypted with unique Key per user, so that it would be possible to search by this User ID. My idea was to use ID itself (known at the time) as a parameter to KDF to get predictable encryption key, and use it to encrypt and perform search (KEYhsm + IDuser -> KEYaes). I do not see how this differs from hashing, but requirements states that encryption should be used. Should I hash this ID before using it as a parameter?
  2. Second case is as stated before, that each record should use distinct Key for encryption. For this my idea was to use record GUID (stored next to encrypted data) as a parameter to KDF to generate symmetric Key (KEYhsm + IDresource -> KEYaes). Again, should I hash it, is this approach secure enough?

It would be really helpful if somebody smarter that me would review my approach and maybe give some hints for algorithms to use (HASH, KDF). I do not know HSM model and maker at the time, but my assumption is that this HSM will be able to use secure private key to generate symmetric AES-256 encryption keys.

encryption – Format for data & symmetric key exchange/storeage

Is there a standard format for storing/exchanging encrypted data along with the key needed to decrypt it (data is encrypted with a single use symmetric key and the symmetric key itself is encrypted with asymmetric key for the receiver)?

We are trying to build an interoperable protocol to exchange large messages between two parties that may not agree on much else besides using asymmetric keys. The best way seems to be using a symmetric single use key to encrypt the data and then encrypt it with the asymmetric key and pass along the whole thing as a package (e.g. RSA wrapped AES). So is there any widely used standard for sharing the encrypted text along with its key, preferably along with some information about the symmetric algorithm used.

The only work that I found in that direction is OpenPGP which is somewhat too implementation specific. I was wondering if there is anything else that has more metadata along with it to describe the alogs and the keys.

co.combinatorics – Eigenvectors of a symmetric sum of tensor products

Let $A$ and $B$ be two (finite-dimensional) Hermitian matrices and $n$ be a positive integer. We define the matrix
L_i = Aotimes dotsotimes Aotimes Botimes Aotimes dotsotimes A~,

where there are $n$ factors and $B$ is the $i$-th factor (with $1 leq i leq n$). We then define the “symmetric sum”
L = sum_{i=1}^n L_i~.

Question: Can we say anything about the eigenvectors of $L$ knowing the eigenvectors of $A$ and $B$?

This question seems simple (and maybe it is) but it has resisted my best attempts. If $A$ and $B$ commute, this is trivial so I am interested in any case where they don’t commute. I am not necessarily asking for a general solution of the problem and would be interested to see any (non-commuting) example. Even the case where $A= begin{pmatrix} a & 0 \ 0 & b end{pmatrix} $ and $B = begin{pmatrix} 0 & 1 \ 1 & 0 end{pmatrix}$ was too complicated (the constraint satisfied by the eigenvectors is hard to solve). I feel like some form of this problem should have appeared before and that’s why I am asking here!

For some background, this problem arises in quantum information theory, when trying to determine the optimal quantum measurement that distinguishes two states, involving $n$ copies of the system. Note that for this application, we have $mathrm{Tr},B = 0$.

Thanks a lot for your help!

gr.group theory – Are the symmetric groups integrable as Hopf algebras?

Let $G$ be a group. For $g,h in G$, let $(g,h)=g^{-1}h^{-1}gh$ be a commutator. The normal subgroup $G’ = langle (g,h) | g,h in G rangle$ is called the commutator subgroup or derived subgroup.

An integral of $G$ is a group $H$ such that $H’simeq G$. The problem of the existence of an integral was first mention by B.H. Neumann is this paper (1956). A group without integral is called non-integrable. The smallest non-integrable finite group is the symmetric group $S_3$; moreover $S_n$ is non-integrable $forall n ge 3$.

Here are two recent references about integrals of groups: Filom-Miraftab (2017) and Ara├║jo-Cameron-Casolo-Matucci (2019).

The commutator subgroup is the smallest normal subgroup for which the quotient is commutative. This notion was extended to semisimple Hopf algebra (Burciu, 2012) and is called commutator subalgebra. It is the smallest normal left coideal subalgebra for which the quotient is commutative. Then let call a semisimple Hopf algebra integrable if it is isomorphic to the commutator subalgebra of a semisimple Hopf algebra.

Question: Are the Hopf algebras $mathbb{C}S_n$ integrable? What if $n=3$?

More generally we can ask whether there exist a non-integrable finite group which is integrable as Hopf algebra, and if so, whether there is one which is not, and if no, whether every finite dimensional semisimple Hopf algebra is integrable.

interpolation – Axial Symmetric 2D Cylindrical Field to Cartesian 3D Field

Previously, I’ve used 2-dimensioned (radius, zeta) output from NDSolve ({InterpolatingFunction, InterpolatingFunction}) and used the FieldTransform to convert the axial-symmetric cylindrical slice to 3D cartesian space for future NDSolve functions. When I try to do that with a multi-dimensioned data list and use Interpolation, it doesn’t structure the result the same. I’m looking for a result like the following:

desired result

Here is my code:

bField = {{{0, -0.0602087}, {0, -0.0950287}, {0, -0.124952}, {0, -0.14618}, {0, -0.155613}, 
           {0, -0.159372}, {0, -0.162784}, {0, -0.168384}, {0, -0.175639}, {0, -0.179005}, 
           {0, -0.175288}, {0, -0.1582}, {0, -0.106581}}, 
          {{0.018056, -0.0599793}, {0.0169553, -0.0970272}, {0.0127164, -0.127771}, 
           {0.00729817, -0.147934}, {0.00259357, -0.157179}, {0.00100155, -0.159464}, 
           {0.00218095, -0.161942}, {0.00361777, -0.168195}, {0.00327177, -0.176241}, 
           {0.000512439, -0.180925}, {-0.00414832, -0.178067}, {-0.0119931, -0.165746}, 
           {-0.0431867, -0.121498}}, 
          {{0.0390599, -0.0595938}, {0.036579, -0.101196}, {0.0270982, -0.136852}, 
           {0.0135705, -0.157793}, {0.00256824, -0.162421}, {-0.000178479, -0.158853}, 
           {0.00408384, -0.158385}, {0.0089802, -0.166161}, {0.0087702, -0.17868}, 
           {0.0020529, -0.18713}, {-0.00678186, -0.184855}, {-0.0130875, -0.176824}, 
           {-0.0100346, -0.184251}}, 
         {{0.0644084, -0.0559792}, {0.0629479, -0.109121}, {0.0453611, -0.154392}, 
          {0.0174589, -0.17746}, {-0.00442631, -0.172572}, {-0.00786805, -0.155377}, 
          {0.00534545, -0.147518}, {0.0192717, -0.160265}, {0.0198827, -0.18429}, 
          {0.00518068, -0.200208}, {-0.0129943, -0.194885}, {-0.0184665, -0.176816},
          {-0.00632971, -0.163783}}};

data = Flatten( 
         Table({{r, zeta}, bField((r + 1, zeta + 7))}, {r, 0, 3}, {zeta, -6,6}),

interpB = Interpolation(data)
bFieldCart3D =  TransformedField("Cylindrical" -> "Cartesian", 
               interpB(r, zeta), {r, th, zeta} -> {x, y, z})

convex geometry – Euclidean volume of symmetric matrices in operator norm

This is a nearly identical question to Euclidean volume of the unit ball of matrices under the matrix norm except in the symmetric case.

Let $mathrm{Sym}_{n times n}(mathbb{R})$ be the space of real-valued $n times n$ symmetric matrices. Let $phi : mathbb{R}^{n(n+1)/2} mapsto mathrm{Sym}_{n times n}(mathbb{R})$ embed $mathbb{R}^{n(n+1)/2}$ into $mathrm{Sym}_{n times n}(mathbb{R})$.
Consider the set $H_n = { v in mathbb{R}^{n(n+1)/2} : | phi(v) | leq 1 }$, where $|M| = max_{|x|=1} |Mx|$ is the $ell_2 mapsto ell_2$ operator norm.

What is the formula for $mathrm{Vol}(H_n)$, where $mathrm{Vol}(cdot)$ is the Lebesgue measure on $mathrm{R}^{n(n+1)/2}$?