## probability theory – First hitting time of a symmetric random walk

Definitions:

Let $$xi_n$$ be a symmetric random walk, i.e.,
$$xi_n=eta_1+eta_2+dots+eta_n,$$
where $${eta_n}$$ is a sequence of i.i.d. random variables such that
$$P{eta_n=1}=P{eta_n=-1}=frac{1}{2}.$$
Furthermore, we define the first hitting time to be $$tau=minleft{n:|xi_n|=Kright},$$
where $$K$$ is a positive integer.

I was reading a book on stochastic processes and here we want to show that $$tau a.s. The book proves this as follows

We want to show that $$P{tau=infty}=0.$$ To this end we shall estimate $$P{tau>2Kn}.$$ Notice that $$P{tau>2Kn}le left(1-frac{1}{2^{2K}}right)^nto0$$ as $$ntoinfty.$$ Thus, we have begin{align} P{tau=infty}&=bigcap_{n=1}^infty P{tau>2Kn} \ &=lim_{ntoinfty} P{tau>2Kn}=0. end{align}

After spending so many time, I could not figure out how to get the inequality $$P{tau>2Kn}le left(1-frac{1}{2^{2K}}right)^n$$ in the first line of the proof. Can someone help me understand why this inequality holds?

## key management – Long-term symmetric key storage

I have some sensitive data that I need to encrypt and retain long-term (i.e., 5+ years at least). I’d ideally like to secure it using multiple hardware devices via a Shamir share. Something like:

``````data_key = gen_symmetric()
s1, s2, s3 = shamir(k=2, n=3, secret=data_key)
k1, k2, k3 = gen_symmetric(), gen_symmetric(), gen_symmetric()
encrypted_secrets = encrypt_and_mac(k1, s1) || encrypt_and_mac(k2, s2) || encrypt_and_mac(k3, s3)
output_to_save = encrypted_data || encrypted_secrets
``````

With the idea to then save `k1`, `k2`, and `k3` in hardware. (Along with some indication of which was the first, 2nd, 3rd secret).

This feels like I’m reinventing the wheel. Is there an existing standard for doing this or a similar crypto scheme? This related question suggests no. In that case, is the above scheme secure against an attacker that obtains one of the 3 devices?

## table – Generate symmetric random tensor

I would like to generate a table $$T$$ of random values of rank $$p$$ such that my table is fully symmetric: If I swap any indices I get the same value. For example when $$p=3$$ I would like $$T_{ijk}$$ to be random with the following symmetry:
$$T_{ijk}=T_{ikj}=T_{jki}=T_{jik}=T_{kij}=T_{kji}$$

For the case $$p=2$$, it boils down to generate random matrices and I can simply take the upper triangular part and take its transpose.

I would like something in those lines for example:

``````T=RandomVariate[NormalDistribution[0, 1],{n,n,n}];
``````

But here $$T$$ is not symmetric. How could I obtain $$T$$ such that for any permutations of its indices I get the same value?

## Spectrum of symmetric Toeplitz matrix

A matrix is Toeplitz if it is constant on the diagonals parallel to the main diagonal.
I am looking for references on the spectrum of finite symmetric Toeplitz matrices over finite fields.

## encryption – symmetric key generation in TLS 1.3

From what I am understanding Diffie-Hellman is used to derive the symmetric key in TLS1.3

I am reading this tls explanation and so many keys are derived

Is the shared secret the symmetric key and from it, we can generate the following keys?

``````early_secret= HKDF-Extract( salt=00,  key=00...)
empty_hash= SHA256("")
derived_secret= HKDF-Expand-Label(key = early_secret,  label = "derived",  context = empty_hash, len = 32)
handshake_secret= HKDF-Extract(salt = derived_secret, key = shared_secret)
client_handshake_traffic_secret= HKDF-Expand-Label(key = handshake_secret, label = "c hs traffic", context = hello_hash, len = 32)
server_handshake_traffic_secret= HKDF-Expand-Label(key = handshake_secret, label = "s hs traffic", context = hello_hash,len = 32)
client_handshake_key=HKDF-Expand-Label(key= client_handshake_traffic_secret, label = "key", context = "", len = 16)
server_handshake_key=HKDF-Expand-Label(key=server_handshake_traffic_secret, label = "key", context = "",  len = 16)
client_handshake_iv= HKDF-Expand-Label( key = client_handshake_traffic_secret,  label = "iv",  context = "", len = 12)
server_handshake_iv=HKDF-Expand-Label(key= server_handshake_traffic_secret,  label = "iv", context = "", len = 12)
``````

So which is the symmetric key that inserts as input in AEAD?

## rt.representation theory – Weakly symmetric Frobenius algebras

Let $$A$$ be a finite dimensional Frobenius algebra and $$e$$ and idempotent of $$A$$.
It is well known that the algebra $$eAe$$ does not have to be a Frobenius algebra. But if $$A$$ is additionally symmetric, then $$eAe$$ is also a symmetric Frobenius algebra for any idempotent $$e$$.

The Frobenius algebra $$A$$ is called weakly symmetric if for every indecomposable projective module $$P$$: $$top(P)=soc(P)$$.

Question: If $$A$$ is just weakly symmetric, is $$eAe$$ also always weakly-symmetric for any idempotent $$e$$?

## co.combinatorics – Number of lattice points in a structural symmetric convex body

Let $$f$$ is a log-concave symmetric function on the interval $$(-a,a)$$, i.e., its logarithm $$log f(x)$$ is concave and $$f(-x)=f(x)$$ for $$forall , xin (-a,a)$$. Then we consider a $$n$$-dimensional convex body in Euclidean space
$$begin{equation} mathscr{R}_n=left{ mathbf{x}=left(x_1,cdots,x_n right)in mathbb{R}^n： -sum_{i=1}^n log f(x_i) le nmathsf{A} right}, end{equation}$$
where the constant $$mathsf{A}>-log f(0)$$. Clearly, $$mathscr{R}_n$$ is symmetric about the origin.
I am interested the following problem relating to the lower bound of numbers of the lattice points inside $$mathscr{R}_n$$:

Does there exit a lattice sequence $$left{ Lambda_n right}$$ such that the number sequence $${N(n)}$$ satisfies
$$begin{equation} N(n)=frac{log left( |Lambda_n cap mathscr{R}_n| right)}{n}ge c, ~text{for sufficiently large n.} end{equation}$$
I also want to know if there exits a best lowerbound $$c$$ for this asymptotic problem.

This problem is motivated by the answer of Geometry interpretation of any continuous random variable

## complexity theory – Symmetric functions in NC¹

A boolean function $$f colon {0,1}^n rightarrow {0,1}$$ is symmetric if $$f(x)$$ depends only on the number of $$1$$s in $$x$$.
It is known that every boolean function is in $$mathrm{NC}^1$$, i.e. there is a circuit of depth $$O(log n)$$ computing it.

What is known about the constant inside the $$O()$$ notation? Specifically, can one construct, for every $$c$$, a symmetric function requiring a circuit of depth at least $$c log n$$? Or there is some constant $$c_0$$ such that every symmetric function has a circuit of depth at most $$c_0log n$$?

## encryption – Whether TLS session resumption reuse the symmetric keys?

I am learning TLS Session Resumption.

What I got is session resumption can reduce 1 RTT for TLS 1.2 by reusing MasterSecret. Both the client and server needn’t to run key exchange algorithm.

My questions are:

1. Whether session resumption reuses symmetric encryption keys (to encrypt TLS records).
2. What factors affect whether to reuse symmetric encryption keys?

I searched around Google, but cannot find a authoritative answer. Here is what I got:

1. Do not reuse encryption keys. Refer to SSL session key usage when browser opens multiple sockets to same server.
2. Reuse encryption keys. Refer to https://wiki.openssl.org/index.php/SSL_and_TLS_Protocols#Session_Resumption

Any ideas are welcome.

## About symmetric rank-1 random matrices

Consider a $$2n-$$dimensional symmetric random matrix $$M$$ of form, $$M = begin{bmatrix} aa^T & ab^T \ ba^T & bb^T end{bmatrix}$$ where $$a$$ and $$b$$ are $$n$$ dimensional random vectors.

• Are there conditions known on $$a$$ and $$b$$ s.t we have the following property : that for any $$hat{x} in S^{2n-1}$$ and any $$R in SO(2n)$$, $$Vert M hat{x} Vert$$ be equidistributed as $$Vert (R M R^T)hat{x} Vert$$ ?