Ferkans — Interactive Telecom Tutor

Theorem: STC Secure Delivery Rate

For the secure coded-caching problem with $K$ users, $N$ files, and cache size $M$ per user with $M \geq 1$ , the achievable secure delivery rate is $R_\text{sec} \;=\; \frac{K(1 - (M-1)/N)} {1 + K(M-1)/N}.$ This is the MAN rate with effective memory $M_\text{eff} = M - 1$ . For $M < 1$ : no secure scheme exists.

Cache splits into two parts: 1 file of random key bits for OTP masking + $(M - 1)$ files of file-content subfiles (MAN structure). Delivery phase XORs each MAN transmission with a fresh key from the cache. Eavesdropper sees only XOR-with-key = uniform noise. Users decode using their cached key + MAN structure.

Proof

Placement

Reserve 1 file of cache for a random key $\mathbf{K}_k$ of size $F$ bits (uniform random, independent of library). Use remaining $(M-1)F$ bits for MAN-style subfile caching with effective memory $M - 1$ .

Delivery

Run MAN delivery on demands with effective memory $M - 1$ : output $X_\text{MAN}$ . Transmit $X = X_\text{MAN} \oplus \mathbf{K}$ , where $\mathbf{K}$ is a key shared by the relevant users (from an appropriate Shamir sharing or reused XOR, see §17.3).

Correctness

Each user $k$ knows $\mathbf{K}$ (constructed to be reconstructible from its cache). It undoes the mask, then decodes via MAN.

Secrecy

$X = X_\text{MAN} \oplus \mathbf{K}$ with $\mathbf{K}$ uniform $\Rightarrow$ $X$ is uniform $\Rightarrow$ $I(\mathcal{W}; X) = 0$ . $\blacksquare$

Definition:
Two-Tier Placement for STC Scheme

STC placement. For secure delivery with cache size $M \geq 1$ :

Tier 1 (key tier): Reserve $F$ bits ( $= 1$ file) of cache for random key material $\mathbf{K}_k$ . The specific key value is user-dependent and shared-randomness-derived so that keys are correlated across users via a $(t, n)$ -secret sharing.
Tier 2 (MAN tier): Use remaining $(M-1)F$ bits for MAN subfile caching. Apply MAN placement with memory $M - 1$ .

In delivery, tier-1 provides the OTP mask; tier-2 provides the file-content subfiles for coded multicast.

The two-tier structure is essential: without tier-1 keys, tier-2 MAN delivery leaks file content to $\mathcal{E}$ . Without tier-2, no caching gain.

STC Secure Delivery

Complexity: Subpacketization:

\binom{K}{t}

. Key material:

\binom{K}{t+1}F/\binom{K}{t} = (K-t)/(t+1) \cdot F

bits of key per delivery round. Amortized:

\leq 1

file of key per user.

Input: Library

\mathcal{W}

; users

1, \ldots, K

; cache size

M \geq 1

; demand vector

\mathbf{d} = (d_1, \ldots, d_K)

.

Output: Broadcast message

X

with

I(\mathcal{W}; X) = 0

.

1. Placement (offline).

2.

\quad

Generate shared random keys

\{\mathbf{K}_\mathcal{S}\}

for each

(t+1)

-subset

\mathcal{S}

, where

t = K(M-1)/N

.

3.

\quad

User

k

stores

\mathbf{K}_\mathcal{S}

iff

k \in \mathcal{S}

(reconstructible via

t

-secret sharing).

4.

\quad

User

k

stores MAN subfiles

W_{n, \mathcal{S}}

for

k \in \mathcal{S}

, with subpacketization

\binom{K}{t}

.

5. Delivery (online).

6.

\quad

Compute MAN XOR

X_\mathcal{S'} = \bigoplus_{k \in \mathcal{S}'} W_{d_k, \mathcal{S}' \setminus \{k\}}

for each

(t+1)

-subset.

7.

\quad

Mask:

\tilde X_{\mathcal{S}'} = X_{\mathcal{S}'} \oplus \mathbf{K}_{\mathcal{S}'}

.

8.

\quad

Broadcast

\tilde X_{\mathcal{S}'}

for all

\mathcal{S}'

.

9. User decoding.

10.

\quad

User

k

reconstructs

\mathbf{K}_{\mathcal{S}'}

for

k \in \mathcal{S}'

from its cache.

11.

\quad

Unmask:

X_{\mathcal{S}'} = \tilde X_{\mathcal{S}'} \oplus \mathbf{K}_{\mathcal{S}'}

.

12.

\quad

Apply MAN decoding to recover

W_{d_k, \mathcal{S}' \setminus \{k\}}

.

13.

\quad

Reassemble

W_{d_k}

.

Each $(t+1)$ -XOR message carries a fresh shared key; shared randomness amortizes over $\binom{K}{t+1}$ messages. Total key material per user stays at $\leq F$ — matches the $M \geq 1$ floor.

STC Rate Gap vs Non-Secure MAN

Absolute rate gap $R_\text{sec} - R_\text{MAN}$ vs number of users $K$ , for several memory ratios $\mu$ . Gap shrinks as $K$ grows — the fixed 1-file secrecy cost becomes negligible at scale.

Parameters

Memory ratio μ = M/N0.25

Max K100

Example: STC Scheme Walkthrough: $K = 3$ , $N = 3$ , $M = 2$

Concrete walkthrough for $K = 3$ users, $N = 3$ files, $M = 2$ per user. Demands $\mathbf{d} = (1, 2, 3)$ .

Solution

Effective memory

$M_\text{eff} = M - 1 = 1$ . MAN parameter: $t = K \cdot M_\text{eff} / N = 3 \cdot 1/3 = 1$ .

Placement

MAN tier: split each file $W_n$ into 3 subfiles $W_{n, \{1\}}, W_{n, \{2\}}, W_{n, \{3\}}$ (each $F/3$ bits). User $k$ stores $W_{n, \{k\}}$ for all $n$ . Key tier: user $k$ also holds random key shares that allow reconstruction of pairwise keys $\mathbf{K}_{\{k,j\}}$ for other $j$ .

MAN delivery

Three XOR messages (one per $(t+1)$ -pair):

$X_{\{1,2\}} = W_{1, \{2\}} \oplus W_{2, \{1\}}$
$X_{\{1,3\}} = W_{1, \{3\}} \oplus W_{3, \{1\}}$
$X_{\{2,3\}} = W_{2, \{3\}} \oplus W_{3, \{2\}}$

Secrecy masking

Broadcast: $\tilde X_\mathcal{S'} = X_\mathcal{S'} \oplus \mathbf{K}_\mathcal{S'}$ for each pair. Eavesdropper sees three masked XORs — uniformly random, zero info about $\mathcal{W}$ .

User 1 decoding

User 1 has $\mathbf{K}_{\{1,2\}}$ and $\mathbf{K}_{\{1,3\}}$ . Unmasks $\tilde X_{\{1,2\}}$ , $\tilde X_{\{1,3\}}$ . Uses cached $W_{2,\{1\}}, W_{3,\{1\}}$ to recover $W_{1,\{2\}}, W_{1,\{3\}}$ . Combined with its cached $W_{1, \{1\}}$ : full $W_1$ .

Rate

$R_\text{sec} = 3(1 - 1/3)/(1 + 3 \cdot 1/3) = 3 \cdot 2/3/2 = 1$ file/use. MAN without secrecy (effective $M = 2$ ) would give $R = 3(1 - 2/3)/(1 + 3 \cdot 2/3) = 1/3$ . Cost of secrecy: 2/3 file per use.

⚠️Engineering Note

STC in Wireless Broadcast Systems

STC is natural for wireless broadcast where the medium is open:

Wi-Fi and cellular. Open-air broadcast; legitimate user decodes via key, eavesdropper gets noise-like signal.
Satellite distribution. DVB-S2 supports CW-C2 (physical layer encryption); STC-style caching could layer on top.
5G multicast (MBMS). Multicast delivery with per-UE keys is specified; adding coded-caching structure is active research.
Comparison to TLS. TLS encrypts unicast with per-session key. STC is information-theoretic: immune to quantum attacks, but requires cache-resident key material.

Limits. Practical keys use pseudorandom generators (PRGs); information-theoretic secrecy requires true randomness. For large-scale deployment, PRGs typically suffice, with standard caveats about cryptographic assumptions.

Practical Constraints

•
Wi-Fi: WPA3 is AES-based (computational security)
•
5G: NAS/AS encryption is AES; info-theoretic uncommon
•
Quantum-safe: STC is natively resistant (no public-key assumption)
•
Key management: coded cache keys need fresh rotation

Key Takeaway

The STC scheme achieves the $\R_\\text{sec} = K(1 - (M-1)/N)/(1 + K(M-1)/N)$ rate by two-tier placement. Tier 1 (1 file per user) provides OTP keys; tier 2 (remaining $M - 1$ ) is standard MAN. Secrecy costs 1 file of memory; the multicasting gain is preserved for the rest.

Sengupta-Tandon-Clancy Secure Delivery Scheme

Theorem: STC Secure Delivery Rate

Placement

Delivery

Correctness

Secrecy

Definition: Two-Tier Placement for STC Scheme

STC Secure Delivery

STC Rate Gap vs Non-Secure MAN

Parameters

Example: STC Scheme Walkthrough: K=3K = 3K=3, N=3N = 3N=3, M=2M = 2M=2

Effective memory

Placement

MAN delivery

Secrecy masking

User 1 decoding

Rate

STC in Wireless Broadcast Systems

Key Takeaway

Definition:
Two-Tier Placement for STC Scheme

Example: STC Scheme Walkthrough: $K = 3$ , $N = 3$ , $M = 2$