Lehrstuhl fr Theoretische Informationstechnik Lehr- und - - PowerPoint PPT Presentation

Lehrstuhl für Theoretische Informationstechnik Lehr- und Forschungseinheit für Nachrichtentechnik

1

Let H and 𝐼 be finite-dimensional complex Hilbert spaces. We consider the channels W: A → S(H) V: A → S(𝐼) (W,V) is called a classical-quantum wiretap channel

• W represents the communication link to the legitimate receiver
• V‘s output is under control of the wiretapper

2

3

4

5

6

7

For classical-quantum channel V, probability distribution P, and quantum states σ and ρ such that supp (ρ) ⊂ supp (σ) S (σ) ≔-tr(σ log σ) χ (P;V) ≔ 𝑇 ∑ 𝑄 𝑦 𝑊 𝑦

∈

− ∑ 𝑄 𝑦 𝑇 𝑊 𝑦

∈

D (σ‖ρ) ≔ tr (σ log σ − log ρ ) D (σ‖ρ) ≔ log tr(σρ)

The strong secrecy capacity of (W,V) is equal to the maximum is taken over finite input sets M, input probability distributions P on M, and classical channels E : M → P(𝑌).

8

The semantic secrecy capacity of (W,V) is equal to its strong secrecy capacity

9

10

This is only an existence statement How to choose the semantically secure message subsets

11

Idea: biregular irreducible functions (BRI functions) Similar to universal hash functions for strong secrecy The channel users share a random seed

• seeds

messages

12

Let S, X, N be finite sets. A function f : S × X → N is called biregular irreducible (BRI) if there exists a subset M of N such that for every m ∈M we have

X S 𝑒 = 4, 𝑒 = 4

13

The modular BRI scheme: (E,D) is a transmission code for W and f is a BRI function. 𝑔

(m)

denotes the random choice of a preimage of f given seed s and message m.

14

ϵ-subnormalized classical-quantum channel V ′ : For all x V ′(x) ≥ 0 1 − ϵ ≤ tr V ′(x) ≤ 1,

15

For ϵ-subnormalized V’≤ V and random variable M independent of S, it holds where λ(f,m) is the second largest singular value of 𝑄

,

16

Define V(X) :=

• || ∑ V(𝑦)
• . For M independent of S

For ϵ-subnormalized V’≤ V and fixed m For ϵ-subnormalized V’≤ V and fixed m

17

For ϵ-subnormalized V’ and fixed m

18

For any probability distribution P over X, there exist BRI modular codes achieving the semantic secrecy rate χ(P;W) − χ (P;V) using transmission codes achieving the transmission rate χ(P;W).

19

transmission codes achieving the transmission rate χ (P;W) with P-typical codewords 𝑦 , there is a V′ such that rank[V′(𝑌)] max∈ V′(𝑦) ≤ 2n(χ(P,V)+δ) Choose |S|≥ 2−n(χ(P,V)+α), |M|≤ 2−n(χ (P;W) − χ(P,V)+β) λ(f,m) ≤ 2−n(χ(P,V)+2δ)

20

The semantic secrecy capacity of (W,V) is equal to its strong secrecy capacity, and can be achieved using transmission codes for W

21

[01100...|110100110000101100111......]

seed message public code BRI modular code

22

23

24

Beware of eavesdropper

#StaySafe