Secret Key Agreement and Secure Omniscience of Tree-PIN Source with Linear Wiretapper
Praneeth Kumar V.
PhD student, ECE, IISc
Joint work with
Navin Kashyap (IISc), Chung Chan (CityU) and Qiaoqiao Zhou (CUHK)
EECS Research Students Symposium - 2022
Multi-terminal Source Model
\mathsf{F}^{(n)}
\mathsf{Z}^n_1
\mathsf{Z}^n_m
\mathsf{Z}^n_{\text{w}}
P_{\mathsf{Z}_1\ldots\mathsf{Z}_m\mathsf{Z}_{\text{w}}}
1
\ldots
m
\text{w}
\mathsf{E}^{(n)}_1
\mathsf{E}^{(n)}_m
Multi-terminal Source Model
\mathsf{F}^{(n)}
\mathsf{Z}^n_1
\mathsf{Z}^n_m
\mathsf{Z}^n_{\text{w}}
P_{\mathsf{Z}_1\ldots\mathsf{Z}_m\mathsf{Z}_{\text{w}}}
1
\ldots
m
\text{w}
\mathsf{E}^{(n)}_1
\mathsf{E}^{(n)}_m
- Secret Key Agreement
\mathsf{K}^{(n)}\approx
\mathsf{K}^{(n)}\approx
\log |\mathcal{K}^{(n)}| - H(\mathsf{K}^{(n)} \mid \mathsf{F}^{(n)},\mathsf{Z}_{\text{w}}^n) \rightarrow 0
(Secrecy condition)
- Wiretap secret key capacity,
C_{\text{W}}=\sup \left\lbrace \liminf_{n \to \infty} \frac{1}{n} \log |\mathcal{K}^{(n)}| \right\rbrace
Multi-terminal Source Model
\mathsf{F}^{(n)}
\mathsf{Z}^n_1
\mathsf{Z}^n_m
\mathsf{Z}^n_{\text{w}}
P_{\mathsf{Z}_1\ldots\mathsf{Z}_m\mathsf{Z}_{\text{w}}}
1
\ldots
m
\text{w}
\mathsf{E}^{(n)}_1
\mathsf{E}^{(n)}_m
- Secure Omniscience
- Secret Key Agreement
\mathsf{Z}_{V}^n\approx
I(\mathsf{Z}_V^n \wedge \mathsf{F}^{(n)} \mid \mathsf{Z}_{\text{w}}^n)
- Wiretap secret key capacity,
C_{\text{W}}
\mathsf{Z}_{V} := (\mathsf{Z}_{1}, \ldots,\mathsf{Z}_{m})
\mathsf{Z}_{V}^n\approx
Multi-terminal Source Model
\mathsf{F}^{(n)}
\mathsf{Z}^n_1
\mathsf{Z}^n_m
\mathsf{Z}^n_{\text{w}}
P_{\mathsf{Z}_1\ldots\mathsf{Z}_m\mathsf{Z}_{\text{w}}}
1
\ldots
m
\text{w}
\mathsf{E}^{(n)}_1
\mathsf{E}^{(n)}_m
- Secure Omniscience
- Secret Key Agreement
\mathsf{Z}_{V}^n\approx
I(\mathsf{Z}_V^n \wedge \mathsf{F}^{(n)} \mid \mathsf{Z}_{\text{w}}^n)
- Minimum leakage rate for omniscience,
\mathsf{Z}_{V} := (\mathsf{Z}_{1}, \ldots,\mathsf{Z}_{m})
R_{\text{L}}=\inf \biggl\lbrace \limsup_{n \to \infty} \frac{1}{n}
\biggr\rbrace
\mathsf{Z}_{V}^n\approx
- Wiretap secret key capacity,
C_{\text{W}}
Multi-terminal Source Model
\mathsf{F}^{(n)}
\mathsf{Z}^n_1
\mathsf{Z}^n_m
\mathsf{Z}^n_{\text{w}}
P_{\mathsf{Z}_1\ldots\mathsf{Z}_m\mathsf{Z}_{\text{w}}}
1
\ldots
m
\text{w}
\mathsf{E}^{(n)}_1
\mathsf{E}^{(n)}_m
- Secure Omniscience
- Secret Key Agreement
- Minimum leakage rate for omniscience,
R_{\text{L}}
- Wiretap secret key capacity,
C_{\text{W}}
C_{\text{W}}
\geq H(\mathsf{Z}_V|\mathsf{Z}_{\text{w}})-
R_{\text{L}}
- How are they related?
- For what sources,
R_{\text{L}}
= H(\mathsf{Z}_V|\mathsf{Z}_{\text{w}})-
C_{\text{W}}
(duality) holds?
Omniscience protocol + Key extraction
Main Result
\mathsf{Z}_{\text{w}}= \mathsf{X} \begin{bmatrix}1 & 0 \\ 1 & 0 \\ 0 & 0 \\ 0 & 1 \\ 0 & 1 \end{bmatrix}= \begin{bmatrix}\mathsf{X}_{a1}+\mathsf{X}_{a2} \!\!\! & \mathsf{X}_{c1} +\mathsf{X}_{d1} \end{bmatrix}
\mathsf{X}= \begin{bmatrix}\mathsf{X}_{a1}\!\!\! & \mathsf{X}_{a2} \!\!\! & \mathsf{X}_{b1} \!\!\! & \mathsf{X}_{c1} \!\!\! & \mathsf{X}_{d1} \end{bmatrix} \\
\text{uniformly dist. over } (\mathbb{F}_q)^5
%\text{unif} (\mathbb{F}_q)^5
\mathsf{Z}_3 = [\mathsf{X}_{b1} \quad \mathsf{X}_{c1}]
\mathsf{Z}_{V}
4
3
2
5
1
\mathsf{X}_{c1}
\mathsf{X}_{b1}
\mathsf{X}_{d1}
[\mathsf{X}_{a1} \quad \mathsf{X}_{a2}]
- Duality holds for Tree-PIN source with linear wiretapper
- Secure omniscience achieves
C_{\text{W}}
- Derivation of the single letter expressions for
and
C_{\text{W}}
R_{\text{L}}
Main Result
\mathsf{Z}_{\text{w}}= \mathsf{X} \boldsymbol{W}
\mathsf{Z}_{V}
- Key features of the proof:
- Reduction to a class called irreducible sources
- Linear (non-interactive)
- Perfectly aligns with ,
\mathsf{Z}_{\text{w}}^n
H(\mathsf{Z}_{\text{w}}^{n} \mid \mathsf{F}^{(n)})=0
- Duality holds for Tree-PIN source with linear wiretapper
- Secure omniscience achieves
C_{\text{W}}
- Existence of an optimal secure omniscience scheme
\mathsf{F}^{(n)}
- Derivation of the single letter expressions for
and
C_{\text{W}}
R_{\text{L}}
Thank you
EECS22 talk
By Praneeth Kumar
EECS22 talk
- 175
