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| <p> $\mathbf{C_{test}}$ </p> |
| <p> $L_{seg}=\mathrm{BCE}(\mathbf{\hat{y}},\mathbf{M_{q}})$ </p> |
| <p> $\mathbf{X}\in\mathbb{R}^{1\times 1\times 2C}$ </p> |
| <p> $\mathbf{P^{\prime}}\in\mathbb{R}^{1\times 1\times C}$ </p> |
| <p> $\mathbf{W}=\mathrm{sigmoid}(f_{2}(\sigma(f_{1}(\mathbf{X}))))$ </p> |
| <p> $\mathbf{F_{r}}\in\mathbb{R}^{H/8\times W/8\times C}$ </p> |
| <p> $\mathbf{F^{\prime}_{q}}\in\mathbb{R}^{H/4\times W/4\times C}$ </p> |
| <p> $\mathbf{\hat{y}}\in\mathbb{R}^{H\times W\times 1}$ </p> |
| <p> $e_{i}=\{\mathbf{S_{i}},\mathbf{Q_{i}}\}$ </p> |
| <p> ${\sf msg^{*}}$ </p> |
| <p> ${\sf sk_{sanit}}$ </p> |
| <p> ${\sf AD}_{i}({\sf MODIFY})=1\}$ </p> |
| <p> $({\sf msg},\sigma)$ </p> |
| <p> ${\sf Fixed_{AD}(msg^{*})}={\sf Fixed_{AD}}({\sf msg}_{i})$ </p> |
| <p> $\{\mathcal{S}_{1},\mathcal{F},\mathcal{S}_{2}\}$ </p> |
| <p> ${\sf msg^{*},\sigma^{*},pk^{*}_{sig},pk_{sanit}}$ </p> |
| <p> $({\sf sk_{sign},pk_{sign}})$ </p> |
| <p> $\sigma_{2}^{{}^{\prime}}$ </p> |
| <p> $({\sf msg_{1}},{\sf MODIFY}_{1})$ </p> |
| <p> ${\sf sec_{k}}$ </p> |
| <p> $({\sf pk*_{sanit},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}^{{\sf |
| Signature% |
| (\cdot,sk_{sign},\cdot,\cdot)}}({\sf pk_{sign}})$ </p> |
| <p> ${\sf msg_{0}=\mathcal{H}(0||{\sf msg_{fix}}||{AD}||{\sf pk_{sanit}})}$ </p> |
| <p> $\displaystyle\mbox{or }{\sf msg^{*}\notin\{MODIFY(msg_{i})\;|\;MODIFY\mbox{ % |
| with }}$ </p> |
| <p> $r_{(1)}\left(\bar{\delta}_{1},\dots,\bar{\delta}_{n}\right)=\dots=r_{(k)}\left% |
| (\bar{\delta}_{1},\dots,\bar{\delta}_{n}\right)=0.$ </p> |
| <p> $({\sf 0,{msg}^{*}_{fixed},AD^{*},pk^{*}_{sanit}})=(0,{\sf msg}^{*}_{{\sf fixed% |
| },i},{\sf AD}^{*}_{i},{\sf pk}^{*}_{{\sf sanit},i})$ </p> |
| <p> $({\sf 0,{msg}^{*}_{fixed},AD^{*},pk^{*}_{sanit}})$ </p> |
| <p> $\{\mathcal{Q},\mathcal{X},\mathcal{Y}\}$ </p> |
| <p> $\sigma^{*}=(\sigma_{1}^{*},\sigma_{2}^{*},{\sf AD}^{*})$ </p> |
| <p> $({\sf msg^{*}_{fixed}},{\sf AD^{*},pk_{sign},pk^{*}_{sanit},\sigma_{1}^{*}})$ </p> |
| <p> $({\sf pk*_{sanit},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}_{\sf sanit}^% |
| {{\sf Signature(\cdot,sk_{sign},\cdot,\cdot)}}({\sf pk_{sign}})$ </p> |
| <p> ${\sf pk_{sanit}}$ </p> |
| <p> ${\sf x_{0}}=\mathcal{S}_{1}^{-1}({\sf y})\in\mathbb{F}_{q}^{m},{\sf x_{1}}=% |
| \mathcal{F}_{1}^{-1}({\sf x_{0}})\in\mathbb{F}_{q}^{n}$ </p> |
| <p> $\mathcal{H}:\{0,1\}^{*}\rightarrow\mathbb{F}^{m}$ </p> |
| <p> $\alpha_{1}=\mathcal{S}^{-1}({\sf msg_{1}}),\beta_{1}=\mathcal{F}^{-1}(\alpha_{% |
| 1})$ </p> |
| <p> $\displaystyle\mbox{and }{\sf Judge({\sf msg^{*},\sigma^{*},pk_{sign},pk*_{% |
| sanit}})=Sig}$ </p> |
| <p> $\forall i=1,2,\ldots,\Delta$ </p> |
| <p> $({\sf pk^{*}_{sign},{\sf msg^{*}},\sigma^{*}})$ </p> |
| <p> ${\sf msg}\in\{0,1\}^{*}$ </p> |
| <p> ${\sf msg_{1}=\mathcal{H}(1||msg||pk_{sanit}||pk_{sign})}$ </p> |
| <p> $(n=160,m=64)$ </p> |
| <p> $(b=0)$ </p> |
| <p> ${\sf msg^{*}_{fixed}}={\sf FIXED_{AD^{*}}(msg^{*})}$ </p> |
| <p> ${\sf Fixed_{AD}}$ </p> |
| <p> $\mathcal{X}:\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}^{m}$ </p> |
| <p> $\mathcal{S}_{2}:\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}^{n}$ </p> |
| <p> ${\sf msg^{\prime}}$ </p> |
| <p> ${\sf msg_{2}=\mathcal{H}(1||msg^{\prime}||pk_{sanit}||pk_{sign})}$ </p> |
| <p> $\delta(\kappa)={\sf Prob}[{\sf EXP\mbox{-}Immutability_{\mathcal{G}}^{SSS}}=1]$ </p> |
| <p> $({\sf msg}_{i},{\sf AD}_{i},{\sf pk_{sign}},{\sf pk_{sanit}}_{i})$ </p> |
| <p> $j=q+1,\dots,r$ </p> |
| <p> $\delta(\kappa)={\sf Prob}[{\sf EXP\mbox{-}Sanitizer\mbox{-}Acc{\mathcal{G}}^{% |
| SSS}}=1]$ </p> |
| <p> ${\sf y}=\mathcal{P}({\sf x})$ </p> |
| <p> ${\sf(msg_{0},{\sf MODIFY}_{0}),(msg_{1},{\sf MODIFY}_{1})}$ </p> |
| <p> ${\sf msg^{\prime}}\in\{{\sf MODIFY(msg)}\;|\;{\sf MODIFY}\mbox{ with }{\sf AD(% |
| MODIFY)}=1\}$ </p> |
| <p> $\displaystyle{\sf msg^{\prime}}\leftarrow{\sf MODIFY(msg)}$ </p> |
| <p> ${\sf(pk_{sanit},sk_{sanit})}\leftarrow{\sf KGen\mbox{-}Sanit(1^{\kappa})}$ </p> |
| <p> $a\leftarrow\mathcal{G}^{{\sf Signature(\cdot,sk_{sign},\cdot,\cdot)},{\sf |
| Sanitization% |
| (\cdot,\cdot,sk_{sanit},\cdot)},{\sf LoRSanit(\cdot,\cdot,\cdot,sk_{sign},sk_{% |
| sanit},b)}}({\sf pk_{sign},pk_{sanit}})$ </p> |
| <p> ${\sf(msg^{\prime}_{j},\sigma^{\prime}_{j})\leftarrow Sanitization(msg_{j,b},% |
| MODIFY_{j,b},\sigma_{j,b},pk_{sign},sk_{sanit})}$ </p> |
| <p> $({\sf msg_{j,0}},{\sf Modify_{j,0}}),({\sf msg_{j,1}},{\sf Modify_{j,0}},)$ </p> |
| <p> ${\sf LoRSanit(\cdot,\cdot,\cdot,sk_{sign},sk_{sanit},b)}$ </p> |
| <p> ${\sf msg_{fixed}}\leftarrow{\sf FIXED}_{{\sf AD}_{i}}({\sf msg}_{i})$ </p> |
| <p> $(\cdot,{\sf sk_{sign}},\cdot,\cdot)$ </p> |
| <p> $\mathcal{R}=\left(r_{(1)}(\delta,\dots,\delta_{n}),\dots,r_{(k)}(\delta_{1},% |
| \dots,\delta_{n})\right)$ </p> |
| <p> ${\sf msg,{\sf MODIFY},\sigma,pk_{sign},}\\ |
| {\sf sk_{sanit}}$ </p> |
| <p> $\mathcal{G}_{\sf sanit}$ </p> |
| <p> $\sigma_{j,b}\leftarrow{\sf Signature({\sf msg_{j,b},sk_{sign},pk_{sanit},AD_{j% |
| }})}$ </p> |
| <p> ${\sf EXP\mbox{-}Unforgeability_{\mathcal{G}}^{SSS}}$ </p> |
| <p> $\mathcal{R}(\sigma_{2})\stackrel{{\scriptstyle?}}{{=}}{\sf msg_{2}}$ </p> |
| <p> $(\bar{\delta}_{1},\dots,\bar{\delta}_{n})\in\mathbb{F}_{q}^{n}$ </p> |
| <p> $\mathcal{S}_{1}\circ\mathcal{F}\circ\mathcal{S}_{2}$ </p> |
| <p> $({\sf msg^{\prime}}_{i},{\sf\sigma^{\prime}}_{i},{\sf pk}_{{\sf sanit},i})$ </p> |
| <p> ${\sf msg_{FIX}=FIXED_{AD}(msg)}$ </p> |
| <p> ${\sf(pk_{sign},sk_{sign})}\leftarrow{\sf KGen\mbox{-}Sign(1^{\kappa})}$ </p> |
| <p> ${\sf pk^{*}_{sign}},{\sf msg^{*}})\neq({\sf pk}_{{\sf sign},i},{\sf msg^{% |
| \prime}}_{i})$ </p> |
| <p> ${\sf\sigma_{1}}$ </p> |
| <p> ${\sf Mul\mbox{-}SAN}$ </p> |
| <p> $GF(16)$ </p> |
| <p> $1,{\sf msg^{*},\sigma^{*},pk^{*}_{sanit},pk_{sign}}$ </p> |
| <p> ${\sf msg_{fix}}$ </p> |
| <p> ${\sf MODIFY}$ </p> |
| <p> $(b=1)$ </p> |
| <p> $\mathcal{Y}:\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}^{n}$ </p> |
| <p> ${\sf msg,sk_{sign},pk_{sanit},AD}$ </p> |
| <p> $\delta(\kappa)$ </p> |
| <p> ${\sf msg}_{i}$ </p> |
| <p> $\mathcal{O}_{\sf LoR}$ </p> |
| <p> ${\sf EXP\mbox{-}Immutability_{\mathcal{G}}^{SSS}}$ </p> |
| <p> ${\sf pub_{k}}$ </p> |
| <p> ${\sf msg,\sigma,pk_{sign},pk_{sanit}}$ </p> |
| <p> ${\sf AD}_{j}$ </p> |
| <p> $\mathcal{R}=\mathcal{Q}\circ\mathcal{X}\circ\mathcal{Y}:\mathbb{F}_{q}^{n}% |
| \rightarrow\mathbb{F}_{q}^{m}$ </p> |
| <p> ${\sf Sig}$ </p> |
| <p> $({\sf pk^{*}_{sanit},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}_{\sf sanit% |
| }({\sf pk_{sign}})$ </p> |
| <p> $i=1,\dots,\Delta$ </p> |
| <p> ${\sf AD(MODIFY)}\in\{0,1\}$ </p> |
| <p> ${\sf(pk_{sanit},sk_{sanit})}\leftarrow$ </p> |
| <p> ${\sf msg^{*}\notin\{MODIFY(msg)\;|\;MODIFY\mbox{ with }}$ </p> |
| <p> $\displaystyle{\sf Verification({\sf msg^{*},\sigma^{*},pk_{sign},pk^{*}_{sanit% |
| }})=true}$ </p> |
| <p> ${\sf Signature(\cdot,sk_{sign},\cdot,\cdot)}$ </p> |
| <p> ${\sf msg_{fix}=Fixed_{AD}({\sf msg})}$ </p> |
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