Interested Article - Псевдослучайная функция Наора — Рейнгольда

??????????????? ??????? ????? — ?????????? — , ????????? ? 1997 ???? ? ??? ?????????? ????????? ????????????????? ?????????? ? ???????????? ?????????? ? ???????????? ? ???????? ?????? . ?????????????? ????????????? ?????? ??????????????? ??????? ???????? ?????? ?????????????? ????????? ? ??????? ????????????????? ????????? . ?????? ???????? ?????? ? ??? ??????, ??? ????????????? ???????? ?????? ??????? ?????? ? ???????????? , ????????? ???????????? ?? ? ???????? ?????? ??? ?????? ????????????????? ???? .

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? ???????????? ??? ?????????? ??????????????? ??????? ???????? ????? ??????? (??????? ????? ???? ?????????? ????????? ?? ?????????????? ?????), ?????????? ????????? ?????????: ????????????? ?? ????? ?????????? ????? ?????-???? ???????????? ???????? ????? ?????????? ??????? ?? ??????? ????????? ? ???????? ????????? ??????? . ????? $p$ — ??? $n$ - ?????? ??????? ????? , $l$ — ??????? ????? , ?????????? ????????? $p-1$ , ? $g\in {\mathbb {F} }_{p}^{*}$ , — ????????? ??????? ? ????????????????? ???????? $l$ ?? ?????? $p$ . ????? ??????? ????? — ?????????? $f_{a}(x)$ ???????????? ????????? $(n+1)$ -?????? ???????? $a=(a_{0},a_{1},...,a_{n})\in ({\mathbb {F} }_{l})^{n+1}$ ??? ????? $\mathbb {F} _{l}$ ? ?????:

f_{a}(x)=g^{a_{0}\cdot a_{1}^{x_{1}}a_{2}^{x_{2}}...a_{n}^{x_{n}}}\in \mathbb {F} _{p}

??? $x=x_{1},...,x_{n}$ — ??? ???????? ????????????? ?????? ????? $x,$ $0\leq x<2^{n}$ , ? ??????????? ??????? ?????, ???? ??? ?????????? .

??????

????? $p=7$ ? $l=3$ . ? ???????? $g\in \mathbb {F} _{7}^{*}$ ? ????????????????? ???????? $3$ ????? ??????? $g=4$ . ????? ??? $n=3$ , $a=(1,1,3,1)$ ? $x=5$ ??????? $f_{a}(5)$ ??????????? ???

f_{a}(x)=g^{a_{0}\cdot a_{1}^{x_{1}}a_{2}^{x_{2}}...a_{n}^{x_{n}}}=4^{1\cdot 1^{1}3^{0}1^{1}}=4^{1}=4\in \mathbb {F} _{7}

??? ??? ???????? ????????????? ????? $5$ — ??? $(1,0,1)$ .

?????????????? ?????????

?????????? ??????????????? ??????? ????? — ?????????? ??????? $n$ ????????? ?? ?????? $l$ ? ???? ?????????? ? ??????? ?? ?????? $p$ , ??????? ????? ???? ??????? ?? $O\left({\frac {n}{\log n}}\right)$ ????????? ?? ????? ?????? .

??? ?????? ??????? ???? ????????, ??? ?????????? ????? ??????? $p(x)$ , ??? ??? ?????? $a=(a_{0},a_{1},...,a_{n})\in ({\mathbb {F} }_{l})^{n+1}$ , $f_{a}(x)$ ????? ???? ??????????? ?????? ?? ????????? ????????? ??????? $d$ ? ???????, ?? ???????????? $p(n)$ . ??? ????????, ??? ??????? ????? — ?????????? ??????????? ? ???????? .

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??????????????? ??????? ????? — ?????????? ????? ???? ???????????? ? ???????? ?????? ?????? ????????????????? ???? , ??????? ???????????? ?????????? , ?????????????? ? ??????????? ??????? .

????? ???? ????????, ??? ?????? ??????? ????? ???? ???????????? ? :

: ???? ???? ????????????? ????? ???????? ????????????? ?????? ? ??????????? ?? ????????, ??????? ??????? ??????????? ????????? ?????????? ??????, ?? ?? ?????? ????????? ??????? ????????.
?????????? ???? ?????????????? ?? ?????? ???????????? , ??????? ??????? ???????? ?? ????.
?????? ????????? ????????????????? ???????, ??????? ????? ???? ????????? ????????????, ?????????????? ???? ????????? ??????? ??????.
?????????? ?????? ????????????????? ??????????? .

????????????

??????????????? ??????? ????? — ?????????? ????? ??????? ????????????????? ????????? . ????? ?????????????? ???????? ???????? ??????? ? ?????????? ??????: $f_{a}(1)=g^{a_{0}a_{1}},f_{a}(2)=g^{a_{0}a_{2}},\dots ,f_{a}(k)=g^{a_{0}a_{1}^{x_{1}}a_{2}^{x_{2}}...a_{n}^{x_{n}}}$ ? ??? ?????????? ????????? $f_{a}(k+1)$ . ???? $x_{1}=0$ , ?? ????? ???????? $f_{a}(k+1)=g^{a_{0}a_{1}a_{2}^{x_{2}}\dots a_{n}^{x_{n}}}$ , ?????????????? ?????????? ?????? ??? $f_{a}(1)=g^{a_{0}a_{1}}$ ? $f_{a}(k)=g^{a_{0}a_{2}^{x_{2}}...a_{n}^{x_{n}}}$ . ?? ?? ?? ?????????? ???????????? ?????????, ?????????? ?????? ?????? ?????? .

????? ?????, ???????????? ??????????????? ???????? , ????? ?????? ???? ???????????????, ?? ????, ?? ?????? ???? ????????? ?? ?????????? ?????????? ?????? ?????. ????? ${\mathcal {A}}^{f}$ ?????????? ????????, ??????? ?????? ? ??????? , ??????? ????????? $f_{a}(x)$ . ???????????, ??? ??????????? ??? $\mathbb {F} _{p}$ . ????? ??? ?????? ?????????????? ??????????????? ????????? ${\mathcal {A}}$ ? ?????????? ???????? $n$ ????????? :

$|{\text{Pr }}[{\mathcal {A}}^{f_{a}(x)}(p,g)=1]-{\text{Pr }}[{\mathcal {A}}^{R}(p,g)=1]|<\epsilon$ , ??? $\epsilon (n)$ — ???????????? ????.

?????? ??????????? ????? ???? ??????? $s=(p,g,a)$ , ?? ??????? ???????? ?????? ???????, ? ?????? ?????????? ?? ?????????? ?????? ???? $(p,g)$ ? ??????? $R_{a}(x)$ ????? ????????? ???? ??????? $\{0,1\}^{n}\to {\mathbb {F} }_{p}$ .

???????? ?????????

????? ?? ???????????? ??????????? ????, ????????? ?????????????????? ????? ???? ???????? ??? ????????????????? ?????, ???????? ?? ???????? ?????????. ???????? ????????? $n$ -?????????? ?????????????????? $W(x),\;x=0,\dots ,n-1$ , ??? ??????? ${\mathcal {R}}$ — ??? ????? $l$ ??????????? ????????? ????????????? ??????????? $W(x+l)=A_{l-1}W(x+l-1)+\dots +A_{0}W(x),\;x=0,\dots ,n-(l-1)$ , ??? $A_{0},\dots ,A_{l-1}\in {\mathcal {R}}$ . ??? ??????? ????? — ?????????? ???? ????????, ??? ?????????? ????? $\gamma >0$ ? $n\geqslant (1+\gamma )\log l$ , ??? ??? ?????? $\delta >0$ ? ?????????? ???????? $l$ ???????? ????????? $L_{a}$ ?????????????????? $f_{a}(x)$ , $0\leq x<2^{n}$ , ????????????? ?????????? ??????????? :

L_{a}\geqslant {\begin{cases}l^{1-\ \delta \,\!}&{\text{, ???? }}\gamma \,\!\geqslant 2\\l^{\left({\tfrac {\ \gamma \,\!}{2-\ \delta \,\!}}\right)}&{\text{, ???? }}\gamma \,\!<2\end{cases}}

??? ????, ????? ?? ????? ??? $3(l-1)^{n-\delta }$ ???????? $a\in (\mathbb {F} _{l})^{n+1}$ .

????????????? ?????????????

?????????????? ????????????? $f_{a}(x)$ ??????????????? ?????? ? ???????????? ????????????? ????? ??? ???? ???????? $a\in ({\mathbb {F} }_{l})^{n+1}$ :

????? ${\mathbf {D} }_{a}$ — ????????? $\{f_{a}(x)|0\leq x<2^{n}\}$ ???, ??????? ???????, ?????????? ????????????? ???????? ??????????????? ??????? ?? ???????????? ????????????? . ???? ????????, ??? ???? $n=\log p$ — ??? ??????? ????? $p$ , ????? ??? ???? ???????? $a$ ? $(\mathbb {F} _{l})^{n+1}$ ??????????? ??????????? ${\mathbf {D} }_{a}\leq \Delta (l,p)$ , ??? :

\Delta (l,p)={\begin{cases}p^{\left({\tfrac {1-\ \gamma \,\!}{2}}\right)}l^{\left({\tfrac {-1}{2}}\right)}\log ^{2}p&{\text{ ???? }}l\geqslant p^{\gamma \,\!}\\p^{\left({\tfrac {1}{2}}\right)}l^{-1}\log ^{2}p&{\text{ ???? }}p^{\gamma \,\!}>l\geqslant p^{\left({\tfrac {2}{3}}\right)}\\p^{\left({\tfrac {1}{4}}\right)}l^{\left({\tfrac {-5}{8}}\right)}\log ^{2}p&{\text{ ???? }}p^{\left({\tfrac {2}{3}}\right)}>l\geqslant p^{\left({\tfrac {1}{2}}\right)}\\p^{\left({\tfrac {1}{8}}\right)}l^{\left({\tfrac {-3}{8}}\right)}\log ^{2}p&{\text{ ???? }}p^{\left({\tfrac {1}{2}}\right)}>l\geqslant p^{\left({\tfrac {1}{3}}\right)}\\\end{cases}}

?

\gamma =2,5-\log 3\approx 0,9150\dots

.

??? ???????? ?? ????? ????????????????? ?????????????????? ????????, ?? ???? ?? ??? ?? ???? ?????????, ??? ????? ?? ??????? ???????????????? ??????????? ??? ?????????? ??????? .

?????????????????? ?? ????????????? ??????

?????? ???? ??????? ?? ????????????? ?????? ????????? ???????? ????????????????? ????????? ??????????????? ???????. ????? $p>3$ — ??????? ????? ? $E$ — ????????????? ?????? ??? $\mathbb {F} _{p}$ . ????? ????? ?????? $a=(a_{0},a_{1},\dots ,a_{n})\in ({\mathbb {F} }_{l}^{*})^{n+1}$ ?????????? ???????? ?????????????????? $f_{a}(x)=(a_{0}a_{1}^{x_{1}}a_{2}^{x_{2}}\dots a_{n}^{x_{n}})G\in \mathbb {F} _{p}^{2}$ ? ??????????? ?????? $\langle G\rangle$ , ??? $x=x_{1}\dots x_{n}$ — ??????? ????????????? $x,\;0\leq x<2^{n}$ , $G\in \mathbb {F} _{p}^{2}$ , — ????????? ??????? ?? $E$ ? ????????????????? ???????? $l$ , ? $(a_{0}a_{1}^{x_{1}}a_{2}^{x_{2}}\dots a_{n}^{x_{n}})G$ — ??? ???????? ?????????? ???????? $G$ ? ??????? $a_{0}a_{1}^{x_{1}}a_{2}^{x_{2}}\dots a_{n}^{x_{n}}$ ???????????? ????????? ???????? ? ???????? ?????? $\mathbb {F} _{p}$ -???????????? ????? ????????????? ?????? . ? ????? ???????????? ?????????????????? ????? — ?????????? ?? ????????????? ?????? ???????????? ??? $u_{k}=X(f_{a}(k))\;$ , ??? $X(P)$ ?????????? ???????? ????? $P\in E$ .

???? ????????????? ????? — ???????? ?????????, ?? ??????? $k$ ?? ?????????? ??? ?????????? $u_{k}$ ?? ?????????????? ?????. ??? ????????????? ?????? ????? — ?????????? ???? ????????, ??? ?????????? ????? $\gamma >0$ ? $n\geqslant (1+\gamma )\log l$ , ??? ??? ?????? $\delta >0$ ? ?????????? ???????? $l$ ???????? ????????? $L_{a}$ ?????????????????? $(u_{k})_{k=0}^{2^{n}-1}$ ????????????? ?????????? ??????????? :

L_{a}\geqslant \min \left(l^{{\frac {1}{3}}-\delta },{\frac {l^{(\gamma -3\delta )}}{\log ^{2}l}}\right)

??? ????, ????? ?? ????? ??? $O((l-1)^{n-\delta })$ ???????? $a\in (\mathbb {F} _{l}^{*})^{n+1}$ .

??. ?????

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