? ???????? , ????? ?????? ??????? (????? ?????? RNG, ???? ????? ?? ????? ? ????? Random number generator) ??? ???? ???? ?? ?????? ?????? ??? ?????? (??????) ???? ????? ?? ??? ?????????? ?????, ?????? ??????? . ?????? ??????? ???? ?????? ???? ?????? ?????? ???????, ?? ?? ?? ???? ?????? ????, ?? ?? ?? ????? ?????? ?????? ?????? ????????? ????? ???? ??? ???? ???? ?????. ???? ????? ???? ?? ?????? ?????? ????? ?????? ???????, ???? ??? ?????? ???? ??? ????? ?????? ????? ???? ??? ???????? ?? ?????? ??????? .

?????? ???????????? ?????? ?????? ??????? ? ??? ?????? ???? , ???? ???? , ????? ????? , ???? ????? ?????? ???? ? ????? ?????? ?????? ????, ????? ????? ????? ??????? . ?? ?? ??????? ???? ????? ???? ????? ?? ?????? ??????? ?????? ?????, ???? ? ????????? ????? ?? ? ??????????? , ??????? ?? ????? ?????? ?????. ?????? ??? ?????, ??? ??? ????, ????? ?? ????? ???????? ??????? ??????? ???? ????? ?????? ???????, ???? ?????? ?????? ????? ??????? ?? ????????, ????? ?????? ???? ?? ??? ??????, ?????? ?????? ????? ? ???? ?'? .

????? ????????? (?????) [ ????? ??? ???? | ????? ]

????? ????? ?????? ?????? ???????, ????? ???? ????? ????? ???? ??? ??????? ??? ?? ??????? ??????? . ????? ?????? ?????? ??????? ??? ????? ??????? ???? ?????? ????, ????? ??? ???? , ??? ????? , ????? ??????????? ?? ?????? "????????" ?????. ??????? ???? ?????? ??????? ????? ?????? ???????, ??? ????? ?????? ???? ?????? ????.

????? ?????? ??? ???? ????? ???? ????? ??????? ?????? ?? ????? ????? ?????? ??????????? ? ???? ??????? ????? ?? ????? ???? ??????? . ?????? ????? ?????? "???? ???" ??????? ?????? ?????? ??????? ???? ????? ?????? ?????. ???? ?????, ????? ?????? ??????? ???? ?????? ??? ???? ??? ????? ???????? (?????) ?????? ??? ???? ????? ?????, ?? ????? ????? ????? ????? ??????? RS-232 . ????? ???? ?????? ???? ?????? ?????? ??????? ?????????, ???? ??????? / ?????? .

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

??????? ??????? ?????? ????? ???? ??????? ??????? ?????. ???? ??????? ?????? ????? ?????? ??? ??? ????? ???????? ?????? ???????. ????? RAND ????? ????? ???????? ????? ???? ????? ??? ????? ?????? ???????. ??????? ????? ?? ???? ????? ????? ? ???????? ?? ????? ???? ????? ?????? ?? ??????? ??????? ??????. ??????? 64 ???? ???? ???? ??????? ????? ??????? ???? ????? ????? ????? ?????? ???? ?????? ????? ??? ?????? ? ??? ??????? ?????? ?????? ???????. ???? ???????? ???? ???? ????? ?????? ?????????, ??????? ???????? ???? ?????? ???? ??? ?????? ????? ????? ?? ????? ???? ????? ????? ?? ????? ???????? ?????? ???????? ?????.

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

?????? ?????? [ ????? ??? ???? | ????? ]

???? ???? ?????? ?????? ???????? ????? ??????? ??????:

• ??? ??? ; ???? ??? ?????? ? ???? ????? . ????? ????? ??? ????? ??? , ???? ????? ?? ??????? ??????? ??????? ????? ?????? ??????? ??? ??????. ???? Atari 8-bit ????? ????? ???.
• ????? ??????????? ; ??? ?? ?????? ??????? ????? ??? ???????. ?????? ????? ??????? ???? ????? ?????? ? ???? . ???? HotBits ????? ?????? ???.
• ??????? ; ?????? ??? ??????? ??? ???? ??? ?????. ???? ????? ?? ???? id Quantique ????? ?? ????????? ????.
• ??? ???? ; ??? ????? ? ??? ?????? ??????? ???? ?????? ???? ??? ?????. ?????? ???? ?? ???? ?????? ??????? ????????? ?? ??????. ?????? ???? ?????? ??????? ?????? ?????? ???????? ??????.
• ??? ?????; ??? ????? ? ????? ????? ?? ? ????? ??? .
• ??? ????????? ?? ??? ??? ; ???? ??????? ???? ???? ?????? ?????. ???? Random.org ?????? ????? ??. ????? ?? ???? ????? ???? ????? ?????? ??????? ???? ????? ???? ?????.
• ????? ??? ; ?????? ????? ?? ????? ?????? ??????? ?? ????? ????? ?????? ???? ?????? ?????? ?????? ??????? ????????. ?????? ?????? Lavarand ?? ???? ??????? ?????? ??? ????? ???.
• ????? ??????? ?? ???? ????? ?? ?????? ?? ???? ???? ?????. ???? ?????????? ?????? ????????? ?????? ? ????? ????? ?????? ????? ???? ????? ??? ????? ?????? ???????.
• ????? ???? (clock drift); ????? ?????? ??????? ??? ????? ?? ?????? ???? ?? ???? ???? ???. ???? ???? ????? ??? ??????? ???? ???? ????? ???? ????? ????????. ?????? ?????? ?? ???? ??????? ?? ????? ?????? ??? ????? ??????. ???? ?????? ?? ????? ????? ????? ???? ????? ????? ??. ?????? ????? ???? ?????? ??? ???? ????? ??? ?????? ????? ?????? ????? ????? ?????? ????? ????? ?? ????? ?????. ??? ??????? ???? ??-??????? ???? ????? ???? ?????. ?? ?????? ???? ???? ????? ????? ?????? ?????? VIA Technologies , ???? ??? ??? ????? ?? ????? ?? ????? ?????? ??????? ?????. ????? XOR ?? ????? ??? ????? ????? ?? ????? ?????? ?????? ???? ?????? ????? ?????? ?????? ?? ??????? ????????. ??????? ????? ???? ?? ????? ?? ?????? ???????? ???????? ??????? ???????? ??? ???????? ???? ????? ?????? ?? ??? ??????? ???? ????? ???????? ?????. ??? ?????? ?? ??????? ???? ????, ???? ?????? ???? ???? ?? ????? ??? ??? ?????? ?????? ????? ?????? ???? ????? ??? ????? ?????? ????? ??????.

????? ???????? ???????? ?? ??????? ??????? ???? ??? ????? ?????? ???????, ????? ????? ????? ???? ??????? ?? ???? ????? ?????? ???????? ??? ??????? ???? ?????? ???????? ??????? ????????. ????? ???, ?????? ????? ???? ????? ??? ??????? ?????? ?????? ?????? ??????? ??? ???????? ?????????? (???????) ?????, ?? ??? ????? ???????? ???? ??????. ????? ????????? ?????? ?????? ???????? ????? ????? ????? ??????? ????; ????????? ??????? ?????? ??????? ???? ?????? ????? ?? ??????????? ???? ?? ????? ????? ???????. ?????????? ????? ????? Yarrow ??????? ??????? ?? ????? ?????? ??????? ??????? ???? "????" (pool), ?? ?? ??????? ??????? ?????????? ???????????? ??? ???????? ????? ?? ???????? ?????, ???? ????? ??? ?????. ??? ???? ????? ???????? ??? ???? ?????? ????. ?????? ?? ????? ??????? ???????, ???? ???? ???????? ???? ?? ??????. ?? ??? ???? ???? ?? ????? ???? ????? ?????? ?????? ??????, ?? ???? ???? ???? ?????? ??? ?????? ????? ??????.

?????? Cryptolib ?????? ???? ????? ?????? ???????????? ???? C++ ‎, ?????? ???????? ????? TrueRand ?????? ?????? ??????? ?????? ?? ????? ? ???? . ????? ??????? ????????? ?????? ??? ????? ???? ??????, ??? ???? ???? ??? ??? ???? ???? ???? ???? ????. ???????? TrueRand ???? ????? ??, ????????? ????? ????? (service) ????? ????? ?? ???? ??? ????. ??-???? ??? ????? ????? ???? ???? ???? ??? (1/60 ?? ?????) ???? ???-???? ???? ????? ?????? ?????. ???? ??????? ?? ????? ????? ???? ???? ???, ?????? ?????? ????? ?? ???? ?????? ??? ????? ?????? ?? ????? ????, ???? ??????. ?????? ????? ?????? ???????? ??? ???? ?????? ??????? ?? ??????? ????? ?????? ?????? ????? ?? ?????? ??????? ??? ???? ????? ?????.

????? ????? ????? [ ????? ??? ???? | ????? ]

??? ????? ? ????? ?????? ?????-???????

????? ????? ????? (PRNG) ??? ???????? ?????? ????? ???? ????? ?? ?????? ?? ??????? ??????? ????? ???????, ?? ?? ??????? ???? ??? ?? ?????? "??????". ??? ?????? ??????? ????? ?????? ????? ??????? ??? ???????? linear congruential generator (?????? LCG) ?????? ?????? ??????: ${\displaystyle \ x_{n+1}=(aX_{n}+c){\mbox{ mod }}m}$. ????? ?? ????? ??? ?????? ????? ??????? ?????? ???? ???? ?? ????? ?? ${\displaystyle \ m}$ ?????????? ???? ????? ?- ${\displaystyle \ 2^{m}}$. ??? ?????? ?????? ???? ?????? ???? ????.

????? ??????? ??????? ????? ?????? ?????? ??????? (???? ?? ????? ?????? "???? ??????") ?????? ?- 1946 ?? ??? ?'?? ??? ????? . ????? ???? ?????? ??????? ?? ?? ?????? ???????? ????? ????? ????? ?????. ????? ???? ??? ?????? ?? ?????? ?????. ????? ???? ?????? ?????? ???????? ?? ????? ????? ??????? ?????? ?????? ??????? ??? ${\displaystyle \ {\mbox{rand}}()}$ ? ??? C (?????? ???????? ?? LCG). ?????? ???????? ?????? ??? ?????? ? ??????? ????? ????? ${\displaystyle \ [0,1]}$ ??????? ???? ??? . ????? ????????? ??????? ????? ???????? ??? ?????, ??????? ?????? ????? ?? ?????? ????? ????? ????? ?????, ??? ???? ???? ?????? ????? ??? ?????? ?? ????? ????? (???? ??????) ?????.

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

????? ?? ???? ??????????? , ?'????? ? ?????????? ?????? ???? ?????? ?? ???? ????? "?????" ?? ???? ??? ??????? ?? ??? ????? ???? ????? ????. ??????? ????? ??????? ?????? ???? ?????? ?????? ??. ??? ?????? ??????? ???? ???? ????? ?? ????? ?? ????? ????? ??????? ????. ???? ??? ?????? ???????? ????????, ???? ????? ?? ???? ???? ????? ???? ?????? ?? ??? ???????? ??? (??? ????? ????' ).

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

????? ????????? ?????? ??????? ??????? ???? ??????? ????? ???????????. ??? ?????? ???????, ???? ?????? ?????? ?????? ????? ?????? ???? ???????? ?????? ?????? ????? ??????? ???????? ?????? ????????????, ??? CryptGenRandom ?? ????????? ?? dev/random ?? ?????? .

??????? ?????? [ ????? ??? ???? | ????? ]

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

???? ????? ????? ????? ???? ? ??????? ?????? ?? ?????? ??????? ?????? ??????? ?????. ?-1995 ???? ????? RAND ??? ????? ????? ?? ?????? ?????? ??????? ?????? ??????? ???????? ????????? ?? ???? ????? ????? ?????? ????? . ?????? ????? ???? ????? ?????? ??????? ?????? ??????? ?????. ??? ?????? ????? ????? ?? ?? ?????? ????? ?????? ????? ?? ????? ?????? ? ????? ????? ????? ?????????? , ???? ?????? ?????? ??? ??? ????? ????? ???? ????.

???????? ???????? ???? ???????? ??????? ??? ?????? ??????? ???????, ?????? ????? ??????? ??????????, ?????? ?????????, ?????? ???? ? ????? ???? , ????????? ?????? ????????????. ???? ????? ???-????? ( ???????? ?????) ????? ????? ?????????? ?????, ??? ????? ????? ????? ???? ????? ???? ?? ??? ????? ????? ?????, ???? ????? ?????. ???? ????? ?????? ?????? ????? ??? ????? ???? ????? ?????? ???? ?????? ??? ??????? ?? ????? ?????. ????? ??? ?????? ????? ???? ??.

?????? ?????? ????? ??????? ??????? ?????? ????????? ? ???? ????? ???? . ?????? ??? ??? ??????? ????? ?? ???? ?????? ??? ???? ?? ??????? ????? ????? ??????.

?????? ??????? ?????? ??????, ?????? ???????????? ?????? ??????? ?????? ????? ????, ?????? ???????? ????? ???? ??? ??????? ???????. ??????? ??????? "??? ????" ?????, ????? ???? ?????? ?????? ?? ???? ?? ??????? ??????? ???? ? ????? ? ??? ?????? , ?? ?????? ???????? ?????? ?????. ???? ???, ???? ???? ???????? ????? ?????? ???? ?????, ?????? ???? ??? ???? ?? ????? ?? ???? ?????? ????, ????? ????? ?????? ?????? ??????? ?? ???????. ???? ???, ????? ?????? ?????? ????? ???? ?????? ???? ?? ??????? ??????? ???? ?? ???? ?? ?????. ?? ???? ??????? ???? ????? ???? ??????? ??????, ????? ???? ??? ?????.

????? ??????? ????????? [ ????? ??? ???? | ????? ]

???? ????? ?????? ??????? ??????? ???? ????? ????? ????? ?????? ????? ?? ???? ??????? ?????. ?????? ??? ????? ????? ?? ????? ????? "?????". ???? ???????? ?????? (???????) ?????? ???? ??? ?????? ?????? ?????? ?????????? ?????? ??? ?????? ???? ????? ?? ???? ?????? ????? ????? ????? ????? ???, ???? ????? ?????-?????. ???????? ????? ????? ???? ????? ???? ????? ?????? ????????? ??????? ????? ????? ???? ??????. ?????? ????? ?????????? ?????? ????? ????? ?????? ??? ??????? ??????? ?????? ???????.

?? ????? ????? ???? ????? ?????? ????????? ???? ????? ???? ????? ?????? ???? ???. ??????? ?????? ?????? ??????? ???????? ??? ????????, ????? ????, ???????? ?? ?????? ????? ??????? ????? ???? ??? ?????? ????? ????? ????, ?? ?????? ??? ???????. ?????? ?????? ?? ????? ???? ????? ???? ??????. ????? ????, ??????? ???????? ????? ??????? ?????? ?? ?? ????? ???? ???? ?????. ???? ????? ??????? ?????? ???????? ?? ??? ?????, ?? ?????? ?? ??? ??? ?????? ?? ???????? ??????? ?? ???? ??-??????, ?? ?? ??? ????? ??? ??? ??? ????? ????? ?? ??. ?? ?????? ???? ????? ???? ????? ??????, ??? ?? ?? ?????? ???????? ??????? ??????? ??????? ???? ???? (unbiased) ??????? ??? ?????.

????? ????? ??? ???? ????? ????? ??????? ??????, ??? ????? "?????" ??? ???? ???? ???. ?? ?? ????? ?????? ?????? ????????? ??? ?? ????? ????? ???? ?????. ????? ????? ???? ???? ?????? ??? (40 ??????):

11100 01100 01000 10100 11101 11100 10010 01001

?? ????? ???? 4 ????? ? ????? ???? ??????? ??? ????? 160 ??????, ????? ??????? ?? ???? ?????? ???. ????? ??? ???? ???????? ?????? ????? ?????? ???? ?????? ?????????. ?? ?? ??????? ?? ????? ????? ????-???????, ???? ????? ?? ??????? ?????????? ???? ?????? ????? ?????? ????????.

????? (whitening) [ ????? ??? ???? | ????? ]

???? ?? ??? ????? ????? ????? ?? ??????? ?????? ?? ??????? ????????, ???? ????? ???????? ????????. ???? ??? ????? ?????? ????? ?? ????? "?????" ??? ????? ? ??? ??? . ???? ???? ????? ?????? ???? ????? ???????, ??????? ?????? ?? ?????????? ????? ????? ??? ?????? ????????.

?'?? ??? ????? ????? ???? ????? ?????? ?????? "??-???????" ????? ???? ?? ????? ???? ????? ??????: ??? ??? ?????? ???? ?????? ???, ?? ?????? ??? ?????? ???? ?????? ????? (?????? ???? ?????), ?? ????? ???? "01" ?????? ?? ???? ?????? "1" ??? ????? ???? "10" ????? ?-"0". ????? ?????? ?? ?? ???? ??? ???? ???? ??????, ????? ?? ????? ??? ??????, ???? ????? ???????? ???????? ?? ?????? ????. ????? ????? ???? ?????? ?????? ?? ?????.

???? ???? ???? ???? ?? ??? ????? ?????? ??????? ??? ????? ?? ??? ????? ??? ?????? ?????, ??? ????? XOR ?? ???? ????? ?? ??? ????? ????? ??????????. ?????? ???? ?????? ????? ???? ?? ????? ???? ???? ???? ??????.

??? ???? ??? ???? ??? ????? ???? '??????? ????' ??????? XOR ?????? ??? ??? ????? ?????? ????? ????. ?? ???????? ?? ????? ??? ???? ?????? ??? ${\displaystyle \ 1/2+e}$ ???? ${\displaystyle \ -1/2\leq e\leq 1/2}$ ??? ${\displaystyle \ e}$ ??? ????? ?? ????. ?? ?????? ?-XOR ?? ??? ?????? ??? ????? ??? ??? ???? ?- ${\displaystyle \ 2e^{2}}$. ?? ?????? ?? ?????? ???? ????? ?? ????? ????? ??????? ?? e ????. ????? ?? ????? ??????? ?? ???????? XOR ? ???? ???????? .

???????? ????????????? ??????? ????????? ????? ??????????? ??? SHA ?? MD5 ??? ???? ???? ????????. ???? ???? ?? ???????? ????? ?????? (CRC) ?? ???????? ?????. ?????? ?? ????????? ???? ?????, ????? ??? ????? ???? ??????, ???? ?????? ????? ??????????, ??????? ?????? ?? ?????????? ???????????? ??????? ?????? ?????, ??? ?????? ?????? ????. ????? ???? ????? ??? ????? ????? ?????? ???????? ?????? ?? ??????. ??? ???? ?????? ??? ????? ????? ??????? ???? ???.

?????? ????????? [ ????? ??? ???? | ????? ]

???? ????? ???? ?? ?????? ????????? ?????? ????? ????? ?????. ??????? ?? ????? ?????? ?????? ????? ??????? ????? ?????, ??? ???? ?????? ????? ?????? ??????? ????? ?? ??????? ?????. ??????? ????? ?? ????? ?? ???? ???????. ?? ???? ??????? ???? ??? ???? ????? ???? ???????? ??????? ????? ???? ???? ????? ?????. ????? ???? ?? ????? ??? ???? ??????????? ??? ?????????. ???? ??? ????? ????? ???? ????? ?????? ??? ???? ????? ???? ??????. ?? ???? ????? ???? ???? ?? ????? ?? ???? ???? ??????? ????? ?? ???? ???? ??????. ????? ??? ?? ???? ??????? ??? ???? ???? ?????? ????? ????? ????? ????, ???? ???? ????? ?????? "?????". ????? ????? ???????? ???? ????? ?? ???? ????????. ?? ?? ???? ???? ?????? ???? ??????? ????, ??? ???? ????? ????? ?????? ????????? ??? ???? ???? ????? ????????.

??? ??????? ?????????? ??????? ?? ??????? ??????? ? ??????? ?? ?????? ( ${\displaystyle \ \chi ^{2}}$). ??????? ??????? ??? ???? ???? ???? ?? ?????? ?????? ???? ?????? ???? ????? ? ????? ????. ????? ???? ${\displaystyle \ X}$ ??? ??? ??????? ??????? ?? ???????? ??????? ??? ??? ??????:

${\displaystyle \ f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left({\frac {-(x-\mu )^{2}}{2\sigma ^{2}}}\right)}$

?????? ??? ${\displaystyle \ X\sim N(\mu ,\sigma ^{2})}$. ????? ?- ${\displaystyle \ X\sim N(0,1)}$ ?????? ??????? ??????? "????????" ??????:

${\displaystyle \ f(x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}}$

?????? ?? "????? ??????" ??? ?????? ???? ???????? ?????? ???? ???? ?- ${\displaystyle \ y}$ ??? ?????? ???? ?????. ????? ???????? ??????? ?? ${\displaystyle \ X\sim N(\mu ,\sigma ^{2})}$ ???????? ??????? ???????? ??? ${\displaystyle \ Z=(X-\mu )/\sigma }$.

??????? ${\displaystyle \ \chi ^{2}}$ ?? ${\displaystyle \ v}$ ????? ???? ?????? ???? ?????? ?????? ${\displaystyle \ v}$ ?????? ?????? ???? ?????? ???? ??????? ??????? ????????. ???? ?????? ???????? ${\displaystyle \ \chi ^{2}}$ ??? ????? ?? ???? ?????? ??? ?????? ??????? ????? ???? ??????? ??????? ??? ???? (???????) ????????? ?????. ???????? ??????? ?? ??????? ?? ?????? ??? ??????:

${\displaystyle \ f(x)={\frac {1}{\Gamma (v/2)2^{v/2}}}x^{(v/2)-1}e^{-x/2}}$, ???? ${\displaystyle \ x}$ ???? ???? ????????? ${\displaystyle \ \Gamma (t)}$ ????? ???????? ??? ???????: ${\displaystyle \ \Gamma (t)=\int _{0}^{\infty }x^{t-1}e^{-x}dx}$ ???? ${\displaystyle \ t}$ ???? ????.

?? ${\displaystyle \ X}$ ??? ????? ???? ??? ??????? ${\displaystyle \ \chi ^{2}}$ ?? ${\displaystyle \ v}$ ????? ????, ??? ${\displaystyle \ P(X>x)=\alpha }$. ?????? ????? ${\displaystyle \ \alpha }$ ????? ?? ???????? ?- ${\displaystyle \ X}$ ????? ????? ????? ?????? ${\displaystyle \ x}$. ?????? ?????? ?????? ?? ???? ??????? ?? ??????, ????? ?????? ??? ????? ???????. ????? ?? ??????? ??????? ???????? ???? ${\displaystyle \ X\sim N(0,1)}$ ?????? ??????? ${\displaystyle \ Z=X^{2}}$ ??? ??? ??????? ?? ?????? ?? ???? ???? ???.

????? ??????? ??? ????? ??????? ?? ????? ????? ?????? ???? ??? ?? ????, ???? ??????? ???????? ?????? ????? ???? (???? ?????? ${\displaystyle \ H_{0}}$) ??????? ???? ????? ???????? ?????? ?? ??????? ????? ? ??? ??????? ${\displaystyle \ \alpha }$. ??? ???????? ?????? ?? ???????? ?????? ???? ????? ?? ???? (?? ?????? ????????). ??? ??? ????? ???? ????? ????? ????? ????? (???? ??? ${\displaystyle \ \alpha }$ ???? ???) ???? " ???? ???? I ". ??????, ???? ???? II ????? ?????? ???? ??? ???????? ????? ???, ?? ????? ???? ?????? ????? ????? ????? ?????, ?????? ????? ?? ???? ????? ?????? ?? ?? ?? ????? ???.

?? ?????? ?????? ?? "????" ??????? ???? ??????? ?? ??????? ??????? (???? ???? ?????? ?? ?????? ??????). ????? ???? ?????? ???????? ???????? ?? ?? ?????? ?? ??? ????? ?????. ?? ???? ????? ?????? ?? ????? ?????? ????. ?????? ?? ${\displaystyle \ X}$ ????? ???????? ?? ?????? ?? ${\displaystyle \ v}$ ????? ???? ????? ?????? ???????? ????? ?????? ???? ???? ????? ?????? ???? ????? ?? ??????? ?????, ?????? ??? "??" ${\displaystyle \ x_{\alpha }}$ ??? ???? ???? ???? (???? ??? ?????? ??????? ?????? ??????? ???????) ??????? ?? ${\displaystyle \ \alpha =P(X_{\alpha })}$, ?? ????? ?????? ????? ????? ???? ?? ??? ????? ???? ??????. ???? ???? ?????? ?? ???? ${\displaystyle \ \alpha }$ ???? ?? ????? ?????? ??????? ?? ????? ?????? ????? ???? ??. ?? ??? ?????? ?????? ?????, ???? ???? ?? ???? ??????. ????? ??? ????? ?????? ???? ???? ?? ??????, ????? ????? ?? ??? ????? ????? ??????? ????? ?????? ????? ??? ??????, ?? ????? ?????? ?? ??????? ?????. ????? ???? ?? ??? ????? ???, ???? ?? ???? ??? ?????? ???????.

???? ??????? ???? ?????? ????????? ??????? (??????? ???? Handbook of Applied Cryptography). ??????? ???????? ${\displaystyle \ s=s_{0},s_{1},s_{2},...,s_{n-1}}$ ??? ???? ?? ?????? ????? ${\displaystyle \ n}$. ???? ${\displaystyle \ n}$ ???? ????? ????, ??? ${\displaystyle \ 10^{3}}$ ?- ${\displaystyle \ 10^{6}}$ (?????? ??????). ??? ???????? ????? ??? ${\displaystyle \ \alpha =0.05}$ (????? ????? ??????? ?? ?????? ?????? ????? ???). ????? ????? ????? ??????? ?? ??????? ??????? ????? ???????? ?????? ???????:

${\displaystyle \ \alpha }$	0.1	0.05	0.025	0.01	0.005	0.0025	0.001	0.0005
${\displaystyle \ x}$	1.2816	1.6449	1.9600	2.3263	2.5758	2.8070	3.0902	3.2905

???? ?????? ${\displaystyle \ \alpha =0.05}$ ????? ${\displaystyle \ x=1.6449}$, ?????? ???? ??? ${\displaystyle \ X}$ ??? ??????? ??????? ???????? ??? ${\displaystyle \ X}$ ???? ????? ${\displaystyle \ 1.6449}$ ?????? ?? ?????? ???? ??????? ??????.

???? ?????? (monobit test) [ ????? ??? ???? | ????? ]

????? ??-??????? ?????? ????? ????? ????? ?????? '1' ?-'0' ??? ??????, ????? ???? ?????. ?? ${\displaystyle \ n_{0}}$ ?- ${\displaystyle \ n_{1}}$ ??????? ?? ?????? '0' ?-'1' ?????? ??? ?????? ???:

${\displaystyle \ {\frac {(n_{0}-n_{1})^{2}}{n}}}$

???? ?????? (two bit test) [ ????? ??? ???? | ????? ]

???? ?????? ????? ??? ???? ?????? 00, 01, 10, 11 ???????? ???? ??? ?????? ????? ???? ?????. ?? ${\displaystyle \ n_{00}}$, ${\displaystyle \ n_{01}}$, ${\displaystyle \ n_{10}}$, ${\displaystyle \ n_{11}}$ ??????? ?? ?????? ??????? ??????, ??? ?????? ???:

${\displaystyle \ {\frac {4}{n-1}}(n_{00}^{2}+n_{01}^{2}+n_{10}^{2}+n_{11}^{2})-{\frac {2}{n}}(n_{0}^{2}+n_{1}^{2})+1}$

???? ???? (poker test) [ ????? ??? ???? | ????? ]

?????? ?? ???? ${\displaystyle \ s}$ ?- ${\displaystyle \ k}$ ????? ?? ??? ????? ${\displaystyle \ \left\lfloor n/m\right\rfloor }$ ?????? ???? ${\displaystyle \ m}$ ????? ${\displaystyle \ \left\lfloor n/m\right\rfloor \geq 5\cdot (2^{m})}$ ????? ${\displaystyle \ n_{i}}$ ????? ?? ???? ??????? ?? ???? ?- ${\displaystyle \ i}$. ???? ???? ???? ??? ???? ?? ??????? ???? ??? ?????? ????? ????? ???? ????? ??????? ??????:

${\displaystyle \ {\frac {2^{m}}{k}}\left(\sum _{i=1}^{2^{m}}n_{i}^{2}\right)-k}$

???? ???? ??? ????? ?? ???? ?????? ?? ???? ?????? ?????? ?????? ?? ??-????? ??????? ?-1.

???? ????? (runs test) [ ????? ??? ???? | ????? ]

???? ?????? ????? ??? ???? ?????? (????? ?? ?????? ??? ?? ??? ??????) ??????? ?????, ??????? ???? ??????? ?????? ???? ?????. ???? ?? ????? ????? ??? block ????? ???? ?? ????? ????? gap. ???? ?????? ????? ${\displaystyle \ i}$ ???? ????? ${\displaystyle \ n}$ ???? ????? ${\displaystyle \ (n-i+3)/2^{i}+2}$ ?? ${\displaystyle \ B_{i}}$ ?- ${\displaystyle \ G_{i}}$ ??????? ?? ????? ???? ????? ??????, ??? ????? ???? ????? ???:

${\displaystyle \ \sum _{i=1}^{k}{\frac {(B_{i}-e_{i})^{2}}{e_{i}}}+\sum _{i=1}^{k}{\frac {(G_{i}-e_{i})^{2}}{e_{i}}}}$

???? ${\displaystyle \ e_{i}=(n-i+3)/2^{i+2}}$. ?- ${\displaystyle \ k}$ ????? ?? ????? ?????? ????? ?? ??????? ${\displaystyle \ e_{i}\leq 5}$.

???? ??????????? (auto-correlation) [ ????? ??? ???? | ????? ]

???? ????? ??? ???? ??????? (??? ????) ??? ????? ?????? ?? ????. ?? ${\displaystyle \ d}$ ??? ??? ????? ?- ${\displaystyle \left\lfloor n/2\right\rfloor }$ ???? ??????? ?- ${\displaystyle \ s}$ ????? ??????? ????? ${\displaystyle \ d}$ ???: ${\displaystyle \ A(d)=\sum _{i=0}^{n-d-1}s_{i}\oplus s_{i+d}}$. ????? ?????-??????? ???:

${\displaystyle \ 2\left(A(d)-{\frac {n-d}{2}}\right)/{\sqrt {n-d}}}$

???? ????????? ?? ????? [ ????? ??? ???? | ????? ]

???? ????? (Ueli M. Maurer) ???? ?????? ???????????? (1992), ???? ????????? ?????? ????? ?????, ??????? ??????? ?????, ????? ???? ?? ??? ?? ????? ???????? ???????? ?? ??? ?????? ???? ????? ?????. ?????? ????? ?? ?????? ??? ??? ?????? ???? ? ????? ????? (??? ????? ????) ????? ???????, ???? ???? ?????. ????? ????? ????? ?? ???? ???, ???????? ????? ???? ?????? ?? ???? ???? ????? ???? ?????. ?????? ?????? ?? ????????? ?????? ????? ????? ?? ??? ???? ????? ??? ?????? ??????? ???????? ????.

?????? ?? ???????? ????? ??? ??????: ?????? ?? ???? s ??????? ?? ?????? ?? ??? ????? ${\displaystyle \ L}$ ?????? ( ${\displaystyle \ L}$ ??? ????? ${\displaystyle \ [6,16]}$), ???? ???? ?????? ????? ???? ????? ${\displaystyle \ Q+K}$. ???? ${\displaystyle \ Q}$ ???? ????? ????? ${\displaystyle \ 10\cdot 2^{L}-1}$ ?- ${\displaystyle \ K}$ ???? ????? ????? ${\displaystyle \ 1000\cdot 2^{L}}$. ???? ${\displaystyle \ b_{i}}$ ????? ?? ????? ???? ??????? ??????? ??? ??? ???? ${\displaystyle \ i}$ ????? ${\displaystyle \ [1,Q+K]}$. ??????? ???? ${\displaystyle \ T}$ ?? ${\displaystyle \ Q+K}$ ??????, ?- ${\displaystyle \ Q}$ ??????? ???????? ?????? ?? ???? ???? ${\displaystyle \ b_{i}}$ ?????? ??????, ????? ???? ${\displaystyle \ i}$ ?-1 ?? ${\displaystyle \ Q}$ ?????? ${\displaystyle \ T[b_{i}]=i}$ ???? ${\displaystyle \ i}$ ????? ?? ???? ???? ${\displaystyle \ b_{i}}$ ????. ${\displaystyle \ K}$ ??????? ??????? ?????? ?????? ???????? ??????. ???? ?? ??? ${\displaystyle \ i}$ ?- ${\displaystyle \ Q+1}$ ?? ${\displaystyle \ Q+K}$ ?????? ?????? ${\displaystyle \ T[b_{i}]}$ ?? ${\displaystyle \ i}$. ?????? ?? ??????? ???? ?? ??????, ????? ?? ???? ?????? ?????? ?????? ?? ???? ?????? ?? ???? ??????, ?????? ?? ? ??????? ????? 2 ?? ????? ????? ??????? ?- ${\displaystyle \ K}$. ??????:

${\displaystyle \ {\frac {1}{K}}\sum _{i=Q+1}^{Q+K}\log _{2}(i-T[b_{i}])}$

????? ???? ??? ?? ???? ?? ???? ?????? ${\displaystyle \ \mu }$ ??????? ${\displaystyle \ \sigma ^{2}}$ ???? ???????? ${\displaystyle \ L}$ ?????? (????? ???? 6 ?-16) ????? ??? ??????? ( ${\displaystyle \ \alpha =0.01}$) ???? ????? ??????????.

??? FIPS PB800-22 ?????? ??????? [ ????? ??? ???? | ????? ]

????? ???????? ??????? ?????????? (NIST) ???? ???? 2001 ?? PB800-22, ????? ????? ?????? ????? ?????? ?????? ????????? ???????? ??????? ?????? ??????? ???? ????????? ????????????. ?????? ????? 16 ????? ??????? ?????, ???????? ?? ?????? ????? ????? ???? ?????. ??? ?? ?????? ????? ???? ????? ?????? ??????, ?????? ????????. ???? ????? 16 ???????:

1. The Frequency (Monobit) Test

???? ?????? ????? ?? ???? ??????? ?? '0' ?-'1' ??? ?????? ????? ???? ?????. ???? ?????? ???? ????? ????? ????? ?????? ?????? ?????. ?? ???? ????? ???? ????? ??, ????? ????? ?? ?????? ??????? ?? ?????? ?????, ???? ???? ???????? ????? ?? ????? ?? ??? ??????? ???????.

2. Frequency Test within a Block

???? ?????? ???? ????? ????? ???? ??? ?????? ????? ????? ?????? '1' ?????? ????? ${\displaystyle \ m}$ ??????. ???? ????? ????? ???? ?????? '1' ?????? ???? ???? ???? ?- ${\displaystyle \ m/2}$ (?? ${\displaystyle \ m=1}$ ????? ???? ??????? ??????).

3. The Runs Test

???? ?????; ???? ?? ?????? ????? ?? ???? ?????? ??????? ????? ???? ??? ????? ???? ?????. "????" ??? ??? ?? ?????? ?????? ???? (0 ?? 1). ????? ???? ?? ???? ????????? ?? ????, ?????? ????? ?????? ?? ??? ?????? ?? ??? ????? ?????. ???? ???? ??? (???? ????? ?????) ?? ??? ??? (??? ????? ??????) ???? ?? ??????? ?????.

4. Test for the Longest-Run-of-Ones in a Block

???? ????? ??????; ???? ????? ?? ????? ?????? ????? ?? ????? (??? ????? ?? ?????) ???? ???? ???? ???? ?? ?????? ????? ???? ?????.

5. The Binary Matrix Rank Test

???? ???? ?????? ???????; ???? ????? ?? ????? ??-??????? ?????? ?? ????. ????? ??? ????? ?????? ????????? ???-????? ????? ???? ????? ????? ??????? ?? ??????? ??? ?????? ?????? ??????? ???? ?????.

.

6. The Discrete Fourier Transform (Spectral) Test

???? ??????? ( ????? ?????? ); ???? ????? ??????? ???????? ?? ??????? ???? ????? ?? ??? ????? ?????? ????? ?? ????. ????? ????? ????? ?"?????" ????????.

7. The Non-overlapping Template Matching Test

???? ????? ?????? ?? ??????; ???? ?????? ?????? ??????? ?? ?????? ????? ?????? ???? ???? ??????. "??????" ?? ????? ???????? ????? ?? ???????? ?????? ????. ?????? ?????? ??????? ???? ????? ${\displaystyle \ m}$ ?????? ??????? ?????? ??????? ????, ?? ?? ????? ????? ??? ??????? ?????? ???? ????? ?????, ?????? ???? ?? ????? ?????? ???. ?? ????? ????? ?????? ????? ?? ${\displaystyle \ m}$ ?????? ???? ????? ?????. ?????? ???? ??? ?? ?????? ???? ?????? ?? ??????? ?????.

8. The Overlapping Template Matching Test

???? ????? ?????? ??????; ????? ????? ?????, ?? ?????? ??. ???? ????? ????? ??? ?????? ???? ?????? ????? ?????? ???? ?? ????? ?? ?????? ???. ??? ??????? ????? ?? ?? ?????? ?????? ????.

9. Maurer's "Universal Statistical" Test

???? ????????? ?? ?????; ???????? ????? ?? ???? ??????? ?? ???? ?????? ??? ??????? ?????? ????. ???? ??????? ?????? ?????? ?? ???? ??????? ????. ??? ????? ???? ???? ?? ???????? ????? ????.

10. The Lempel-Ziv Compression Test

???? ????? ????-??? ; ???? ????? ????? ?? ???? ??????? ???? ?? ??? ????? ??????? ?? ???????? ???? ??? ?????? ??????. ????? ????? ????? ????? ??? ???? ????? ???? ?????? ???? ?? ???????? ????? ????. ??????? ??? ????? ????? ?? ?? ?????? (words) ??? ??????? (?? ???? ?????) ???????? ????. ???? ?????? ????? ?????? ????? ?? ??? ????????. ???? ???? ???? ?? ??????? ?????.

11. The Linear Complexity Test

???? ???????? ???????? ; ????? ??????? LFSR . ?? ??? ????? ???? ????? ????? ??????? ????? ?? ??? ??? ????. ???? ??? ????? L ?????? ???? ???? ?????? ??? ????? ???? ???? ?? ??? ????? ?????? ?? ????? ?? ??? ???????? ??????? ????? ??? ??? ?????. ???????? ??????-??? ???? ???????? ???????? ?? ??? ?????? ?? ??? ????? ???? ????? ???? ????? ??????, ??? ???? ????? ?? ???? ????. ??????? ??? ??-LFSR ???? ???? ?? ???? ???????? ????? ????.

12. The Serial Test

???? ??????; ???? ?????? ???????? ?????? ?? ?????? ?????? ?? m ?????? ???? ????. ???? ????? ???? ???????? ???? ?? ${\displaystyle \ m}$ ?????? ????? ???? ???????? ?? ?? ??? ???. ????? ??? ${\displaystyle \ m=1}$ ????? ???? ??????? ??????. ????? ??? ?????? ?? ?????? ?????? ?? ????? ?????? ???? ?-1 (??? ${\displaystyle \ m=3}$). ???? ??? ????? ${\displaystyle \ n}$ ?????? ${\displaystyle \ m}$ ???? ????? ???? ?- ${\displaystyle \ \left\lfloor \log {n}\right\rfloor -2}$.

13. The Approximate Entropy Test

???? ???????? ??????; ????? ????? ??????? ????? ?? ?????? ?????? ?????? ?? ?? ???????? ???????? ????? ${\displaystyle \ m}$ ?????? ????? ?? ???? ??? ???? ?????? ?? ?????? ????? ?????? ?????? ?????? ??? ${\displaystyle \ m}$ ?- ${\displaystyle \ m+1}$ ?????? ??? ?????? ?????? ????? ?????? ???????. ???? ?? ????? ???? ??????? ?? ??-??????? ?? ????.

14. The Cumulative Sums (Cusums) Test

???? ???? ?????; ???? ????? ?? ???? ?????? ???? ???? ? ????? ????? ????? ?? "????? ?????". ?????? ?????? ??? ????? ?? ??? ????? ????? ?????? ?? ?????? ???? ???? ??????? ?- ${\displaystyle \ 1}$ ?- ${\displaystyle \ (-1)}$ ??????. ???? ????? ?????? ????? ???? ????? ?????, ???? ?????? ???????? ?????? ???? ??? ????. ????? ??????? ?? ????????? ?? ???? ????? ????? ?? ??? ????????. ????? ????? ??? ?? ?????? ??? ????? ?? ??????? ?????.

15. The Random Excursions Test

???? ????? ?????; ????? ?????? ??????, ??? ??????? ????? ??????? ?? ??????? ?????? ?????? ????? ?????? ?? ????? ??????. "????" ?? ????? ????? ??? ???? ????? ??????? ??????? ???? ??????? ?????? ???? ???? ????? ?? ????? ?????? ?????. ?????? ?? ????? ??? ????? ?? ???? ???????? (??????) ??????? ???? ????? ??? ??????? ?????? ???? ?????. ????? ????? ????? ?????? ???? ?? 8 ?????? ???? 4 ??? ??? ???? ????? ???? ????.

16. The Random Excursions Variant Test

???? ????? ????? (???? ????); ?????? ?? ????? ?????, ????? ?? ?????? ?? ???? ???????? ????? ?????? ???? ????? ????? ?????? ?? ?????? ??????. ????? ????? ????? ?-18 ????? (?????? ????) ?????.

??????? ???????? ???????? ???????? [ ????? ??? ???? | ????? ]

???? ???? ???? ??????? ??????? ???? ? ??????? ??????? , ???? ?????? ??????? ??????? ???? ??????? ????? ????? ???? ??? ???? ??? ????? ?????? ??? ??????? ????? ?? ??? ?????? ???????? ?????? ?? ???????? ??????? ?????? , ?? ???????? ??????. ??? ????? ??? ?????? ??????? ${\displaystyle \ (x,y)}$ ???? ?????? ???? ??????? ???????, ???? ????? ????? ??????????? ??????? ${\displaystyle \ (r,\theta )}$ ???? ${\displaystyle \ r~X_{2}^{2}}$ ?- ${\displaystyle \ \theta }$ ??? ???????? ????? ????? ${\displaystyle \ (0,2\pi )}$. ??? ???????????? Box-Muller .

???? ????? ?????? ??????? ?? ??????? ???????? ??????? ???????? ??? ???????? ??????? ?? ???????? ?????? ?? ????? ???? ??????. ???? ??? ??? ???? ?????? ?????????? ??? ??????? ????? ???? ???? ??????? ???? ?? ??????? ????? ?????? ???????? ???????? ???????, ????? ?? ?????? ?? ???????? ?? ??? ????? ????? ?? ???? ????? ????? ?????? ?????? (0,1). ???? ???? ????? "???? ?????-????", ?????? ?? ??? ?????? ????? ??????? ??????? ???????? ?????. ?????? ??? ????? x,y ??????? ?? ???????? ?? x ????? ?? ???? ???? y. ?? ?? ???? ?? x "?????" ???? ???? "????" ????? ?????.

??? ?? [ ????? ??? ???? | ????? ]

• ????? ????
• ???????

??????? ???????? [ ????? ??? ???? | ????? ]

• Handbook of Applied Cryptography ??? 5.
• SP800-22b A Statistical Test Suite for Random And Pseudorandom Number Generators For Cryptographic Applications
• Ueli M. Maurer, A Universal Statistical Test for Random Bit Generators, Journal of Cryptology, vol. 5, no. 2, 1992, pp. 89-105