ρ /( ρ)

+ ( q, v ) : ( q, v ), L < q < q < q < L 0 0 ( t) ( q ( t), v ( t)) dq ( t) v ( t) lmr + 0 Φ( r) dt lmr + 0 Φ ( r) dv ( t) Φ ( q ( t) q ( t)) + Φ ( q+ ( t) q ( t)) dt ( ) < 0 ( q (0), v (0)) ( q, v ) Φ(r) :,, lm lm r r Φ( r) 0 Φ ( r) 0 Φ r ( q (0), v (0)) ( q, v )

[, ] X γ N( ; ) #{ q } { ( ) L < q ( t) < q ( t) < q+ ( t) < L, () ( q ( t), v ( t)) X, () sup γ q q or q + H ( ; ) v + Φ( q+ δ > 0 s.t. t lm σ + oloσ + o σ ( t) lm σ olo σ o σ ( k k k k t) σ ( t) f{ s t q+ ( s) q ( s) δ} + q ) γ,, L() N( ; ) <, sup,, L() H ( ; ) < }, 0 < γ γ ( q, v ) X γ s t, q + ( s) q ( s) δ K [ K, K] ( K,, L) K { q ( t)} t [,0] t [,0] Ascol-Arzla

+ Z X {0, } L ) X ( 0 L 0 [ a La j ] j {( L 0 L) X l al, l j } a L j a j X P(, [ a La j ] j ) t t+ [ a L a ] j j X (MP) X µ µ ( A ) µ ( d) P(, A) X

β, 0 < γ < α ( β )( βγ ) ( 0 < β < α) X π γ π γ 0 < α / ( βγ )( β / α) f ( ) γ ( β ) ( βγ / α) (ral masur),3,l π γ βγ + γ π π π γ 0 Θ + π γ ( [ a La π γ ( [ a La j j, 0 < γ <, γ 0] j ) + γ ] j ) + γ ( β ) ( β ) # 00 # 00 γ γ # 0 # 0 # βγ γ # βγ γ βγ α βγ α j j (MP) π γ # ([ a La ]) #{ l a a uv} uv j l l+ ( ) f ( βγ)( β / α) γ ( β ) ( βγ / α) ( ) f (3) f () f () f (),3,L / < α <

coupld Markov t t + -

Z X {0, } + L ) X ( 0 L 0 F( t) +, - ( Ωf )( ) Z χab( uv) 0 λ f f uv ab uv ab µ α β chagabl masur : α β λt X (MP) c { αχ α β ( ) + βχ0( ) + αχ( + + ) + βχ0 α > β ( coupld Markov + +, + ( L 0 LL + + )}[ f ( L) α < β, + ( Ω f )( ) dµ ( ) 0 X ) f ( )]

0 < ρ < q, (0, ) q qq /[( q)( q )] β /α ( q ) /( q ) ( ρ ) / ρ µ ρ ([]) ρ µ ρ ([0]) ρ X µ ρ µ ρ ([ a L a j00]) qµ ρ ([ a La j0]) µ ([ a ]) ρ La j q µ ρ ([ a La j]) (3) f ρ f () f () () ρ ρ f ρ µ ρ, 0 < ρ <, (MP) c q ( α / β ) q q ρ ( k) k (MP) c k k,3,l 0 µ ρ, 0 < ρ < [, ] µ 0 µ H ( ν ) ν ( ζ ) logν (( ν ( ζ ) / µ ( ζ )) t t t ρ µ ρ f (ral masur) µ ρ ρ ν ν ζ [ ala ]

s 3 3! + 5 5! 7 7! + 9 9! L + ( ) + L ( )! : s 3 3! + 5 5! 7 7! + 9 9! L + ( ) ( )! F ( ) + F ( ) + F3 ( ) + L + F ( ) 0 0 6

s F ( ) + F ( ) + F3 ( ) + L + F30( ) F.0000000000000.0000000000000 F -774.6666597656-774.66666666667 F3 4946.9335937500 4946.9333333333 F4-4949.850000-4949.79365079 F5 33690.5000000 33690.339859 F6-4638405.0000000-4638405.485538 F7 4546588.0000000 454659.703848 F8-0467444.000000-0467449.73649 F9 8658896.000000 865890.735463 F0-63594464.000000-6359448.859545 F 3037648.000000 3037660.047666 F -9055440.000000-9055448.7446 F3 3438046.000000 3438047.37938 F4-659566.000000-65956.53393 F5 9630536.0000000 9630543.335989 F6-50808.0000000-5080.737439 F7 97538.0000000 975384.07736 F8-9344609.00000000-9344609.9943968 F9 3395488.50000000 3395488.96889 F0-0898.00000000-0898.078079 F 3766.0350000 3766.075789 F -87705.8437500000-87705.8586090578 F3 439.0703500 439.09884 F4-4799.563678750-4799.5709666588 F5 987.657653808594 987.65778689893 F6-87.46884548-87.46309 F7 3.936005869 3.93640865984 F8-5.36496577898-5.36496303633455 F9 0.834843038 0.83484370708 F30-0.505739390850-0.50574036735-6.80646896363-0.0378806339486 () F F ( ) + L + F30( ) F8, L, F4 cacllato rror

s 30) ( ) )( ( 3) 4)( ( 3 G )! ( ) ( 9! 7! 5! 3! 9 7 5 3 + + + L s, ) )( ( + k k G k k G 0, G 9 30 G G G + 5 4 3 5! 3! 5 3

, s 30 6~7 G0.0.0 G 0.858563063843 0.858564430578 G 0.86987085089 0.8698700305709 3 ( 4)( 3) ( )( ) G3 0.85850484943 0.85850439466 G4 0.8497648305899 0.8497649556607 G5 0.838804006576538 0.838803994065 G G6 0.87388944669 0.87388974557504 G7 0.84775099077 0.8477508676766 G G8 0.8008374843086 0.800837567346 G9 0.785380363464355 0.78538036863305 G0 0.76870787347 0.7687035955 G 0.74994746399 0.74938489 G 0.7780048867340 0.7780039383445 k 6,7,8 G3 0.70398706977386 0.7039870667065 G4 0.67733955809784 0.67733960974 G5 0.647499938507 0.6474955584 G6 0.64056348800659 0.640563403069 G G7 0.5766335973785 0.576633503589 k Gk ( k)( k + ) G8 0.53484898805683 0.5348489604439 G9 0.4884053706800 0.4884053437499 G0 0.43770636 0.43770986406034 G 0.383353590866 0.3833574793800 G 0.34860486595 0.348638969856 G3 0.59050846099854 0.5905099399384 G4 0.967839948845 0.967768530889 G5 0.36376870535 0.363788466789 ( )0.0???????? G6 0.0834896544806 0.0833553640355 { G7 0.040654778480598} 0.0408095387754 G8 0.065436006 0.04093043738067 G9-0.303837793965-0.0007948707606 G30-6.6686043739388-0.037888756735

s Horr 6 9999 s Horr 4-6.6686043739388 8604-7.800577636969 0577-7.645578863657 5578 4.753845484375 3845... Ω { 0,,, L, k, L, } X, 0,,, L P ( X k, X k, L, X l kl ) P ( X k) P( X k ) L P( X l kl ) P( X k), k

G k ( k)( k + ) Gk ( )0.0???????? { 0.0?? 0.0?? (5 ) 0.0034567 3.34567 0 0.4567?? 0 [,) 0.0 0

Smplfd Shft-Ral radom umbr grator (t) SSR Φ t, [,) t ) ) ( ) (5 ) 0...0 0 ) 0 [,) 4 Φ ( 0 ) Φ ( Φ ( Φ ( 0 )) ) 444 4444 3 4.775845 5845,, 3, L t t + f Φ ( t) t t + f t < t < 4

SSRΦ t t + ( t) t t + f f t < t < 4 Φ () Φ 3 () Φ 7 () 4 () Φ SSR 4 Φ (. ) Φ, 3,,L [,) 4 (.788L) 4 4 Φ (. ) Φ (. π ) (SSRK)

SSR SSI [,) t Φ ( t) t t t + + f f t < t < 4 SSI Smplfd Shft-Itgr (t) M β : [,) [,) M β M β βt +, ( t) βt β > βt mod [,) t M β ( t), t [,) β 3.78L [,) +, 0,, L,9999 0000 y 3 β y ( M 3 ) 6 () y ( M 3 ) 6 0,,,L (),, 0,,L M M 3 M 3 3 4444444 44 444444444 3 6 y ( M 3 ) 6 ()

[,), y SSI3K p, q, r( < p), s( < q) k 0,,,3, ( r k, s k ) ( rk mod p, sk mod q ), k ( y k k ( k XOR r ) y s ) 3 M ( 0 ) k 3 M ( v 0 ) y k b L k 7 b 48 SSIK (3 bts) 644744 8 ooooooo ooooooo LLLLL b b6 b7 b b b 48 49 64 SSIK prod of { } { y pq.8 0 k k } p 34359738337 q 3435973839

4 0.4 0.56 0.9 0.05 57 78 3

MD5 : 8 SHA : 60 RIPEMD-60 : 60 hash fucto / mssag dgst fucto / hash algorthm

B B B 0!! 3! LLL N- N- N (byt) B N B N. + + + + + L. 78L MB3hash MB3hash BN out put MB3hash.78L ζ : MBhash, 60,..., 4096 Stag. B XOR., M ( B ), k. k 0 3 k y N N. 0 B M 3. B M 3. B 3 M 3. B N N M 3. N y Stag. y y y M 3 y M 3 M y 3 y 4444444 44 444444444 3 6 tms 6 z ( M ) ( y) 3 ζ 6 ( M ) ( y) 3 y ( z z ) 3 z y

MB3hash. 0 B M 3. B M 3. B B^ ˆ N N ˆ ˆ N N ˆ B 3 M 3. B N N M 3. N y γ 3. : ~ N N t ~ γ (. + ) + γ ( ~ p + t ) + L + ~ γ ( ~ p ~ Nˆ Nˆ (. tˆ ) ~ γ + + γ ( q ~ + tˆ ) + L + ~ γ ( q~ γ 3 + + + BB^ y, yˆ N + tn ) + ~ pn + tˆ ) + q~ Nˆ Nˆ + + L 0!! 3! ˆ N N ˆ γ ~ Nˆ t ˆ ad t j ar of th form ± 0.0000000b8b 9 Lb ~ p ~ ad q j ar { 9, 0, L, 9}. 8 N + Nˆ N + Nˆ ( totally, ( ) quatos) 000L,, 6 z 3 ( M ) ( y) ζ z z 3 y ζ ˆ ζ ( ) y yˆ > y, yˆ 9 y, yˆ y, yˆ y, yˆ B B^ 5

MB3hash k B k +η y > 9 η y ζ 3 7 6 8 5 8 7 ) ( ) (. b b b c b c b b b L L L (MBhash) ζ ζ ˆ y y ˆ, ζ ζ ˆ 4 0 49 0 8 38 85 0) 0)(8 38( 85 8 38 85 09 890 38 85 09 890 38 785 09 890 38 6785 44 0) 0)( 44( 44 ac d df a f a f a f f a f f a f a cb a cb a cb a B shft B cut mul. cut OR mul. LL ) ( ˆ ) 8( 38 85 6) 3)( 38(9 85 ˆ 38 9 85 50 643 38 9 85 50 643 389 785 50 643 389 6785 440 ) 0)( 44( 44 ˆ 4 ˆ 0 a f f f c f c c cb a cb a cb a B t B L mul. cut OR shf mul. B ) ( ) (. 8 3 4 3 c b c b b b L L B00000 B^0036 ζ ζ ˆ

β > T β [,) : [0,) [0,), T ( s) ( β s mod ) β β s β s T β Lar mod β >, 0 α < T [0,) [0,), T ( ) ( β + α mod ) β + α β, α : β, α + β α T β,α β > M β β t : [,) [,), M ( t) ( β t mod [,) ) β t + β M β β β + ˆ β M t) T ( t ) +, t [, ). β (, ˆ β β T β,α X [ 0,), ( X, B, λ) : hβ, α ( ) β β ν < Tβ, α (), 0 β, α ( β, α λ E ) h ( ) d ( ) E < Tβ, α (0), T β,α, ( t),,,3, L, h β, ( ) α h β, α ( ) T β α