A known plaintext attack on the ISAAC keystream generator
|
|
- Holger Bak
- 6 år siden
- Visninger:
Transkript
1 A known plinex ck on he ISAAC keysrem generor Mrin Pudovkin Moscow Engineering Physics Insiue (Technicl Universiy) Deprmen of Crypology nd Discree Mhemics Absrc. Srem ciphers re ofen used in pplicions where high speed nd low dely re requiremen. The ISAAC keysrem generor is fs sofwre-oriened encrypion lgorihm. In his ppers he securiy of he ISAAC keysrem generor is invesiged. Crypnlyic lgorihm is developed for known plinex ck where only smll segmen of plinex is ssumed o be known. Keywords. ISAAC. Keysrem generor. Crypnlysis. 1 Inroducion Srem ciphers re n imporn clss of encrypion lgorihms. They encryp individul chrcers of plinex messge one ime, using n encrypion rnsformion, which vries wih ime. By conrs, block ciphers end o simulneously encryp groups of chrcers of plinex messge using fixed encrypion rnsformion. Srem ciphers re generlly fser hn block ciphers in hrdwre, nd hve less complex hrdwre circuiry. There is vs body of heoreicl knowledge on srem ciphers, nd vrious design principles for srem ciphers hve been proposed nd exensively nlyzed. The mos of srem ciphers proposed in open lierure re bsed on LFSRs (liner feedbck shif regisers). For sofwre implemenion, few keysrem generors hve been designed which re no bsed on shif regisers. One of hese generors is ISAAC. The ISAAC (Indirecion, Shif, Accumule, Add, nd Coun) keysrem generor ws inroduced in [1] by R. Jenkins s fs sofwre-oriened encrypion lgorihm. The im of his pper is o derive some crypnlyic lgorihm h find he correc iniil se of he ISAAC srem cipher using only smll segmen of oupu srem, nd o give precise esimes for he complexiy of he ck. Our resuls re inrinsic o he design principles of ISAAC nd re independen of he size of he key. The pper is orgnized s follows. In secion 2 we give generl descripion of ISAAC. In secion 3 we discuss some properies of ISAAC. Secion 4 describes ck on ISAAC. We conclude in secion 5. 2 Descripion of ISAAC ISAAC is in fc fmily of lgorihms indexed by prmeer m, which is posiive ineger. The inernl se of ISAAC ime consiss of ble S ={s [0],.,s [m-1]} of m=2 n K-bi words nd of wo K-bi words nd i. Le z denoe he oupu K-bi word of ISAAC ime. Le iniilly i 0 = 0 =0. K=2n+, >0. The key of ISAAC is he iniil ble S 0. 1
2 Jenkins kes m=256, n=8, K=32, p0=13, p1=6, p2=2, p3=16, θ 1 =θ 2 =2. Le θ 1,θ 2 <n. (( -1 <<p0) -1 ) if =0 (mod 4 ). G( -1,, p())= (( -1 >>p1) -1 ) if =1 (mod 4 ). (( -1 <<p2) -1 ) if =2 (mod 4 ). (( -1 >>p3) -1 ) if =3 (mod 4 ). where >> nd << indice roion o he righ nd lef, nd p0 if =0 (mod 4). p()= p1 if =1 (mod 4). p2 if =2 (mod 4). p3 if =3 (mod 4). The nex-se funcion F ) i = i (mod m ). b) = (G( -1,, p())+ s [(+ m/2 )(mod m )]) (mod 2 K ). c) s [i ]= (s -1 [(s -1 [i ]>>θ 1 ) (mod m )]+ + z -1 )(mod 2 K ). The oupu funcion f Oupu: z =(s [(s [i ]>>(n+θ 2 ))(mod m)]+s -1 [i ]) (mod 2 K ). 3 Properies of ISAAC In his secion we describe some properies of ISAAC h re used in he descripion of our ck. We will ssume h he oupu sequence z1, z2,, zm+1 is known. Le =(,K-1,,K-2,,,i,,,1,,0 ) be binry represenion of Z K, {, j } Z 2. Proposiion 1 1. The rnsformion F<< (,p)=((<<p) )= (K-1 K-p-1, K-2 K-p-2,, p+i i,, p 0, p-1, p-2,, 1, 0). 2. The rnsformion F >> (,p)=((>>p) )=( K-1, K-2,, p, K-1 K-p-1, K-2 K-p-2,, p+i i,, p 0 ). Proof Noe h (<<p)= ( K-p-1, K-p-2,, i,, 1, 0,0,,0 ). Thus, F<<(,p)=( (<<p) )= (K-1 K-p-1, K-2 K-p-2,, p+i i,, p 0, p-1, p-2,, 1, 0). Noe h (>>p)= (0,, 0, K-1, K-2,, p+1, p ). Therefore, F >> (,p)=( (>>p) )= ( K-1, K-2,, p, K-1 K-p-1, K-2 K-p-2,, p+i i,, p 0 ) The proposiion is proved. Denoe by q =s [s [i ] >>(n+θ 2 ) (mod m)]) (mod m) nd α = (s -1 []>> θ 1 )(mod m), =1,2.. 2
3 In proposiions given below we will ssume h j, α re known. Proposiion 2 If we know s m [0] (mod 2 ), s 1 [1] (mod 2 ),, s m-1 [m-1] (mod 2 ) nd z 1, z 2,, z m+1, hen s 0 [0] (mod 2 ), s0[1] (mod 2 ),, s0[m-1] (mod 2 ), cn be found for =m, m-1, m-2,, 2, 1 s follows. If j =0, m-1,,+1, hen [ s 0 ](mod 2 ) = ( z (mod 2 ) s0[ j ](mod 2 ))(mod 2 ). If 0<j<+1, hen s 0[ ](mod 2 ) = ( z (mod 2 ) s j [ j ](mod 2 ))(mod 2 ), Proof. Noe h nd Then z (mod 2 )=(s [j ] (mod 2 )+s -1 [i ] (mod 2 )) (mod 2 ), (s m [j m ] (mod 2 )+ s 0 [0] (mod 2 )) (mod 2 )= z m (mod 2 ), s j ] = s [ j ]. m[ m j m m s 0[0](mod 2 ) = ( zm (mod 2 ) s j [ jm ](mod 2 ))(mod 2 ). m Le us consider =m-1. If j m =0 (mod m), hen we ge s0[ m 1](mod 2 ) = ( zm 1 (mod 2 ) s0[0](mod 2 ))(mod 2 ). If 0<j m-1 <m, hen s m 1](mod 2 ) = ( z (mod 2 ) s [ j ](mod 2 ))(mod 2 ). 0[ m 1 j m 1 m 1 Now we consider =m-2 1. Assume h s 0 [0] (mod 2 ), s 0 [m-1] (mod 2 ),, s 0 [+1] (mod 2 ) hve been deermined. Then for j m =0, m-1,,+1 we hve [ s 0 ](mod 2 ) = ( z (mod 2 ) s0[ j ](mod 2 ))(mod 2 ). If 0<j <+1, hen we obin s 0[ ](mod 2 ) = ( z (mod 2 ) s j [ j ](mod 2 ))(mod 2 ). The proposiion is proved. Proposiion 3 If we know s m [0](mod 2 ), s 1 [1](mod 2 ),,s m-1 [m-1](mod 2 ) nd z 1, z 2,, z m+1, hen 1 (mod 2 ), 2 (mod 2 ),., m+1 (mod 2 ), cn be found s follows. If j >, hen (mod 2 )=( s [] (mod 2 )- s 0 [α ](mod 2 ) - z -1 (mod 2 )) (mod 2 ). If j, hen (mod 2 ) = (s [](mod 2 ) s [ α ](mod 2 ) z (mod 2 ))(mod2 ), α 1 where =1 m+1. Proof. Noe h for =1, 2,. we hve Whence, s [i ]= s -1 [α ]+ +z -1 (mod 2 K ). (mod 2 )=( s [i ] (mod 2 )- s -1 [α ](mod 2 ) - z -1 (mod 2 )) (mod 2 ). 3
4 Using proposiion 2 we cn find s 0 [0] (mod 2 ), s 0 [1] (mod 2 ),, s 0 [m-1] (mod 2 ). Le us remrk h for ny, d if d>, hen s [d]= s 0 [d] nd if d, hen s [d]= s d [d]. This implies h if α>, hen we ge (mod 2 )=( s [] (mod 2 )- s 0 [α ](mod 2 ) - z -1 (mod 2 )) (mod 2 ), if α, hen The proposiion is proved. α 1 (mod 2 ) = (s [](mod 2 ) s [ α ](mod 2 ) z (mod 2 ))(mod 2 ). Le q be he smlles number of p1, p3, i.e. q=min(p1, p3). Proposiion 4 Le τ 2n+θ2. If we know 1 (mod 2 τ ), 2 (mod 2 τ ),, i (mod 2 τ ),, m (mod 2 τ ) è s 2 [2+m/2] (mod 2 τ ), s 4 [4+m/2] (mod 2 τ ),., s 2i [(m/2+2i)(mod m)](mod 2 τ ),,s m [m](mod 2 τ ), hen 1 (mod 2 τ+q ), 3 (mod 2 τ+q ),, 2i+1 (mod 2 τ+q ),. m-1 (mod 2 τ+q ) cn be deermined s follows.,q-1+τ = b +1,τ-1,τ-1,,q+τ-j = b +1,τ-j,τ-j,.,τ = b +1,τ-q,τ-q. where =1 (mod 2), b +1 =( +1 (mod 2 τ )-s +1 [+1+m/2] (mod 2 τ )) (mod 2 τ ). Proof. By proposiion 1 F >> (,p)=((>>p) )=( K-1, K-2,, p, K-1 K-p-1, K-2 K-p-2,, p+i i,, p 0 ). From (mod 2 τ )=( F << ( -1,p()) (mod 2 τ )+s [+m/2] (mod 2 τ )) (mod 2 τ ), where =0 (mod 2), i follows h F >> ( -1,p()) (mod 2 τ )= b = ( (mod 2 τ )- s [+m/2] (mod 2 τ )) (mod 2 τ ). Hence, b,τ-1 = -1,p()-1+τ -1, τ-1,,b,τ-p() = -1,τ -1,τ-p(). Thus, we hve found unknown p() bis -1,p()-1+τ = b,τ-1-1,τ-1, -1,τ=b,τ-p() -1,τ-p(). Therefore, we hve compued 1 (mod 2 τ+p1 ), 3 (mod 2 τ+p3 ),, 4i+1 (mod 2 τ+p1 ), 4i+3 (mod 2 τ+p3 ),, m-1 (mod 2 τ+p3 ). This shows h 1 (mod 2 τ+q ),, 2i+1 (mod 2 τ+q ),. m-1 (mod 2 τ+q ) re found. The proposiion is proved. Le σ (j) be crry bi in j h -bi of he sum (G( -1,, p())+ s [(+ m/2 ) (mod m )]) (mod 2 K ), σ z (j) be crry bi in j h -bi of he sum (s [(s []>>(n+θ 2 ))(mod m)]+s -1 []) (mod 2 K ) nd σ s (j) be crry bi in j h -bi of he sum (s -1 [(s -1 []>>θ 1 ) (mod m )]+ + z -1 ) (mod 2 K ). Le 4
5 nd 0 δ (, k) = if if > k k 0 if > k ρ (, k) =. 1 if k Proposiion 5 If j<p(), hen 2i+1,j=2i,j sδ(2i+1,2i+1+m/2),j[2i+1+m/2] σ2i +1(j). If j p(), hen 2i+1,j = 2i,j 2i,j - p(2i+1) s δ(2i+1,2i+1+m/2),j [2i+1+m/2] σ 2i +1 (j). This proposiion cn esily be proved if noe h (mod 2 j )= (G( -1,, p()) (mod 2 j )+ s [(+ m/2 )(mod m )] (mod 2 j )) (mod 2 j ) nd j h -bi of G(2i,, p(2i+1)) is 2i,j if j<p(), G( 2i,, p(2i+1)) j = 2i,j 2i,j - p(2i+1) if j p(). Proposiion 6 If we know s [] (mod 2 j-1 ), s 0 [] (mod 2 j-1 ), α, j (mod 2 j-1 ), z -1 (mod 2 j-1 ), -1 (mod 2 j-1 ), hen σ s (j), σ z (j) nd σ 2i +1 (j) cn be compued s follows. σ s (j)= 1 if (s δ(, á )[α ](mod 2 j-1 )+z -1 (mod 2 j-1 )+,j-1 (mod 2 j-1 )) (mod 2 j+1 ) 2 j, 0 oherwise. σ z (j)= 1 if (s δ(, j ),j-1[j ] (mod 2 j-1 )+ s 0,j -1 [] (mod 2 j-1 )) (mod 2 j+1 ) 2 j, 0 oherwise. If j<p() nd =1 (mod 2), hen σ2i +1(j)= 1 if ( 2i(mod 2 j-1 )+s2i+1[2i+1+m/2](mod 2 j-1 )) (mod 2 j+1 ) 2 j, 0 oherwise. If j p() nd =1 (mod 2), hen σ j (j)= 1, if 1 ( 2 k + -1(mod 2 p()-1 )+sδ(,+m/2),j[+m/2] (mod 2 j-1 )) (mod 2 j+1 ) 2 j k= p() -1,k -1 p(),k ) 0, oherwise. Proof. Noe h σ s (j), σ (j) nd σ z (j) re equl o 1 if nd only if (G( -1,, p())+ s [(+ m/2 )(mod m)]) (mod 2 j+1 ) 2 j, (s [(s []>>(n+θ 2 ))(mod m)]+s -1 []) (mod 2 j+1 ) 2 j, (s -1 [(s -1 []>>θ 1 ) (mod m) ] + + z -1 ) (mod 2 j+1 ) 2 j. Thus, σ (j)= 1 if (G(-1,, p()) (mod 2 j-1 )+s[+m/2](mod 2 j-1 )) (mod 2 j+1 ) 2 j, 0 oherwise. 5
6 nd G( -1,, p() (mod 2 j-1 )= 2i (mod 2 j-1 ) j 1 k= p() -1,k -1 p(),k ) By he bove noes we obin he proof of he proposiion. if j<p(), ( 2 k + -1 (mod 2 p()-1 ) oherwise. Theorem 1 If we know α, j, 2i+1(mod 2 j ), 2i(mod 2 j-1 ), σ s (j), σ (j), σ z (j), z,j, =1,,m, i=0 m/2-1 nd j mx(p(1), p(3)), hen s,j [], s 0,j [], =1,,m, cn be found by solving he following sysem of equions. s0,j[0] sδ(m, j ),j[jm] = zm,j σm z (j), m s 0,j [1] s δ(1, j 1 ),j [j 1 ]= z 1,j σ z 1 (j), s0,j[] sδ(, j ),j[j]=z,j σ z (j), s 0,j [1+m/2]= 1,j σ 1 (j), s 0,j [] s δ(, j ),j[j ]=z,j σ z (j), s0,j[m-1] sδ(m-1, j ),j[jm-1]= zm-1,j σm z -1(j), m-1 s 1,j [1] s δ(1, α 1 ),j[α 1 ] = 1,j σ s 1 (j), s 2,j [2] s δ(2, α 2 ),j[α 2 ] s 0,j [3+m/2]= z 1,j 3,j 2,j - p(3) σ 3 (j) σ s 2 (j), s 3,j [3] s δ(3, α3 ),j[α 3 ] = z 2,j 3,j σ s 3 (j), (1) s 2i,j [2i] s δ(2i+1,2i+1+m/2),j [2i+1+m/2] s δ(2i, α2 i ),j[α 2i ]=z 2i-1,j 2i+1,j 2i,j - p(2i+1) σ 2i +1 (j) σ s 2i (j), s 2i+1,j [2i+1] s δ(2i+1, α 2 i+ 1 ),j s [α 2i+1 ] = z 2i,j 2i+1,j σ 2i +1 (j), s m-2,j [m-2] s δ(m-2, α m - 2 s m-1,j [m-1] s δ(m-1,m/2-1),j [m/2-1] s δ(m-1, α 1 ),j [α m-2 ]= z m-3,j m-2,j σ m s -2 (j), m ),j [α m-1 ]= z m,j m-1,j m-2,j - p(3) σ m -1 (j) σ m s -1 (j), s m,j [m] s δ(m, α m ),j[αm] = z m-1,j m,j σm s (j). 2i,j, i=0 m/2 cn be found s follows 2i,j = 2i+1,j 2i,j - p(2i+1) s δ(2i+1,2i+1+m/2),j [2i+1+m/2] σ 2i +1 (j). Proof Consider j h -bi of = (G( -1,, p())+ s [(+ m/2 )(mod m )]) (mod 2 K ), =1 (mod 2) s []= (s -1 [α ]+ + z -1 )(mod 2 K ), =1 m. z = (s [j ]+s 0 []) (mod 2 K ), =1 m. Thus, we obin he following sysem of equions. 1,j=s0,j[1+m/2] σ1 (j), s 1,j [1]= s δ(1, α 1 ),j[α 1 ] 1,j σ 1 s (j), 6
7 z 1,j = s δ(1, j 1 ),j[j 1 ] s 0,j [1] σ z 1 (j), s 2,j [2]= s δ(2, α 2 ),j[α 2 ] z 1,j 2,j σ s 2 (j), z 2,j = s δ(2, j 2 ),j [j 2 ] s 0,j [2] σ z 2 (j), 3,j = 2,j 2,j - p(3) s 0,j [3+m/2] σ 3 (j), s 3,j [3]= s δ(3, α3 ),j[α 3 ] z 2,j 3,j σ s 3 (j), z 3,j = s δ(3, j 3 ),j [j 3 ] s 0,j [3] σ z 3 (j), (2) s 2i,j [2i]= s δ(2i, α 2 i ),j[α 2i ] z 2i-1,j 2i,j σ s 2i (j), z 2i,j = s δ(2i, j 2i),j[j 2i ] s 0,j [2i] σ z 2i (j), 2i+1,j=2i,j 2i,j - p(2i+1) sδ(2i+1,2i+1+m/2),j[2i+1+m/2] σ2i +1(j), s 2i+1,j [2i+1]= s δ(2i+1, α 2 i+ 1 ),j s [α 2i+1 ] z 2i,j 2i+1,j σ 2i +1 (j), z 2i+1,j = s δ(2i+1, j 2i+ 1 ),j z [j 2i+1 ] s 0,j [2i+1] σ 2i +1 (j), s m-2,j [m-2]= s δ(m-2, α m - 2 ),j s [α m-2 ] z m-3,j m-2,j σ m -2 (j), z z m-2,j = s δ(m-2, j m-2 ),j[j m-2 ] s 0,j [m-2] σ m -2 (j), m-1,j=m-2,j m-2,j - p(m-1) sδ(m-1,m/2-1),j[m/2-1] σm -1(j), s m-1,j [m-1]= s δ(m-1, α m 1 ),j s [α m-1 ] z m,j m-1,j σ m -1 (j), z z m-1,j = s δ(m-1, j m-1),j[j m-1 ] s 0,j [m-1] σ m -1 (j), s m,j [m]= s δ(m, α m ),j[α m ] z m-1,j m,j σ s m (j), zm,j= sδ(m, j ),j[jm] s0,j[0] σm z (j). m Noe h s,j [], s 0,j [], 2i,j, =1,, m, i=0 m/2, re unknown nd he number of unknown vlues in (2) is 5m/2. By proposiion 6 we hve 2i,j = 2i+1,j 2i,j - p(2i+1) s δ(2i+1,2i+1+m/2),j [2i+1+m/2] σ 2i +1 (j), where i=1 m/2-1. If we replce 2i,j by 2i+1,j 2i,j - p(2i+1) s δ(2i+1,2i+1+m/2),j [2i+1+m/2] σ 2i +1 (j) in (2), we ge 1,j =s 0,j [1+m/2] σ 1 (j), s 1,j [1]= s δ(1, α 1 ),j[α 1 ] 1,j σ s 1 (j), z 1,j = s δ(1, j 1 ),j[j 1 ] s 0,j [1] σ1 z (j), s 2,j [2]= s δ(2, α 2 ),j[α 2 ] z 1,j 3,j 2,j - p(3) s 0,j [3+m/2] σ 3 (j) σ s 2 (j), z 2,j = s δ(2, j 2 ),j [j 2 ] s 0,j [2] σ z 2 (j), s 3,j [3]= s δ(3, α3 ),j[α 3 ] z 2,j 3,j σ s 3 (j), z3,j= sδ(3, j ),j[j3] s0,j[3] σ3 z (j), 3 (3) s 2i,j [2i]= s δ(2i, α 2 i ),j[α 2i ] z 2i-1,j 2i+1,j 2i,j - p(2i+1) s δ(2i+1,2i+1+m/2),j [2i+1+m/2] σ 2i +1 (j) σ s 2i (j), z 2i,j = s δ(2i, j ),j[j 2i 2i ] s 0,j [2i] σ2i z (j), s 2i+1,j [2i+1]= s δ(2i+1, α 2 i+ 1 ),j s [α 2i+1 ] z 2i,j 2i+1,j σ 2i +1 (j), z 2i+1,j = s δ(2i+1, j 2i+ 1 ),j z [j 2i+1 ] s 0,j [2i+1] σ 2i +1 (j), 7
8 s m-2,j [m-2]= s δ(m-2, α m -2 ),j s [α m-2 ] z m-3,j m-2,j σ m -2 (j), z z m-2,j = s δ(m-2, j m-2 ),j[j m-2 ] s 0,j [m-2] σ m -2 (j), s m-1,j [m-1]= s δ(m-1, α m 1 ),j s [α m-1 ] z m,j m-1,j m-2,j - p(3) s δ(m-1,m/2-1),j [m/2-1] σ m -1 (j) σ m -1 (j), zm-1,j= sδ(m-1, j ),j[jm-1] s0,j[m-1] σm z -1(j), m-1 s m,j [m]= s δ(m, α m ),j[α m ] z m-1,j m,j σ s m (j), z m,j = s δ(m, j m ),j[j m ] s 0,j [0] σ m z (j), We sress h he number of equions nd he number of unknown vlues in (3) re 2m. If we rewrie (3) such h unknown elemens in he equions re on he lef, nd known vlues on he righ, hen we hve s 0,j [0] s δ(m, j m ),j [j m ] = z m,j σ z m (j), s 0,j [1] s δ(1, j 1 ),j [j 1 ]= z 1,j σ z 1 (j), s 0,j [] s δ(, j ),j[j ]=z,j σ z (j), s0,j[1+m/2]= 1,j σ1 (j), s 0,j [] s δ(, j ),j[j ]=z,j σ z (j), z s 0,j [m-1] s δ(m-1, j m-1),j[j m-1 ]= z m-1,j σ m -1 (j), s 1,j [1] s δ(1, α 1 ),j[α 1 ] = 1,j σ s 1 (j), s 2,j [2] s δ(2, α 2 ),j[α 2 ] s 0,j [3+m/2]= z 1,j 3,j 2,j - p(3) σ 3 (j) σ s 2 (j), s3,j[3] sδ(3, α 3 ),j[α3] = z2,j 3,j σ3 s (j), (1) s 2i,j [2i] s δ(2i+1,2i+1+m/2),j [2i+1+m/2] s δ(2i, α 2 i ),j[α 2i ]=z 2i-1,j 2i+1,j 2i,j - p(2i+1) σ 2i +1 (j) σ s 2i (j), s2i+1,j[2i+1] sδ(2i+1, α ),j[α2i+1] = z2i,j 2 i+ 1 2i+1,j σ2is +1(j), s m-2,j [m-2] s δ(m-2, α m - 2 ),j s [α m-2 ]= z m-3,j m-2,j σ m -2 (j), sm-1,j[m-1] sδ(m-1,m/2-1),j[m/2-1] sδ(m-1, α ),j[αm-1]= zm,j m 1 m-1,j m-2,j - p(3) σm -1(j) σm s -1(j), s m,j [m] s δ(m, α m ),j[α m ] = z m-1,j m,j σ s m (j), The heorem is proved. 4 Ack on ISAAC In his secion we describe known plinex ck on he ISAAC keysrem generor. Firs le us crry ou n esimion of he uniciy disnce DISAAC of ISAAC. Recll h he uniciy disnce is he number of keysrem symbols h need o be observed in known plinex ck before he key cn be uniquely deermined. 8
9 Noe h he number of vrious ses of he ISAAC is equl o m 2 K 2 Km. Then we ge h K D ISAAC K+ Km ( 2 ) = m 2. Therefore, D ISAAC m+2. Le us denoe wih mrk * guessed elemens of S * nd elemens of he oupu sequence {z i * } produced on he guessed iniil se. The mehod consiss of four seps. Sep 1. Guess s m [0] (mod 2 2n+θ2 ),, s [] (mod 2 2n+θ2 ),, s m-1 [m-1] (mod 2 2n+θ2 ). Sep 2 Le =2n+θ2. 1. Use proposiion 2 o compue s 0 [], =0,1,,m Use proposiion 3 o compue (mod 2 2n+θ2 ), =1 m Le τ=. Use proposiion 4 o compue 2j+1(mod 2 τ+q ), j=0 m/ To find s m [0](mod 2 τ+q ), s 1 [1] (mod 2 τ+q ),, s m-1 [m-1] (mod 2 τ+q ), s 0 [0] (mod 2 τ+q ), s 0 [1] (mod 2 τ+q ),,s 0 [m-1] (mod 2 τ+q ), 2i (mod 2 τ+q ), i=0 m/2, we do he following. ) Le j=τ+1. b) While j τ+q do. Use heorem 1 o compue sm[0]( mod 2 j ),,sm-1[m-1] ( mod 2 j ), s0[0] ( mod 2 j ),, s 0 [m-1] ( mod 2 j ), 2i ( mod 2 j ), i=0 m/2. Tke j=j+1. Sep 3 Le τ=2n+θ2+q. While τ<k. 1. Use proposiion 4 o compue 2j+1 (mod 2 τ+q ), j=0 m/ To find s m [0] (mod 2 τ+q ), s 1 [1] (mod 2 τ+q ),, s m-1 [m-1] (mod 2 τ+q ), s 0 [0] (mod 2 τ+q ), s 0 [1] (mod 2 τ+q ),,s 0 [m-1] (mod 2 τ+q ), 2i (mod 2 τ+q ), i=0 m/2, we do he following. ) Le j=τ+1. b) While j τ+q do. Use o heorem 1 o compue s m [0]( mod 2 j ),,s m-1 [m-1] ( mod 2 j ), s 0 [0] ( mod 2 j ),, s 0 [m-1] (mod 2 j ), 2i ( mod 2 j ), i=0 m/2. Tke j=j+1. Sep 4 Compue he firs L = D ISAAC of elemens of he oupu sequence z 1 *, z 2 *, z L *. If z 1 * = z 1, z 2 * = z2, zl * = zl hen we hve found he correc iniil se of he cryposysem, oherwise reurn o sep 1. Le us esime he complexiy of he mehod. We my ssume h he probbiliy P{s * 0 [0] (mod 2 (2n+θ2) )= s 0 [0] (mod 2 (2n+θ2) ),,s * 0 [m-1] (mod 2 (2n+θ2) )= s 0 [m-1] (mod 2 (2n+θ2) )} = 1/ 2 (2n+θ2)m. Then he verge of guessed elemens is equl o 2 (2n+θ2)m-1. The complexiy of soluion of sysems of equions seps 2, 3 cn be esimed (K-2n-θ2) (2m)/3. Therefore, he complexiy of he mehod is equl o T me =2 (2n+θ2)m-1 (K-2n-θ2) (2m)/3. Noe h he complexiy of he brue force ck is equl o T br =2 K m-1. For m=256, n=8, K=32, p0=13, p1=6, p2=2, p3=16, θ 1 =θ 2 =2, we ge T me = , T br =
10 5 Conclusion We hve described crypnlyic lgorihm on he ISAAC srem cipher. The lgorihm ries o deduce he iniil se in known plinex ck. The described mehod depends on difference K-2n. If K-2n-θ2 2n, hen he complexiy of he ck is pproximed o be less hn ime of serching hrough he squre roo of ll possible iniil ses. For vlues used in he cryposysem we ge he complexiy T me = ISAAC remins secure cipher for prcicl pplicions. References. [1] R.J. Jenkins, ISAAC, Fs Sofwre Encrypion Cmbridge 1996, vol. 1039, D. Gollmnn ed., Springer-Verlg. [2] R. J. Jenkins ISAAC hp://ourworld.compuserve.com/homepges/ bob_jenkins/isc.hm [3] Vrfolomeev A.A., Zhukov A.E., Pudovkin M., ''Anlysis of Srem Ciphers '', Moscow, [4] Pudovkin M. A Cycle Srucure of he Alleged RC4 Keysrem Generor. Journl of "Securiy of informion echnologies", Moscow, 4,
Hermite-Hadamard-Fejer Type Inequalities for s Convex Function in the Second Sense via Fractional Integrals
Filom 30: 06), 33 338 DOI 0.98/FIL63S Published by Fculy o Sciences nd Mhemics, Universiy o Niš, Serbi Avilble : hp://www.pm.ni.c.rs/ilom Hermie-Hdmrd-Fejer Type Ineuliies or s Convex Funcion in he Second
Læs mereBasic statistics for experimental medical researchers
Basic statistics for experimental medical researchers Sample size calculations September 15th 2016 Christian Pipper Department of public health (IFSV) Faculty of Health and Medicinal Science (SUND) E-mail:
Læs mereLarge time behavior of solutions for a complex-valued quadratic heat equation
Nonlinear Differ. Equ. Appl. 5, 5 45 c 5 Springer Basel -97/5/55-4 published online March 5, 5 DOI.7/s3-5-3-7 Nonlinear Differenial Equaions and Applicaions NoDEA Large ime behavior of soluions for a complex-valued
Læs mereGeneralized Probit Model in Design of Dose Finding Experiments. Yuehui Wu Valerii V. Fedorov RSU, GlaxoSmithKline, US
Generalized Probit Model in Design of Dose Finding Experiments Yuehui Wu Valerii V. Fedorov RSU, GlaxoSmithKline, US Outline Motivation Generalized probit model Utility function Locally optimal designs
Læs mereThe LWR Model in Lagrangian coordinates
The LWR Model in Lagrangian coordinaes ACI-NIM: Mah Models on Traffic Flow Ludovic Leclercq, LICIT (ENTPE/INRETS) Jorge Laval, Georgiaech Universiy 31 ocober 2007 INRETS Ouline Lagrangian resoluion of
Læs mereVores mange brugere på musskema.dk er rigtig gode til at komme med kvalificerede ønsker og behov.
På dansk/in Danish: Aarhus d. 10. januar 2013/ the 10 th of January 2013 Kære alle Chefer i MUS-regi! Vores mange brugere på musskema.dk er rigtig gode til at komme med kvalificerede ønsker og behov. Og
Læs mereProbabilistic properties of modular addition. Victoria Vysotskaya
Probabilistic properties of modular addition Victoria Vysotskaya JSC InfoTeCS, NPK Kryptonite CTCrypt 19 / June 4, 2019 vysotskaya.victory@gmail.com Victoria Vysotskaya (Infotecs, Kryptonite) Probabilistic
Læs mereVina Nguyen HSSP July 13, 2008
Vina Nguyen HSSP July 13, 2008 1 What does it mean if sets A, B, C are a partition of set D? 2 How do you calculate P(A B) using the formula for conditional probability? 3 What is the difference between
Læs mereThe X Factor. Målgruppe. Læringsmål. Introduktion til læreren klasse & ungdomsuddannelser Engelskundervisningen
The X Factor Målgruppe 7-10 klasse & ungdomsuddannelser Engelskundervisningen Læringsmål Eleven kan give sammenhængende fremstillinger på basis af indhentede informationer Eleven har viden om at søge og
Læs mereOn the complexity of drawing trees nicely: corrigendum
Acta Informatica 40, 603 607 (2004) Digital Object Identifier (DOI) 10.1007/s00236-004-0138-y On the complexity of drawing trees nicely: corrigendum Thorsten Akkerman, Christoph Buchheim, Michael Jünger,
Læs mereDoodleBUGS (Hands-on)
DoodleBUGS (Hands-on) Simple example: Program: bino_ave_sim_doodle.odc A simulation example Generate a sample from F=(r1+r2)/2 where r1~bin(0.5,200) and r2~bin(0.25,100) Note that E(F)=(100+25)/2=62.5
Læs mereSign variation, the Grassmannian, and total positivity
Sign variation, the Grassmannian, and total positivity arxiv:1503.05622 Slides available at math.berkeley.edu/~skarp Steven N. Karp, UC Berkeley FPSAC 2015 KAIST, Daejeon Steven N. Karp (UC Berkeley) Sign
Læs mereEngelsk. Niveau D. De Merkantile Erhvervsuddannelser September Casebaseret eksamen. og
052431_EngelskD 08/09/05 13:29 Side 1 De Merkantile Erhvervsuddannelser September 2005 Side 1 af 4 sider Casebaseret eksamen Engelsk Niveau D www.jysk.dk og www.jysk.com Indhold: Opgave 1 Presentation
Læs mereHelp / Hjælp
Home page Lisa & Petur www.lisapetur.dk Help / Hjælp Help / Hjælp General The purpose of our Homepage is to allow external access to pictures and videos taken/made by the Gunnarsson family. The Association
Læs mereHenstock-Kurzweil Laplace Transform
Chper 5 Hensock-Kurzweil Lplce Trnsform 5.1 Inroducion The Lplce rnsform of funcion f : [, ) R, s C, is defined s he inegrl [15] e s f() d. (5.1) Mny uhors, in some previous decdes, showed heir ineres
Læs mereResource types R 1 1, R 2 2,..., R m CPU cycles, memory space, files, I/O devices Each resource type R i has W i instances.
System Model Resource types R 1 1, R 2 2,..., R m CPU cycles, memory space, files, I/O devices Each resource type R i has W i instances. Each process utilizes a resource as follows: request use e.g., request
Læs merePrivat-, statslig- eller regional institution m.v. Andet Added Bekaempelsesudfoerende: string No Label: Bekæmpelsesudførende
Changes for Rottedatabasen Web Service The coming version of Rottedatabasen Web Service will have several changes some of them breaking for the exposed methods. These changes and the business logic behind
Læs mereEngelsk. Niveau C. De Merkantile Erhvervsuddannelser September 2005. Casebaseret eksamen. www.jysk.dk og www.jysk.com.
052430_EngelskC 08/09/05 13:29 Side 1 De Merkantile Erhvervsuddannelser September 2005 Side 1 af 4 sider Casebaseret eksamen Engelsk Niveau C www.jysk.dk og www.jysk.com Indhold: Opgave 1 Presentation
Læs mereEric Nordenstam 1 Benjamin Young 2. FPSAC 12, Nagoya, Japan
Eric 1 Benjamin 2 1 Fakultät für Matematik Universität Wien 2 Institutionen för Matematik Royal Institute of Technology (KTH) Stockholm FPSAC 12, Nagoya, Japan The Aztec Diamond Aztec diamonds of orders
Læs mereE-PAD Bluetooth hængelås E-PAD Bluetooth padlock E-PAD Bluetooth Vorhängeschloss
E-PAD Bluetooth hængelås E-PAD Bluetooth padlock E-PAD Bluetooth Vorhängeschloss Brugervejledning (side 2-6) Userguide (page 7-11) Bedienungsanleitung 1 - Hvordan forbinder du din E-PAD hængelås med din
Læs mereAktivering af Survey funktionalitet
Surveys i REDCap REDCap gør det muligt at eksponere ét eller flere instrumenter som et survey (spørgeskema) som derefter kan udfyldes direkte af patienten eller forsøgspersonen over internettet. Dette
Læs mereBesvarelser til Lineær Algebra Reeksamen Februar 2017
Besvarelser til Lineær Algebra Reeksamen - 7. Februar 207 Mikkel Findinge Bemærk, at der kan være sneget sig fejl ind. Kontakt mig endelig, hvis du skulle falde over en sådan. Dette dokument har udelukkende
Læs mereSkriftlig Eksamen Beregnelighed (DM517)
Skriftlig Eksamen Beregnelighed (DM517) Institut for Matematik & Datalogi Syddansk Universitet Mandag den 31 Oktober 2011, kl. 9 13 Alle sædvanlige hjælpemidler (lærebøger, notater etc.) samt brug af lommeregner
Læs mereFejlbeskeder i SMDB. Business Rules Fejlbesked Kommentar. Validate Business Rules. Request- ValidateRequestRegist ration (Rules :1)
Fejlbeskeder i SMDB Validate Business Rules Request- ValidateRequestRegist ration (Rules :1) Business Rules Fejlbesked Kommentar the municipality must have no more than one Kontaktforløb at a time Fejl
Læs mereÅbenrå Orienteringsklub
Åbenrå Orienteringsklub Velkommen til det ægte orienteringsløb på Blå Sommer 2009 Din gruppe har tilmeldt spejdere til at deltage i det ægte orienteringsløb på Blå Sommer 2009. Orienteringsløbet gennemføres
Læs mereATEX direktivet. Vedligeholdelse af ATEX certifikater mv. Steen Christensen stec@teknologisk.dk www.atexdirektivet.
ATEX direktivet Vedligeholdelse af ATEX certifikater mv. Steen Christensen stec@teknologisk.dk www.atexdirektivet.dk tlf: 7220 2693 Vedligeholdelse af Certifikater / tekniske dossier / overensstemmelseserklæringen.
Læs mereUserguide. NN Markedsdata. for. Microsoft Dynamics CRM 2011. v. 1.0
Userguide NN Markedsdata for Microsoft Dynamics CRM 2011 v. 1.0 NN Markedsdata www. Introduction Navne & Numre Web Services for Microsoft Dynamics CRM hereafter termed NN-DynCRM enable integration to Microsoft
Læs mereLinear Programming ١ C H A P T E R 2
Linear Programming ١ C H A P T E R 2 Problem Formulation Problem formulation or modeling is the process of translating a verbal statement of a problem into a mathematical statement. The Guidelines of formulation
Læs mereDET KONGELIGE BIBLIOTEK NATIONALBIBLIOTEK OG KØBENHAVNS UNIVERSITETS- BIBLIOTEK. Index
DET KONGELIGE Index Download driver... 2 Find the Windows 7 version.... 2 Download the Windows Vista driver.... 4 Extract driver... 5 Windows Vista installation of a printer.... 7 Side 1 af 12 DET KONGELIGE
Læs mereProject Step 7. Behavioral modeling of a dual ported register set. 1/8/ L11 Project Step 5 Copyright Joanne DeGroat, ECE, OSU 1
Project Step 7 Behavioral modeling of a dual ported register set. Copyright 2006 - Joanne DeGroat, ECE, OSU 1 The register set Register set specifications 16 dual ported registers each with 16- bit words
Læs mereExercise 6.14 Linearly independent vectors are also affinely independent.
Affine sets Linear Inequality Systems Definition 6.12 The vectors v 1, v 2,..., v k are affinely independent if v 2 v 1,..., v k v 1 is linearly independent; affinely dependent, otherwise. We first check
Læs mereBusiness Rules Fejlbesked Kommentar
Fejlbeskeder i SMDB Validate Business Request- ValidateRequestRegi stration ( :1) Business Fejlbesked Kommentar the municipality must have no more than one Kontaktforløb at a time Fejl 1: Anmodning En
Læs mereSkriftlig Eksamen Beregnelighed (DM517)
Skriftlig Eksamen Beregnelighed (DM517) Institut for Matematik & Datalogi Syddansk Universitet Mandag den 7 Januar 2008, kl. 9 13 Alle sædvanlige hjælpemidler (lærebøger, notater etc.) samt brug af lommeregner
Læs mereTrolling Master Bornholm 2015
Trolling Master Bornholm 2015 (English version further down) Sæsonen er ved at komme i omdrejninger. Her er det John Eriksen fra Nexø med 95 cm og en kontrolleret vægt på 11,8 kg fanget på østkysten af
Læs mereFejlbeskeder i Stofmisbrugsdatabasen (SMDB)
Fejlbeskeder i Stofmisbrugsdatabasen (SMDB) Oversigt over fejlbeskeder (efter fejlnummer) ved indberetning til SMDB via webløsning og via webservices (hvor der dog kan være yderligere typer fejlbeskeder).
Læs mereSome results for the weighted Drazin inverse of a modified matrix
International Journal of Applied Mathematics Computation Journal homepage: www.darbose.in/ijamc ISSN: 0974-4665 (Print) 0974-4673 (Online) Volume 6(1) 2014 1 9 Some results for the weighted Drazin inverse
Læs merePARALLELIZATION OF ATTILA SIMULATOR WITH OPENMP MIGUEL ÁNGEL MARTÍNEZ DEL AMOR MINIPROJECT OF TDT24 NTNU
PARALLELIZATION OF ATTILA SIMULATOR WITH OPENMP MIGUEL ÁNGEL MARTÍNEZ DEL AMOR MINIPROJECT OF TDT24 NTNU OUTLINE INEFFICIENCY OF ATTILA WAYS TO PARALLELIZE LOW COMPATIBILITY IN THE COMPILATION A SOLUTION
Læs mereTrolling Master Bornholm 2015
Trolling Master Bornholm 2015 (English version further down) Panorama billede fra starten den første dag i 2014 Michael Koldtoft fra Trolling Centrum har brugt lidt tid på at arbejde med billederne fra
Læs mereSkriftlig Eksamen Diskret matematik med anvendelser (DM72)
Skriftlig Eksamen Diskret matematik med anvendelser (DM72) Institut for Matematik & Datalogi Syddansk Universitet, Odense Onsdag den 18. januar 2006 Alle sædvanlige hjælpemidler (lærebøger, notater etc.),
Læs mereOur activities. Dry sales market. The assortment
First we like to start to introduce our activities. Kébol B.V., based in the heart of the bulb district since 1989, specialises in importing and exporting bulbs world-wide. Bulbs suitable for dry sale,
Læs mereAdaptive Algorithms for Blind Separation of Dependent Sources. George V. Moustakides INRIA, Sigma 2
Adaptive Algorithms for Blind Separation of Dependent Sources George V. Moustakides INRIA, Sigma 2 Problem definition-motivation Existing adaptive scheme-independence General adaptive scheme-dependence
Læs mereMultivariate Extremes and Dependence in Elliptical Distributions
Multivariate Extremes and Dependence in Elliptical Distributions Filip Lindskog, RiskLab, ETH Zürich joint work with Henrik Hult, KTH Stockholm I II III IV V Motivation Elliptical distributions A class
Læs mereTM4 Central Station. User Manual / brugervejledning K2070-EU. Tel Fax
TM4 Central Station User Manual / brugervejledning K2070-EU STT Condigi A/S Niels Bohrs Vej 42, Stilling 8660 Skanderborg Denmark Tel. +45 87 93 50 00 Fax. +45 87 93 50 10 info@sttcondigi.com www.sttcondigi.com
Læs mereSkriftlig Eksamen Kombinatorik, Sandsynlighed og Randomiserede Algoritmer (DM528)
Skriftlig Eksamen Kombinatorik, Sandsynlighed og Randomiserede Algoritmer (DM58) Institut for Matematik og Datalogi Syddansk Universitet, Odense Torsdag den 1. januar 01 kl. 9 13 Alle sædvanlige hjælpemidler
Læs mereName: Week of April 15 MathWorksheets.com
Get a fidget spinner! Spin it. I needed to spin time(s) to finish. Spin again. Add. Complete each number bond. I needed to spin time(s) to finish. How many times do you need to spin? I needed to spin time(s)
Læs mereUser Manual for LTC IGNOU
User Manual for LTC IGNOU 1 LTC (Leave Travel Concession) Navigation: Portal Launch HCM Application Self Service LTC Self Service 1. LTC Advance/Intimation Navigation: Launch HCM Application Self Service
Læs mereMeasuring Evolution of Populations
Measuring Evolution of Populations 2007-2008 5 Agents of evolutionary change Mutation Gene Flow Non-random mating Genetic Drift Selection Populations & gene pools Concepts a population is a localized group
Læs mereFractional Wavelet Transform in Terms of Fractional Convolution
Progr. Frc. Differ. Appl., No. 3, 20-20 205 20 Progress in Frcionl Differeniion nd Applicions An Inernionl Journl hp://dx.doi.org/0.2785/pfd/00305 Frcionl Wvele Trnsform in Terms of Frcionl Convoluion
Læs mereUniversity of Copenhagen Faculty of Science Written Exam April Algebra 3
University of Copenhagen Faculty of Science Written Exam - 16. April 2010 Algebra This exam contains 5 exercises which are to be solved in hours. The exercises are posed in an English and in a Danish version.
Læs mereDanish and English. Standard Field Analysis (Diderichsen) Standard Field Analysis (Diderichsen)
Danish and English Some major poins of synacic conrass [Righ click for speaker s noes] Sandard Field nalysis (Diderichsen) Main clause able fel Forfel (Fundamenfel) Nexusfel ndholdsfel og Hvorfor Søren
Læs mereBusiness Opening. Very formal, recipient has a special title that must be used in place of their name
- Opening English Danish Dear Mr. President, Kære Hr. Direktør, Very formal, recipient has a special title that must be used in place of their name Dear Sir, Formal, male recipient, name unknown Dear Madam,
Læs mereBusiness Opening. Very formal, recipient has a special title that must be used in place of their name
- Opening Danish English Kære Hr. Direktør, Dear Mr. President, Very formal, recipient has a special title that must be used in place of their name Kære Hr., Formal, male recipient, name unknown Kære Fru.,
Læs mereUniversity of Copenhagen Faculty of Science Written Exam - 3. April Algebra 3
University of Copenhagen Faculty of Science Written Exam - 3. April 2009 Algebra 3 This exam contains 5 exercises which are to be solved in 3 hours. The exercises are posed in an English and in a Danish
Læs merehow to save excel as pdf
1 how to save excel as pdf This guide will show you how to save your Excel workbook as PDF files. Before you do so, you may want to copy several sheets from several documents into one document. To do so,
Læs mereUnitel EDI MT940 June 2010. Based on: SWIFT Standards - Category 9 MT940 Customer Statement Message (January 2004)
Unitel EDI MT940 June 2010 Based on: SWIFT Standards - Category 9 MT940 Customer Statement Message (January 2004) Contents 1. Introduction...3 2. General...3 3. Description of the MT940 message...3 3.1.
Læs mereTrolling Master Bornholm 2016 Nyhedsbrev nr. 5
Trolling Master Bornholm 2016 Nyhedsbrev nr. 5 English version further down Kim Finne med 11 kg laks Laksen blev fanget i denne uge øst for Bornholm ud for Nexø. Et andet eksempel er her to laks taget
Læs mereTrolling Master Bornholm 2014
Trolling Master Bornholm 2014 (English version further down) Den ny havn i Tejn Havn Bornholms Regionskommune er gået i gang med at udvide Tejn Havn, og det er med til at gøre det muligt, at vi kan være
Læs mereTrolling Master Bornholm 2013
Trolling Master Bornholm 2013 (English version further down) Tilmeldingen åbner om to uger Mandag den 3. december kl. 8.00 åbner tilmeldingen til Trolling Master Bornholm 2013. Vi har flere tilmeldinger
Læs mereKurver og flader Aktivitet 15 Geodætiske kurver, Isometri, Mainardi-Codazzi, Teorema Egregium
Kurver og flader Aktivitet 15 Geodætiske kurver, Isometri, Mainardi-Codazzi, Teorema Egregium Lisbeth Fajstrup Institut for Matematiske Fag Aalborg Universitet Kurver og Flader 2013 Lisbeth Fajstrup (AAU)
Læs mereLæs vejledningen godt igennem, før du begynder at samle vuggen. Please read the instruction carefully before you start.
00 Samlevejledning på Vugge ssembly instruction for the cradle Læs vejledningen godt igennem, før du begynder at samle vuggen. Please read the instruction carefully before you start. www.oliverfurniture.dk
Læs mereF o r t o l k n i n g e r a f m a n d a l a e r i G I M - t e r a p i
F o r t o l k n i n g e r a f m a n d a l a e r i G I M - t e r a p i - To fortolkningsmodeller undersøgt og sammenlignet ifm. et casestudium S i g r i d H a l l b e r g Institut for kommunikation Aalborg
Læs mereEksamen i Signalbehandling og matematik
Opgave. (%).a. Figur og afbilleder et diskret tid signal [n ] og dets DTFT. [n] bruges som input til et LTI filter med en frekvens amplitude respons som vist på figur. Hvilket af de 4 output signaler (y
Læs merePlease report absence, also if you don t plan to participate in dinner to Birgit Møller Jensen Telephone: /
Annex 01.01 Board Meeting - Draft Agenda Wednesday, 24 th April 2013 at 15.00-20.00 in the Meetery, AADK, Fælledvej 12, 2200 Copenhagen N Agenda Status Time (proposed) Annex Comments 1. Welcome and approval
Læs mereDårlig litteratur sælger - Trykkekultur i 1800-tallets Storbritannien og idag. Maria Damkjær Post.doc. i Engelsk Litteratur
Dårlig litteratur sælger - Trykkekultur i 1800-tallets Storbritannien og idag Maria Damkjær Post.doc. i Engelsk Litteratur Horace Engdahl i interview i Politiken Bøger, 7. december 2014: [Hos os i Norden]
Læs mereUdbud på engelsk i UCL. Skabelon til beskrivelse
Udbud på engelsk i UCL Skabelon til beskrivelse Indhold 1. Forord... 3 2. What to do... 3 3. Skabelon... 4 3.1 Course Overview... 4 3.2 Target Group... 4 3.3 Purpose of the module... 4 3.4 Content of the
Læs mereTrolling Master Bornholm 2014
Trolling Master Bornholm 2014 (English version further down) Ny præmie Trolling Master Bornholm fylder 10 år næste gang. Det betyder, at vi har fundet på en ny og ganske anderledes præmie. Den fisker,
Læs mereUNISONIC TECHNOLOGIES CO.,
UNISONIC TECHNOLOGIES CO., 3 TERMINAL 1A NEGATIVE VOLTAGE REGULATOR DESCRIPTION 1 TO-263 The UTC series of three-terminal negative regulators are available in TO-263 package and with several fixed output
Læs mereNovember hilsner fra NORDJYSKE Medier, Distributionen
Uret er stillet til vintertid, og det betyder, at der nu er mørkt både morgen og aften. Det er vigtigt, at du er synlig i trafikken i vintermørket, og derfor opfordrer vi dig til at bruge din refleksvest,
Læs merePædagogisk vejledning
Titel: Tema: Fag: Målgruppe: Diary of a Wimpy Kid Dog Days Growing up, Teenagers, Holidays, Relationships, USA Engelsk 6.-8.kl. Det er sommerferie og Greg Heffley aner problemer. Gregs far forbyder ham
Læs mereTrolling Master Bornholm 2016 Nyhedsbrev nr. 3
Trolling Master Bornholm 2016 Nyhedsbrev nr. 3 English version further down Den første dag i Bornholmerlaks konkurrencen Formanden for Bornholms Trollingklub, Anders Schou Jensen (og meddomer i TMB) fik
Læs mereTrolling Master Bornholm 2013
Trolling Master Bornholm 2013 (English version further down) Trolling Master Bornholm 2013 Husk at tjekke jeres reservationer! Vi ved, at der er nogen, som har lavet reservationer af overnatning, og at
Læs mereFORVANDLENDE FORBINDELSER: Et studie af migranters forbindelser til hjemstavnen og deres visioner for at deltage i lokal udvikling
FORVANDLENDE FORBINDELSER: Et studie af migranters forbindelser til hjemstavnen og deres visioner for at deltage i lokal udvikling Ditte Brøgger PhD studerende / Geografi Agergaard, J. and Brøgger, D.
Læs mereParticle-based T-Spline Level Set Evolution for 3D Object Reconstruction with Range and Volume Constraints
Particle-based T-Spline Level Set for 3D Object Reconstruction with Range and Volume Constraints Robert Feichtinger (joint work with Huaiping Yang, Bert Jüttler) Institute of Applied Geometry, JKU Linz
Læs mereName: Week of April 22 MathWorksheets.com
Cross off the number that does NOT belong. 45, 54, 63, 72, 81, 90, 93, 99, 108 Why does not belong in the pattern? Cross off the number that does NOT belong. 2, 2, 12, 9, 20, 22, 16, 32, 23, 42, 30, 52,
Læs mereUniversity of Copenhagen Faculty of Science Written Exam - 8. April 2008. Algebra 3
University of Copenhagen Faculty of Science Written Exam - 8. April 2008 Algebra 3 This exam contains 5 exercises which are to be solved in 3 hours. The exercises are posed in an English and in a Danish
Læs mereWIKI & Lady Avenue New B2B shop
WIKI & Lady Avenue New B2B shop Login Login: You need a personal username and password Du skal bruge et personligt username og password Only Recommended Retail Prices Viser kun vejl.priser! Bestilling
Læs mereVirkningsfulde bønner og påkaldelser
Virkningsfulde bønner og påkaldelser (4) Den Store Invokation (billeder + tekst) DEN STORE INVOKATION - THE GREAT INVOCATION Fra lysets kilde i Guds sind Lad lys strømme ind i menneskers tanker Lad lyset
Læs mereSkriftlig Eksamen Automatteori og Beregnelighed (DM17)
Skriftlig Eksamen Automatteori og Beregnelighed (DM17) Institut for Matematik & Datalogi Syddansk Universitet Odense Campus Lørdag, den 15. Januar 2005 Alle sædvanlige hjælpemidler (lærebøger, notater
Læs mereAngle Ini/al side Terminal side Vertex Standard posi/on Posi/ve angles Nega/ve angles. Quadrantal angle
Mrs. Valentine AFM Objective: I will be able to identify angle types, convert between degrees and radians for angle measures, identify coterminal angles, find the length of an intercepted arc, and find
Læs mereGusset Plate Connections in Tension
Gusset Plate Connections in Tension Jakob Schmidt Olsen BSc Thesis Department of Civil Engineering 2014 DTU Civil Engineering June 2014 i Preface This project is a BSc project credited 20 ECTS points written
Læs mereMeget formel, modtager har en meget speciel titel som skal bruges i stedet for deres navne
- Åbning Engelsk Dansk Dear Mr. President, Kære Hr. Direktør, Meget formel, modtager har en meget speciel titel som skal bruges i stedet for deres navne Dear Sir, Formel, mandelig modtager, navn ukendt
Læs mereListen Mr Oxford Don, Additional Work
57 (104) Listen Mr Oxford Don, Additional Work Listen Mr Oxford Don Crosswords Across 1 Attack someone physically or emotionally (7) 6 Someone who helps another person commit a crime (9) 7 Rob at gunpoint
Læs mereIPTV Box (MAG250/254) Bruger Manual
IPTV Box (MAG250/254) Bruger Manual Når din STB (Set top Box) starter op, bliver der vist en pop up boks på skærmen, hvor du kan åbne EPG ved at trykke på F2 (Nogle bokse kan fortælle at den har brug for
Læs merePopular Sorting Algorithms CHAPTER 7: SORTING & SEARCHING. Popular Sorting Algorithms. Selection Sort 4/23/2013
Popular Sorting Algorithms CHAPTER 7: SORTING & SEARCHING Introduction to Computer Science Using Ruby Computers spend a tremendous amount of time sorting The sorting problem: given a list of elements in
Læs mereMeget formel, modtager har en meget speciel titel som skal bruges i stedet for deres navne
- Åbning Dansk Engelsk Kære Hr. Direktør, Dear Mr. President, Meget formel, modtager har en meget speciel titel som skal bruges i stedet for deres navne Kære Hr., Formel, mandelig modtager, navn ukendt
Læs mereLæs venligst Beboer information om projekt vandskade - sikring i 2015/2016
Læs venligst Beboer information om projekt vandskade - sikring i 2015/2016 Vi er nødsaget til at få adgang til din lejlighed!! Hvis Kridahl (VVS firma) har bedt om adgang til din/jeres lejlighed og nøgler,
Læs mereLarge Scale Sequencing By Hybridization. Tel Aviv University
Large Scale Sequencing By Hybridization Ron Shamir Dekel Tsur Tel Aviv University Outline Background: SBH Shotgun SBH Analysis of the errorless case Analysis of error-prone Sequencing By Hybridization
Læs mereBrug sømbrættet til at lave sjove figurer. Lav fx: Få de andre til at gætte, hvad du har lavet. Use the nail board to make funny shapes.
Brug sømbrættet til at lave sjove figurer. Lav f: Et dannebrogsflag Et hus med tag, vinduer og dør En fugl En bil En blomst Få de andre til at gætte, hvad du har lavet. Use the nail board to make funn
Læs mereVEDLIGEHOLDELSE AF SENGE
DK VEDLIGEHOLDELSE AF SENGE VEDLIGEHOLDELSE AF SENGE Sengen er typisk det møbel i hjemmet som bruges i flest timer gennem døgnet. Det betyder at sengen udsættes for et stort slid, og det er derfor vigtigt
Læs mereECE 551: Digital System * Design & Synthesis Lecture Set 5
ECE 551: Digital System * Design & Synthesis Lecture Set 5 5.1: Verilog Behavioral Model for Finite State Machines (FSMs) 5.2: Verilog Simulation I/O and 2001 Standard (In Separate File) 3/4/2003 1 ECE
Læs mereChapter 6. Hydrogen Atom. 6.1 Schrödinger Equation. The Hamiltonian for a hydrogen atom is. Recall that. 1 r 2 sin 2 θ + 1. and.
Chapter 6 Hydrogen Atom 6. Schrödinger Equation The Hamiltonian for a hydrogen atom is Recall that Ĥ = h e m e 4πɛ o r = r ) + r r r r sin θ sin θ ) + θ θ r sin θ φ and [ ˆL = h sin θ ) + )] sin θ θ θ
Læs merePortal Registration. Check Junk Mail for activation . 1 Click the hyperlink to take you back to the portal to confirm your registration
Portal Registration Step 1 Provide the necessary information to create your user. Note: First Name, Last Name and Email have to match exactly to your profile in the Membership system. Step 2 Click on the
Læs mereWe hope you have enjoyed your holiday and that you are willing to help us improve our holiday support programme by completing this questionnaire.
Dear holiday-maker We hope you have enjoyed your holiday and that you are willing to help us improve our holiday support programme by completing this questionnaire. The information and answers you provide
Læs mereBilag. Resume. Side 1 af 12
Bilag Resume I denne opgave, lægges der fokus på unge og ensomhed gennem sociale medier. Vi har i denne opgave valgt at benytte Facebook som det sociale medie vi ligger fokus på, da det er det største
Læs mereNyhedsmail, december 2013 (scroll down for English version)
Nyhedsmail, december 2013 (scroll down for English version) Kære Omdeler Julen venter rundt om hjørnet. Og netop julen er årsagen til, at NORDJYSKE Distributions mange omdelere har ekstra travlt med at
Læs mereStatistik for MPH: 7
Statistik for MPH: 7 3. november 2011 www.biostat.ku.dk/~pka/mph11 Attributable risk, bestemmelse af stikprøvestørrelse (Silva: 333-365, 381-383) Per Kragh Andersen 1 Fra den 6. uges statistikundervisning:
Læs mereOn the Relations Between Fuzzy Topologies and α Cut Topologies
S Ü Fen Ed Fak Fen Derg Sayı 23 (2004) 21-27, KONYA On the Relations Between Fuzzy Topologies and α Cut Topologies Zekeriya GÜNEY 1 Abstract: In this study, some relations have been generated between fuzzy
Læs mereSunlite pakke 2004 Standard (EC) (SUN SL512EC)
Sunlite pakke 2004 Standard (EC) (SUN SL512EC) - Gruppering af chasere igen bag efter. På den måde kan laves cirkelbevægelser og det kan 2,787.00 DKK Side 1 Sunlite pakke 2006 Standard (EC) LAN (SUN SL512EC
Læs mereMatematik 2 AL. Opgave 2 (20p)
Opgver til besvrelse i 4 timer. Alle sædvnlige hjælpemidler må medbringes. Sættet består f 6 opgver. Opgve 1, 2, 3, 4, 5 og 6 er for de studerende, der hr læst efter nyt pensum. Opgve 1, 2, 3, 4, 5, og
Læs mereANNONCERING AF CYKELTAXAHOLDEPLADSER I RØD ZONE OG LANGELINIE
KØBENHAVNS KOMMUNE Teknik- og Miljøforvaltningen Byens Anvendelse CYKELTAXA I RØD ZONE 5. oktober 2018 ANNONCERING AF CYKELTAXAHOLDEPLADSER I RØD ZONE OG LANGELINIE English version Der er nu mulighed for
Læs mere