THE FAMILY OF BLOCK CIPHERS SDn k S
THE FAMILY OF BLOCK CIPHERS “SD-(n, k)” S. Markovski D. Gligoroski V. Dimitrova A. Mileva
Outline l l l Introduction Block ciphers Quasigroups Encryption/Decryption Algorithms Conclusion Future work 2 NATO ARW, Velingrad 21 -25 October 2006
Introduction l l l We present a new family of block ciphers “SD-(n, k)“ is based on the properties of quasigroup operations and quasigroup string transformations. This design allows choosing different level of security and different kind of performances. 3 NATO ARW, Velingrad 21 -25 October 2006
Block ciphers l Block cipher is a symmetric key cipher which operates on fixed-length groups of bits, termed blocks, with an unvarying transformation. Plaintext Key E Ciphertext Key Ciphertext D Plaintext 4 NATO ARW, Velingrad 21 -25 October 2006
Block ciphers l l To encrypt messages longer than block size a mode of operation is used Basic mode of operation: ECB, CBC, OFB, CFB Typical key size in bits are: 40, 56, 64, 80, 128, 192, 256, . . . From 2001 standard is AES witch use – – 128 bits for SECRET 192 bits, 256 bits for TOP SECRET 5 NATO ARW, Velingrad 21 -25 October 2006
ECB – Electronic Code Book M 0 M 1 . . . Mn E E . . . E C 0 C 1 . . . Cn 6 NATO ARW, Velingrad 21 -25 October 2006
CBC – Cipher Block Chaining IV M 0 M 1 . . . Mn E E . . . E C 0 C 1 . . . Cn 7 NATO ARW, Velingrad 21 -25 October 2006
OFB – Output Feed. Back M 0 M 1 . . . Mn IV E E . . . C 0 E C 1 . . . Cn 8 NATO ARW, Velingrad 21 -25 October 2006
CFB – Cipher Feed. Back M 0 M 1 E E Mn . . . E IV C 0 C 1 . . . Cn 9 NATO ARW, Velingrad 21 -25 October 2006
Quasigroup l l l Quasigroup (Q, *) is a groupoid satisfying the law: ( u, v Q)( !x, y Q) (x*u=v & u*y=v). Q is a finite set. * is quasigroup oparation. * 0 1 2 3 0 2 1 3 0 1 0 3 1 2 2 1 0 2 3 3 3 2 0 1 10 NATO ARW, Velingrad 21 -25 October 2006
Latin square l Releated combinatorial structure is Latin square. 2 1 3 0 0 3 1 2 l Latin square is an nxn matrix with elements from Q such that each row and column is a permutation of Q. 1 0 2 3 3 2 0 1 11 NATO ARW, Velingrad 21 -25 October 2006
Quasigroup operations l Given a quasigroup (Q, *) two new operations, can be derived and / defined by: x*y=z y=xz x=z/y. l The algebra (Q, *, , /) satisfies the identities: x(x*y)=y, x*(xy)=y, (x*y)/y=x, (x/y)*y=x. l (Q, ), (Q, /) are qusigroups too. 12 NATO ARW, Velingrad 21 -25 October 2006
Quasigroup operations * 0123 21230 0 1 2 30321 3 0 3 2 1 02103 13012 0 2 1 3 1 1 2 0 3 1 3 3 0 2 / 0 1 2 0 3 2 0 1 1 0 2 2 0 1 3 3 2 3 1 3 2 0 13 NATO ARW, Velingrad 21 -25 October 2006
Quasigroup string transformations l l We consider: – an alphabet A (finite set); – the set A+ of all nonempty finite words; – quasigroup operation *; – element l A (leader); – =a 1 a 2. . . an, where ai A. We define: – 4 functions: el, *, dl, *, e’l, *, d’l, *: A+ A+. NATO ARW, Velingrad 21 -25 October 2006 14
Quasigroup string transformations l el, *( )= b 1 b 2. . . bn b 1=l*a 1, b 2=b 1*a 2, . . . bn=bn-1*an l a 1 a 2 . . . an-1 an b 1 b 2 . . . bn-1 bn 15 NATO ARW, Velingrad 21 -25 October 2006
Quasigroup string transformations l dl, *( )= c 1 c 2. . . cn c 1=l*a 1, c 2=a 1*a 2, . . . cn=an-1*an l a 1 a 2 . . . an-1 an c 1 c 2 . . . cn-1 cn 16 NATO ARW, Velingrad 21 -25 October 2006
Quasigroup string transformations l e’l, *( )= b 1 b 2. . . bn b 1=a 1*l, b 2=a 2*b 1, . . . bn=an*bn-1 l a 1 a 2 . . . an-1 an b 1 b 2 . . . bn-1 bn 17 NATO ARW, Velingrad 21 -25 October 2006
Quasigroup string transformations l d’l, *( )= c 1 c 2. . . cn c 1=a 1*l, c 2=a 2*a 1, . . . cn=an*an-1 l a 1 a 2 . . . an-1 an c 1 c 2 . . . cn-1 cn 18 NATO ARW, Velingrad 21 -25 October 2006
Quasigroup string transformations l * Example: – – – A={0, 1, 2, 3}, l=0, (A, *) and (A, ) 0 1 2 3 0 1 2 3 0 2 1 0 3 1 3 0 1 2 1 1 2 3 0 1 2 3 0 3 2 1 - =1021000000000112102201010300 ’= e 0, *( ) 132213021011211133013130 ’’=d 0, ( ’) 102100000112102201010300 NATO ARW, Velingrad 21 -25 October 2006 19
Quasigroup string transformations l l Proposition 1: For each string M A+ and each leader l Q it holds that dl, (el, *(M))=M=el, *(dl, (M)), i. e. el, * and dl, are mutually inverse permutations of A+ ((el, *)-1= dl, ). Proposition 2: For each string M A+ and each leader l Q it holds that d’l, /(e’l, *(M))=M=e’l, *(d’l, /(M)), i. e. e’l, * and d’l, / are mutually inverse permutations of A+ ((e’l, *)-1= d’l, /). 20 NATO ARW, Velingrad 21 -25 October 2006
Encryption/Decryption functions of “SD-(n, k)” l We use: – – Blocks with length of n letters; Key K=K 0 K 1. . . Kn+4 k-1, Ki A , where k is number of repeating of four different quasigroup string transformations in encryption/decryption functions; Input: plaintext m 0 m 1. . . mn-1, mi A Output: ciphertext c 0 c 1. . . cn-1, ci A 21 NATO ARW, Velingrad 21 -25 October 2006
Encryption algorithm EA 1: For i=0 to n-1 do bi=Ki*mi EA 2: For j=0 to k-1 do b 0 Kn+4 j*b 0 For i=0 to n-1 do bi bi-1*bi bn-1 Kn+4 j+1*bn-1 For i=n-1 down to 1 do bi-1 bi*bi-1 b 0 *Kn+4 j+2 For i=1 to n-1 do bi bi*bi-1 bn-1 * Kn+4 j+3 For i=n-1 down to 1 do bi-1*bi EA 3: For i=0 to n-1 do ci=Ki*bi NATO ARW, Velingrad 21 -25 October 2006 22
Decryption algorithm DA 1: For i=0 to n-1 do bi=Kici DA 2: For j=k-1 down to 0 do For i=1 to n-1 do bi-1/bi bn-1 /Kn+4 j+3 For i=n-1 down to 1 do bi bi/bi-1 b 0 /Kn+4 j+2 For i=1 to n-1 do bi-1 bibi-1 bn-1 Kn+4 j+1 bn-1 For i=n-1 down to 1 do bi bi-1bi b 0 Kn+4 jb 0 DA 3: For i=0 to n-1 do mi=Kibi NATO ARW, Velingrad 21 -25 October 2006 23
Encryption/Decryption algorithms l The algorithms EAK and DAK for fixed K can be considered as transformations of the set An l l l EAK(DAK(m 0 m 1. . . mn-1))=m 0 m 1. . . mn-1 DAK(EAK(m 0 m 1. . . mn-1))=m 0 m 1. . . mn-1. Theorem: The transformations EAK and DAK are permutations of the set An. 24 NATO ARW, Velingrad 21 -25 October 2006
Conclusion – – This is a new family of block ciphers. Very flexible design. Easy implementation. It has a large range of applications. 25 NATO ARW, Velingrad 21 -25 October 2006
Future Work – – – Cryptanalysis of “SD-(n, k)”. Practical implementation. Design improvement. 26 NATO ARW, Velingrad 21 -25 October 2006
THANK YOU FOR YOUR ATTENTION 27 NATO ARW, Velingrad 21 -25 October 2006
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