THE FAMILY OF BLOCK CIPHERS “ SD-(n,k)”

THE FAMILY OF BLOCK CIPHERS“SD-(n,k)” S. Markovski D. Gligoroski V. Dimitrova A. Mileva

Outline • Introduction • Block ciphers • Quasigroups • Encryption/Decryption Algorithms • Conclusion • Future work NATO ARW, Velingrad 21-25 October 2006

Introduction • We present a new family of block ciphers “SD-(n,k)“. • “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. NATO ARW, Velingrad 21-25 October 2006

Block ciphers • Block cipher is a symmetric key cipher which operates on fixed-length groups of bits, termed blocks, with an unvarying transformation. Plaintext Ciphertext Key E Key D Ciphertext Plaintext NATO ARW, Velingrad 21-25 October 2006

Block ciphers • 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 NATO ARW, Velingrad 21-25 October 2006

ECB – Electronic Code Book M0 M1 ... Mn E E ... E C0 C1 ... Cn NATO ARW, Velingrad 21-25 October 2006

CBC – Cipher Block Chaining M0 M1 ... Mn IV    E E ... E C0 C1 ... Cn NATO ARW, Velingrad 21-25 October 2006

OFB – Output FeedBack M0 M1 ... Mn IV E E ... E    C0 C1 ... Cn NATO ARW, Velingrad 21-25 October 2006

CFB – Cipher FeedBack M0 M1 Mn ...    E E ... E IV C1 Cn C0 ... NATO ARW, Velingrad 21-25 October 2006

Quasigroup • 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. NATO ARW, Velingrad 21-25 October 2006

Latin square • Releated combinatorial structure is Latin square. • Latin square is an nxn matrix with elements from Q such that each row and column is a permutation of Q. NATO ARW, Velingrad 21-25 October 2006

Quasigroup operations • Given a quasigroup (Q,*) two new operations, can be derived \ and / defined by: x*y=z  y=x\z  x=z/y. • The algebra (Q,*,\,/) satisfies the identities: x\(x*y)=y, x*(x\y)=y, (x*y)/y=x, (x/y)*y=x. • (Q,\), (Q,/) are qusigroups too. NATO ARW, Velingrad 21-25 October 2006

Quasigroup operations NATO ARW, Velingrad 21-25 October 2006

Quasigroup string transformations • We consider: • an alphabet A (finite set); • the set A+ of all nonempty finite words; • quasigroup operation *; • element lA(leader); • =a1a2...an, where aiA. • We define: • 4 functions: el,*, dl,*, e’l,*,d’l,*:A+ A+. NATO ARW, Velingrad 21-25 October 2006

Quasigroup string transformations • el,*()= b1b2...bn  b1=l*a1, b2=b1*a2, ... bn=bn-1*an NATO ARW, Velingrad 21-25 October 2006

Quasigroup string transformations • dl,*()= c1c2...cn  c1=l*a1, c2=a1*a2, ... cn=an-1*an NATO ARW, Velingrad 21-25 October 2006

Quasigroup string transformations • e’l,*()= b1b2...bn  b1=a1*l, b2=a2*b1, ... bn=an*bn-1 NATO ARW, Velingrad 21-25 October 2006

Quasigroup string transformations • d’l,*()= c1c2...cn  c1=a1*l, c2=a2*a1, ... cn=an*an-1 NATO ARW, Velingrad 21-25 October 2006

Quasigroup string transformations • Example: • A={0,1,2,3}, • l=0, • (A,*) and (A,\) - =1021000000000112102201010300 NATO ARW, Velingrad 21-25 October 2006

Quasigroup string transformations • 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,/). NATO ARW, Velingrad 21-25 October 2006

Encryption/Decryption functions of “SD-(n,k)” • We use: • Blocks with length of n letters; • Key K=K0K1...Kn+4k-1, KiA, where k is number of repeating of four different quasigroup string transformations in encryption/decryption functions; • Input: plaintext m0m1...mn-1, miA • Output: ciphertext c0c1...cn-1, ciA NATO ARW, Velingrad 21-25 October 2006

Encryption algorithm EA1: For i=0 to n-1 do bi=Ki*mi EA2: For j=0 to k-1 do b0Kn+4j*b0 For i=0 to n-1 do bibi-1*bi bn-1Kn+4j+1*bn-1 For i=n-1 down to 1 do bi-1bi*bi-1 b0b0 *Kn+4j+2 For i=1 to n-1 do bibi*bi-1 bn-1bn-1 *Kn+4j+3 For i=n-1 down to 1 do bi-1bi-1*bi EA3: For i=0 to n-1 do ci=Ki*bi NATO ARW, Velingrad 21-25 October 2006

Decryption algorithm DA1: For i=0 to n-1 do bi=Ki\ci DA2: For j=k-1 down to 0 do For i=1 to n-1 do bi-1bi-1/bi bn-1bn-1 /Kn+4j+3 For i=n-1 down to 1 do bibi/bi-1 b0b0 /Kn+4j+2 For i=1 to n-1 do bi-1bi\bi-1 bn-1Kn+4j+1 \ bn-1 For i=n-1 down to 1 do bibi-1\bi b0Kn+4j\b0 DA3: For i=0 to n-1 do mi=Ki\bi NATO ARW, Velingrad 21-25 October 2006

Encryption/Decryption algorithms • The algorithms EAKand DAKfor fixed Kcan be considered as transformations of the set An • EAK(DAK(m0m1...mn-1))=m0m1...mn-1 • DAK(EAK(m0m1...mn-1))=m0m1...mn-1. • Theorem: The transformations EAK and DAK are permutations of the set An. 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. NATO ARW, Velingrad 21-25 October 2006

Future Work • Cryptanalysis of “SD-(n,k)”. • Practical implementation. • Design improvement. NATO ARW, Velingrad 21-25 October 2006

THANK YOU FOR YOUR ATTENTION NATO ARW, Velingrad 21-25 October 2006

THE FAMILY OF BLOCK CIPHERS “ SD-(n,k)”