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

2. NATO ARW, Velingrad 21-25 October 2006 2 Outline Introduction Block ciphers Quasigroups Encryption/Decryption Algorithms Conclusion Future work

3. NATO ARW, Velingrad 21-25 October 2006 3 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.

4. NATO ARW, Velingrad 21-25 October 2006 4 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 E Key Ciphertext Plaintext D Key

5. NATO ARW, Velingrad 21-25 October 2006 5 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

6. NATO ARW, Velingrad 21-25 October 2006 6 ECB – Electronic Code Book M0M0... MnMn M1M1 C0C0 CnCn C1C1 E EE

7. NATO ARW, Velingrad 21-25 October 2006 7 CBC – Cipher Block Chaining M0M0... MnMn M1M1 C0C0 CnCn C1C1 E EE IV 

8. NATO ARW, Velingrad 21-25 October 2006 8 OFB – Output FeedBack M0M0... MnMn M1M1 C0C0 CnCn C1C1 E EE IV 

9. NATO ARW, Velingrad 21-25 October 2006 9 CFB – Cipher FeedBack M0M0... MnMn M1M1 C0C0 CnCn C1C1 E EE IV 

10. NATO ARW, Velingrad 21-25 October 2006 10 Quasigroup Quasigroup (Q,*) is a groupoid satisfying the law: (  u,v  Q)(  !x,y  Q) (x*u=v & u*y=v). *0123 02130 10312 21023 33201 Q is a finite set. * is quasigroup oparation.

11. NATO ARW, Velingrad 21-25 October 2006 11 Latin square Releated combinatorial structure is Latin square. Latin square is an n x n matrix with elements from Q such that each row and column is a permutation of Q. 2130 0312 1023 3201

12. NATO ARW, Velingrad 21-25 October 2006 12 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.

13. NATO ARW, Velingrad 21-25 October 2006 13 Quasigroup operations *0123 02103 13012 21230 30321 \0123 02103 11230 23012 30321 /0123 03102 12013 20231 31320

14. NATO ARW, Velingrad 21-25 October 2006 14 Quasigroup string transformations 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...a n, where a i  A. We define: – 4 functions: e l,*, d l,*, e’ l,*,d’ l,* :A +  A +.

15. NATO ARW, Velingrad 21-25 October 2006 15 Quasigroup string transformations e l,* (  )= b 1 b 2...b n  b 1 =l*a 1, b 2 =b 1 *a 2,... b n =b n-1 *a n a1a1 a2a2...a n-1 anan lb1b1 b2b2...b n-1 bnbn

16. NATO ARW, Velingrad 21-25 October 2006 16 Quasigroup string transformations d l,* (  )= c 1 c 2...c n  c 1 =l*a 1, c 2 =a 1 *a 2,... c n =a n-1 *a n la1a1 a2a2...a n-1 anan c1c1 c2c2...c n-1 cncn

17. NATO ARW, Velingrad 21-25 October 2006 17 Quasigroup string transformations e’ l,* (  )= b 1 b 2...b n  b 1 =a 1 *l, b 2 =a 2 *b 1,... b n =a n *b n-1 a1a1 a2a2...a n-1 anan lb1b1 b2b2...b n-1 bnbn

18. NATO ARW, Velingrad 21-25 October 2006 18 Quasigroup string transformations d’ l,* (  )= c 1 c 2...c n  c 1 =a 1 *l, c 2 =a 2 *a 1,... c n =a n *a n-1 la1a1 a2a2...a n-1 anan c1c1 c2c2...c n-1 cncn

19. NATO ARW, Velingrad 21-25 October 2006 19 Quasigroup string transformations Example: – A={0,1,2,3}, – l=0, – (A,*) and (A,\)  1021000000000112102201010300  ’= e 0,* (  ) 1322130213021011211133013130  ’’=d 0,\ (  ’) 1021000000000112102201010300 *0123 02103 13012 21230 30321 -  =1021000000000112102201010300 \0123 02103 11230 23012 30321

20. NATO ARW, Velingrad 21-25 October 2006 20 Quasigroup string transformations Proposition 1: 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,\ ). 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,/ ).

21. NATO ARW, Velingrad 21-25 October 2006 21 Encryption/Decryption functions of “SD-(n,k)” We use: – Blocks with length of n letters; – Key K=K 0 K 1...K n+4k-1, K i  A, where k is number of repeating of four different quasigroup string transformations in encryption/decryption functions; – Input: plaintext m 0 m 1...m n-1, m i  A – Output: ciphertext c 0 c 1...c n-1, c i  A We use: – Blocks with length of n letters; – Key K=K 0 K 1...K n+4k-1, K i  A, where k is number of repeating of four different quasigroup string transformations in encryption/decryption functions; – Input: plaintext m 0 m 1...m n-1, m i  A – Output: ciphertext c 0 c 1...c n-1, c i  A

22. NATO ARW, Velingrad 21-25 October 2006 22 Encryption algorithm EA1: For i=0 to n-1 do b i =K i *m i EA2: For j=0 to k-1 do b 0  K n+4j *b 0 For i=0 to n-1 do b i  b i-1 *b i b n-1  K n+4j+1 *b n-1 For i=n-1 down to 1 do b i-1  b i *b i-1 b 0  b 0 *K n+4j+2 For i=1 to n-1 do b i  b i *b i-1 b n-1  b n-1 * K n+4j+3 For i=n-1 down to 1 do b i-1  b i-1 *b i EA3: For i=0 to n-1 do c i =K i *b i EA1: For i=0 to n-1 do b i =K i *m i EA2: For j=0 to k-1 do b 0  K n+4j *b 0 For i=0 to n-1 do b i  b i-1 *b i b n-1  K n+4j+1 *b n-1 For i=n-1 down to 1 do b i-1  b i *b i-1 b 0  b 0 *K n+4j+2 For i=1 to n-1 do b i  b i *b i-1 b n-1  b n-1 * K n+4j+3 For i=n-1 down to 1 do b i-1  b i-1 *b i EA3: For i=0 to n-1 do c i =K i *b i

23. NATO ARW, Velingrad 21-25 October 2006 23 Decryption algorithm DA1: For i=0 to n-1 do b i =K i \c i DA2: For j=k-1 down to 0 do For i=1 to n-1 do b i-1  b i-1 /b i b n-1  b n-1 /K n+4j+3 For i=n-1 down to 1 do b i  b i /b i-1 b 0  b 0 /K n+4j+2 For i=1 to n-1 do b i-1  b i \b i-1 b n-1  K n+4j+1 \ b n-1 For i=n-1 down to 1 do b i  b i-1 \b i b 0  K n+4j \b 0 DA3: For i=0 to n-1 do m i =K i \b i DA1: For i=0 to n-1 do b i =K i \c i DA2: For j=k-1 down to 0 do For i=1 to n-1 do b i-1  b i-1 /b i b n-1  b n-1 /K n+4j+3 For i=n-1 down to 1 do b i  b i /b i-1 b 0  b 0 /K n+4j+2 For i=1 to n-1 do b i-1  b i \b i-1 b n-1  K n+4j+1 \ b n-1 For i=n-1 down to 1 do b i  b i-1 \b i b 0  K n+4j \b 0 DA3: For i=0 to n-1 do m i =K i \b i

24. NATO ARW, Velingrad 21-25 October 2006 24 Encryption/Decryption algorithms The algorithms EA K and DA K for fixed K can be considered as transformations of the set A n EA K (DA K (m0m1...m n-1 ))=m 0 m 1...m n-1 DA K (EA K (m 0 m 1...m n-1 ))=m 0 m 1...m n-1. Theorem: The transformations EA K and DA K are permutations of the set A n. The algorithms EA K and DA K for fixed K can be considered as transformations of the set A n EA K (DA K (m0m1...m n-1 ))=m 0 m 1...m n-1 DA K (EA K (m 0 m 1...m n-1 ))=m 0 m 1...m n-1. Theorem: The transformations EA K and DA K are permutations of the set A n.

25. NATO ARW, Velingrad 21-25 October 2006 25 Conclusion – This is a new family of block ciphers. – Very flexible design. – Easy implementation. – It has a large range of applications. – This is a new family of block ciphers. – Very flexible design. – Easy implementation. – It has a large range of applications.

26. NATO ARW, Velingrad 21-25 October 2006 26 Future Work – Cryptanalysis of “SD-(n,k)”. – Practical implementation. – Design improvement. – Cryptanalysis of “SD-(n,k)”. – Practical implementation. – Design improvement.

27. NATO ARW, Velingrad 21-25 October 2006 27 THANK YOU FOR YOUR ATTENTION