THE FAMILY OF BLOCK CIPHERS

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

• Introduction
• Block ciphers
• Quasigroups
• Encryption/Decryption Algorithms
• Conclusion
• Future work

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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.

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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
D
Key
Ciphertext
Plaintext
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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

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ECB Electronic Code Book
M0
...
Mn
M1
E
...
E
E
C0
...
Cn
C1
7
CBC Cipher Block Chaining
M0
...
Mn
M1
IV
?
?
?
E
...
E
E
C0
...
Cn
C1
8
OFB Output FeedBack
M0
...
Mn
M1
IV
E
...
E
E
?
?
?
C0
...
Cn
C1
9
CFB Cipher FeedBack
M0
Mn
M1
...
?
?
?
E
...
E
E
IV
C0
Cn
C1
...
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Quasigroup

• Quasigroup (Q,) is a groupoid satisfying the
  law
• (?u,v?Q)(?!x,y?Q)
• (xuv uyv).

0 1 2 3
0 2 1 3 0
1 0 3 1 2
2 1 0 2 3
3 3 2 0 1

• Q is a finite set.
• is quasigroup oparation.

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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.

2 1 3 0
0 3 1 2
1 0 2 3
3 2 0 1
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Quasigroup operations

• Given a quasigroup (Q,) two new operations, can
  be derived \ and / defined by
• xyz ? yx\z ? xz/y.
• The algebra (Q,,\,/) satisfies the identities
• x\(xy)y, x(x\y)y, (xy)/yx, (x/y)yx.
• (Q,\), (Q,/) are qusigroups too.

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Quasigroup operations
0 1 2 3
0 2 1 0 3
1 3 0 1 2
2 1 2 3 0
3 0 3 2 1
\ 0 1 2 3
0 2 1 0 3
1 1 2 3 0
2 3 0 1 2
3 0 3 2 1
/ 0 1 2 3
0 3 1 0 2
1 2 0 1 3
2 0 2 3 1
3 1 3 2 0
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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,, el,,dl,A? A.

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Quasigroup string transformations

• el,(?) b1b2...bn ? b1la1, b2b1a2, ...
  bnbn-1an

a1 a2 ... an-1 an
l b1 b2 ... bn-1 bn
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Quasigroup string transformations

• dl,(?) c1c2...cn ? c1la1, c2a1a2, ...
  cnan-1an

l a1 a2 ... an-1 an
c1 c2 ... cn-1 cn
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Quasigroup string transformations

• el,(?) b1b2...bn ? b1a1l, b2a2b1, ...
  bnanbn-1

a1 a2 ... an-1 an
l b1 b2 ... bn-1 bn
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Quasigroup string transformations

• dl,(?) c1c2...cn ? c1a1l, c2a2a1, ...
  cnanan-1

l a1 a2 ... an-1 an
c1 c2 ... cn-1 cn
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Quasigroup string transformations
0 1 2 3
0 2 1 0 3
1 3 0 1 2
2 1 2 3 0
3 0 3 2 1
\ 0 1 2 3
0 2 1 0 3
1 1 2 3 0
2 3 0 1 2
3 0 3 2 1

• Example
• A0,1,2,3,
• l0,
• (A,) and (A,\)

- ?1021000000000112102201010300
? 1021000000000112102201010300
? e0,(?) 1322130213021011211133013130
?d0,\(?) 1021000000000112102201010300
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Quasigroup string transformations

• Proposition 1 For each string M?A and each
  leader l?Q it holds that dl,\(el,(M))Mel,(dl,\
  (M)), i.e. el, and dl,\ are mutually inverse
  permutations of A ((el,)-1 dl,\).
• Proposition 2 For each string M?A and each
  leader l?Q it holds that dl,/(el,(M))Mel,(d
  l,/(M)), i.e. el, and dl,/ are mutually
  inverse permutations of A ((el,)-1 dl,/).

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Encryption/Decryption functions of SD-(n,k)

• We use
• Blocks with length of n letters
• Key KK0K1...Kn4k-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

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Encryption algorithm

• EA1 For i0 to n-1 do biKimi
• EA2 For j0 to k-1 do
• b0?Kn4jb0
• For i0 to n-1 do bi?bi-1bi
• bn-1?Kn4j1bn-1
• For in-1 down to 1 do bi-1?bibi-1
• b0?b0 Kn4j2
• For i1 to n-1 do bi?bibi-1
• bn-1?bn-1 Kn4j3
• For in-1 down to 1 do bi-1?bi-1bi
• EA3 For i0 to n-1 do ciKibi

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Decryption algorithm

• DA1 For i0 to n-1 do biKi\ci
• DA2 For jk-1 down to 0 do
• For i1 to n-1 do bi-1?bi-1/bi
• bn-1?bn-1 /Kn4j3
• For in-1 down to 1 do bi?bi/bi-1
• b0?b0 /Kn4j2
• For i1 to n-1 do bi-1?bi\bi-1
• bn-1?Kn4j1 \ bn-1
• For in-1 down to 1 do bi?bi-1\bi
• b0?Kn4j\b0
• DA3 For i0 to n-1 do miKi\bi

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Encryption/Decryption algorithms

• The algorithms EAK and DAK for fixed K can 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.

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Conclusion

• This is a new family of block ciphers.
• Very flexible design.
• Easy implementation.
• It has a large range of applications.

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Future Work

• Cryptanalysis of SD-(n,k).
• Practical implementation.
• Design improvement.

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• THANK YOU
• FOR
• YOUR ATTENTION