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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. Introduction. We present a new family of block ciphers “SD-(n,k)“.

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

NATO ARW,

Velingrad 21-25 October 2006

introduction
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 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 ciphers1
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
ECB – Electronic Code Book

M0

M1

...

Mn

E

E

...

E

C0

C1

...

Cn

NATO ARW,

Velingrad 21-25 October 2006

cbc cipher block chaining
CBC – Cipher Block Chaining

M0

M1

...

Mn

IV

E

E

...

E

C0

C1

...

Cn

NATO ARW,

Velingrad 21-25 October 2006

ofb output feedback
OFB – Output FeedBack

M0

M1

...

Mn

IV

E

E

...

E

C0

C1

...

Cn

NATO ARW,

Velingrad 21-25 October 2006

cfb cipher feedback
CFB – Cipher FeedBack

M0

M1

Mn

...

E

E

...

E

IV

C1

Cn

C0

...

NATO ARW,

Velingrad 21-25 October 2006

quasigroup
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
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
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 operations1
Quasigroup operations

NATO ARW,

Velingrad 21-25 October 2006

quasigroup string transformations
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 transformations1
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 transformations2
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 transformations3
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 transformations4
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 transformations5
Quasigroup string transformations
  • Example:
    • A={0,1,2,3},
    • l=0,
    • (A,*) and (A,\)

- =1021000000000112102201010300

NATO ARW,

Velingrad 21-25 October 2006

quasigroup string transformations6
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
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
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
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
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
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
Future Work
  • Cryptanalysis of “SD-(n,k)”.
  • Practical implementation.
  • Design improvement.

NATO ARW,

Velingrad 21-25 October 2006

slide27
THANK YOU

FOR

YOUR ATTENTION

NATO ARW,

Velingrad 21-25 October 2006