Using quasigroups for secure encoding of file system

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Using quasigroups for secure encoding of file
system

• Elika Ochodková, Václav Snáel
• eliska.ochodkova_at_vsb.cz, vaclav.snasel_at_vsb.cz
• Department of Computer Science
• Faculty of Electrical Engineering and Computer
  ScienceVB Technical University of
  OstravaOstrava / Czech Republic

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Contents

• Some necessary concepts
• Constructing a stream cipher based upon
  quasigroups
• Properties of the method
• Installable File Systems
• Conclusions

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Some necessary concepts

• Let Aa1,a2 ,...,an, n?1 be an alphabet, a k x
  n Latin rectangle is a matrix with entries aij ?
  A, i1,2,k, j1,2,,n, such that each row and
  each column consists of different elements of A.
  If kn we say a Latin square instead of a Latin
  rectangle.

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• A grupoid (Q, ) is said to be a quasigroup
  satisfying the law
• (? u, v ? Q) (?? x, y ? Q) (u x v ? y u
  v)
• We can associate to the operation a new
  operation \ on Q, called right inverse of ,
  by
• x y z ? x \ z y

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• We say that (Q, \) is inverse quasigroup to (Q,
  ). The quasigroup (Q, , \) satisfies the
  following identities
• x \ (x y) y, x (x \ y) y

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Constructing a stream cipher

• Let a finite set Aa1,a2 ,...,an, n?1 be an
  alphabet and let (A, , \) be the quasigroup. Let
  A is the set of all nonempty words formed by
  elements of A. The elements of A will be denoted
  by elements of A.

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• Definition Let ui?A, k?1. Then
• f(u1u2...uk) v1v2 ...vk
• ltgt v1 l u1, vi1 vi ui1, i1,2,,k-1,
• f\(u1u2...uk) v1v2 ...vk
• ltgt v1 l \ u1, vi1 ui \ ui1, i1,2,,k-1.
• We say that the sextuple (A,,\,l, f , f\) is a
  quasigroup cipher over the alphabet A. A fixed
  element l is called leader.

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Properties of the method
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It is resist to the brute force attack.

• The Hall algorithm there is at least n! (n
  1)!2! Latin squares. Let A0,,255 (i.e. data
  are represented by 8 bits), there are at least
  256! 255! 2!gt1058000 quasigroups.
• Suppose that intruder knows a cipher text
  vv1v2vk, he has to recover the quasigroup
  (A,). But there is no algorithm of the
  exhaustive search of all quasigroups that can be
  generated.

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Numbers of reduced Latin rectangles

• n Ln
• 1 1
• 2 1
• 3 1
• 4 4
• 5 56
• 6 9,408

• n Ln
• 7 16,942,080
• 8 535,281,401,856
• 9 377,597,570,964,258,816
• 10 7,580,721,483,160,132,811,489,280

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It is resist to the statistical attack.

• Let (Q, ) be a quasigroup of q elements. Among
  the set of all possible cipher of certain length,
  all possible element of Q occurs with equal
  probability, i.e., each element of quasigroup Q
  should occur as often as any other in each
  position.

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• It is proved that each element occurs exactly q
  times among the products of two elements of Q,
  q2 times among the products of three elements of
  Q and, generally qt-1 among the products of t
  elements of Q.

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Distribution of characters

• In a common plaintext.
• In a plaintext that contains only a, b and
  a new line.

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A common text
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Just a and b and new line
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It produces a cipher text with the same length as
the plaintext and encryption is of a stream
nature.
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Example

• Table 1. The quasigroup (A, , \)
• a b c \ a b c
• a b c a a c a b
• b c a b b b c a
• c a b c c a b c
• Example 1. Let Aa, b, c and let the quasigroup
  (A,), i.e. (A, \) be defined by Tab.1. Let la
  and ubbcaacba. Then the cipher text of u is
  vf(u)cbbcaaca. Applying of decoding function
  on v we get f\(v)bbcaacbau.

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It is also robust on errors.
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Proposed method, being very simple, offers very
fast implementation of encrypting and decrypting
procedures.
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Installable file system

• Example Windows 9x and Windows NT directly
  support a variety of file systems, such as hard
  disks, CD-ROMs, floppy disks and network
  redirectors, and in addition permit third parties
  to create their own so-called installable file
  systems - - file system that can be installed in
  place of the usual file allocation table file
  system.
• Figure Windows98 file system architecture

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• Installable File System allows complete
  protection of data, thus it seems to be very
  useful complete presented method as a new feature
  of it. It appears to be especially convenient for
  laptops.

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Conclusions

• Quasigroups, in spite of their simplicity, have
  various applications.
• Many other encrypting algorithms can be formed on
  the basis of quasigroups.

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• In future works well continue with applications
  of non-associative algebraic systems in
  cryptography.
• Such algebraic systems exist for higher orders,
  they offer simple construction and implementation
  and very fast procedures of encrypting and
  decrypting, too.