Using quasigroups for secure encoding of file system

Security and Protection of Information, Brno 9.-11.5.2001

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Using quasigroups for secure Using quasigroups for secure encoding of file systemencoding of file system

Eliška Ochodková, Václav Snášel

[email protected], [email protected]

Department of Computer Science

Faculty of Electrical Engineering and Computer ScienceVŠB Technical University of Ostrava

Ostrava / Czech Republic


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ContentsContents

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


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Some necessary conceptsSome necessary concepts

Let A={a1,a2 ,...,an}, n1 be an alphabet, a k x n Latin rectangle is a matrix with entries aij A, i=1,2,…k, j=1,2,…,n, such that each row and each column consists of different elements of A. If k=n 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 cipherConstructing a stream cipher

Let a finite set A={a1,a2 ,...,an}, n1 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 uiA, k1. Then

f*(u1u2...uk) = v1v2 ...vk

<=> v1= l * u1, vi+1= vi* ui+1, i=1,2,…,k-1,

f\(u1u2...uk) = v1v2 ...vk

<=> v1= l \ u1, vi+1= ui \ ui+1, i=1,2,…,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 methodProperties of the method


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It is resist to the brute force attack.It is resist to the brute force attack.

The Hall algorithm: there is at least n! (n – 1)!…2! Latin squares. Let A={0,…,255} (i.e. data are represented by 8 bits), there are at least 256! 255! …2!>1058000 quasigroups.

Suppose that intruder knows a cipher text v=v1v2…vk, 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 Numbers of reduced Latin rectanglesrectangles

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 It is resist to the statistical attackthe 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 charactersDistribution of characters

In a common plaintext. In a plaintext that contains only ‘a’, ‘b’ and

“a new line”.


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A common textA common text

0

20000

40000

60000

80000

100000

120000

140000

160000

180000

200000

0 50 100 150 200 250 300

cipher text

plaintext


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Just Just ‘a’ and ‘b’ and new line‘a’ and ‘b’ and new line

-10000

0

10000

20000

30000

40000

50000

60000

-50 0 50 100 150 200 250 300

cipher text

plaintext


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It produces a cipher text with It produces a cipher text with the same length as the the same length as the

plaintext and encryption is of a plaintext and encryption is of a stream naturestream nature..


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ExampleExample 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 A={a, b, c} and let the quasigroup (A,*), i.e. (A, \) be defined by Tab.1. Let l=a and u=bbcaacba. Then the cipher text of u is v=f*(u)=cbbcaaca. Applying of decoding function on v we get f\(v)=bbcaacba=u.


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It is also robust on errorsIt is also robust on errors. .


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Proposed method, being very Proposed method, being very simple, offers very fast simple, offers very fast

implementation of encrypting implementation of encrypting and decrypting proceduresand decrypting procedures..


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Installable file systemInstallable 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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ConclusionsConclusions

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 we’ll 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.

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Using quasigroups for secure encoding of file system

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