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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
eliska.ochodkova@vsb.cz, vaclav.snasel@vsb.cz
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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