Using quasigroups for secure encoding of file system

1. Using quasigroups for secure encoding 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 OstravaOstrava / Czech Republic Security and Protection of Information, Brno 9.-11.5.2001

2. Contents • Some necessary concepts • Constructing a stream cipher based upon quasigroups • Properties of the method • Installable File Systems • Conclusions Security and Protection of Information, Brno 9.-11.5.2001

3. Some necessary concepts • Let A={a1,a2 ,...,an}, n1 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. Security and Protection of Information, Brno 9.-11.5.2001

4. 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 Security and Protection of Information, Brno 9.-11.5.2001

5. We say that (Q, \) is inverse quasigroup to (Q, *). The quasigroup (Q, *, \) satisfies the following identities: x \ (x * y) = y, x * (x \ y) = y Security and Protection of Information, Brno 9.-11.5.2001

6. Constructing a stream cipher • Let a finite set A={a1,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. Security and Protection of Information, Brno 9.-11.5.2001

7. Definition: Let uiA, k1. 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. Security and Protection of Information, Brno 9.-11.5.2001

8. Properties of the method Security and Protection of Information, Brno 9.-11.5.2001

9. 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. Security and Protection of Information, Brno 9.-11.5.2001

10. 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 Numbers of reduced Latin rectangles Security and Protection of Information, Brno 9.-11.5.2001

11. 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. Security and Protection of Information, Brno 9.-11.5.2001

12. It is proved that each element occurs exactly q times among the products of two elements of Q, q2times among the products of three elements of Q and, generally qt-1 among the products of t elements of Q. Security and Protection of Information, Brno 9.-11.5.2001

13. Distribution of characters • In a common plaintext. • In a plaintext that contains only ‘a’, ‘b’ and “a new line”. Security and Protection of Information, Brno 9.-11.5.2001

14. A common text Security and Protection of Information, Brno 9.-11.5.2001

15. Just ‘a’ and ‘b’ and new line Security and Protection of Information, Brno 9.-11.5.2001

16. It produces a cipher text with the same length as the plaintext and encryption is of a stream nature. Security and Protection of Information, Brno 9.-11.5.2001

17. 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 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. Security and Protection of Information, Brno 9.-11.5.2001

18. It is also robust on errors. Security and Protection of Information, Brno 9.-11.5.2001

19. Proposed method, being very simple, offers very fast implementation of encrypting and decrypting procedures. Security and Protection of Information, Brno 9.-11.5.2001

20. 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 Security and Protection of Information, Brno 9.-11.5.2001

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22. 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. Security and Protection of Information, Brno 9.-11.5.2001

23. Conclusions • Quasigroups, in spite of their simplicity, have various applications. • Many other encrypting algorithms can be formed on the basis of quasigroups. Security and Protection of Information, Brno 9.-11.5.2001

24. 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. Security and Protection of Information, Brno 9.-11.5.2001