1. Security and Protection of Information, Brno 9.-11.5.2001 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 Science VŠB Technical University of Ostrava Ostrava / Czech Republic

2. 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

3. Security and Protection of Information, Brno 9.-11.5.2001 3 Some necessary concepts Let A={a 1,a 2,...,a n }, n  1 be an alphabet, a k x n Latin rectangle is a matrix with entries a ij  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.

4. 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

5. 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

6. Security and Protection of Information, Brno 9.-11.5.2001 6 Constructing a stream cipher Let a finite set A={a 1,a 2,...,a n }, 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.

7. Security and Protection of Information, Brno 9.-11.5.2001 7 Definition: Let u i  A, k  1. Then f * (u 1 u 2...u k ) = v 1 v 2...v k v 1 = l * u 1, v i+1 = v i * u i+1, i=1,2,…,k-1, f \ (u 1 u 2...u k ) = v 1 v 2...v k v 1 = l \ u 1, v i+1 = u i \ u i+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.

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

9. 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!>10 58000 quasigroups. Suppose that intruder knows a cipher text v=v 1 v 2 …v k, he has to recover the quasigroup (A,*). But there is no algorithm of the exhaustive search of all quasigroups that can be generated.

10. Security and Protection of Information, Brno 9.-11.5.2001 10 Numbers of reduced Latin rectangles n L n 1 2 1 3 1 4 5 56 6 9,408 n L n 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

11. 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.

12. 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, q 2 times among the products of three elements of Q and, generally q t-1 among the products of t elements of Q.

13. 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”.

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

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

16. 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.

17. 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 aa c a b b c a bb b c ac 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.

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

19. 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.

20. 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

21. Security and Protection of Information, Brno 9.-11.5.2001 21

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

23. 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.

24. 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.