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Using quasigroups for secure encoding of file system

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Eliška Ochodková, Václav Snášel

[email protected], [email protected]

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

- 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

- 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.

Security and Protection of Information, Brno 9.-11.5.2001

- 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

- 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

- 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.

Security and Protection of Information, Brno 9.-11.5.2001

- 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.

Security and Protection of Information, Brno 9.-11.5.2001

Security and Protection of Information, Brno 9.-11.5.2001

- 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

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

Security and Protection of Information, Brno 9.-11.5.2001

- 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

- 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

- 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

Security and Protection of Information, Brno 9.-11.5.2001

Security and Protection of Information, Brno 9.-11.5.2001

Security and Protection of Information, Brno 9.-11.5.2001

- Table 1. The quasigroup (A, *, \)
* a b c\ a b c

a b c aa c a b

b c a bb b c a

c a b cc 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

Security and Protection of Information, Brno 9.-11.5.2001

Security and Protection of Information, Brno 9.-11.5.2001

- 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

Security and Protection of Information, Brno 9.-11.5.2001

- 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

- 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

- 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