A. I. Lobanovs transformations

Cybernetics and Systems Analysis, Vol. 40, No. 5, 2004

SOFTWARE-HARDWARE SYSTEMS A. I. LOBANOV’S TRANSFORMATIONS A. P. Velikiia and A. F. Turbinb

UDC 512.8

The concept of a Lobanov transformation is introduced. Its distinctive features and application to information coding are studied. It is shown that Lobanov transformations belong to the class of geometric transformation. Keywords: cryptography, finite groups, barymetric, barycentric norm, baryelliptic norm, hypercomplex number, continuous signal. 1. INTRODUCTION Let X n = ( x1 , x 2 , K , x n ) be a collection (array) of integers interpreted as a digital message of length n. In cryptography, reversible (not necessarily unique) transformations G ( X n ) = Y n = ( y1 , K , y n ) are connected with application of the so-called unidirectional function (with a loophole) if definite conditions are fulfilled whose exact formulation is given, for example, in [1]. The simplest nontrivial transformation of the message X n consists of permutation of all or some components x i , i = 1, n , for example, Y n , j = Õ j ( X n ) = ( x j1 , x j2 , K , x jn ) ,

where

æ 1 è j1

Õ j ( × ) = çç

2 K j2 K

(1)

nö ÷ is some permutation from the permutation group S n . j n ÷ø

Such a transformation is called a permutation code [1]. The order of the group S n equals n !. For a large n (for example, for n ~1021 and larger, which is characteristic of bank or textual information in digital format), the number of alternative permutations becomes rather large (~ 10000 and larger). Such a transformation is potentially attractive from the viewpoint of information privacy protection against third-party threats. However, permutation algorithms completely ignore the numerical nature of the message X n and, moreover, after their practical realization in which all X j change places, they require overexpenditures of time and computational resources. In 1997, professor A. I. Lobanov (Amurskii State University and Taras Shevchenko University in Kiev) consulted a group of representatives of Amurbank and proposed a new class of transformations of digital information with the unique arithmetic subtraction operation. For a fixed i = 1, 2, K , n, we assume that Li ( X n ) = ( x1 - x i , x 2 - x i , K , x i -1 - x i ,- x i , x i + 1 - x i , K , x n - x i ) .

(2)

Then the double application of the L j -transformation decodes the message Li ( Li ( X n )) = L2i ( X n ) = X n , which calls into question cryptographic protective capabilities of such transformations of digital information. a

State Scientific-Research Institute of Informatization and Modeling of Economy, Kiev, Ukraine. bInstitute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 160-168, September-October 2004. Original article submitted August 8, 2002.

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1060-0396/04/4005-0764

©

2004 Springer Science+Business Media, Inc.

Our more careful and all-round analysis leads to opposite and quite optimistic conclusions. Note that the realization of any L j -transformation is very simple. The result obtained below (Theorem 1) claims that any of n ! transformations of the type (1) can be obtained as a definite composition (successive application) of pairwise different L j -transformations. It also turned out that L j -transformations and their compositions have a profound mathematical meaning, which gave ground to call them Lobanov transformations. To be more exact, we assume that L0 ( X n ) = X n is the identical transformation, L j1 , j2 ,K, jk ( X n ) = L j1 ( L j2 (K ( L jk ( X n ))K )) , k = 1, n, j i = 1, 2, K , n, and < j > = { j1 , K , j k } is a multiindex. We call a transformation L< j > ( × ) = L j1 ( L j2 (K ( L jk ( × ))K )) a Lobanov transformation and the multiindex < j > its key. An exact matrix representation for L< j > ( × ) in the group of reversible n ´ n matrices GLn ( F3 ) is constructed in Sec. 2, where F3 = {0, 1, - 1} is a Galois field. It is shown that the set of Lobanov transformations forms a group that is isomorphic not to the group S n but to its extension, i.e., to the group S n+ 1 of order ( n + 1) ! . Moreover, several new metrics are introduced in centroaffine spaces that are interesting in themselves and invariant with respect to L< j >-transformations. In Secs. 3 and 4, practically determined generalizations are considered. Some of the results considered below are formulated in [3–5]. 2. EXACT MATRIX REPRESENTATION IN GLn ( F3 ) A digital message X n = ( x1 , x 2 , K , x n ) of length n is conveniently and naturally represented geometrically as a vector x of an n-dimensional vector space V n with a fixed basis t1 , t 2 , K , t n , i.e., n

V n ® x = å xj t j. j =1

n

If P j ( × ) is some permutation and P j ( x ) = å x jk t k is the transformation (of the vector x) that is obtained by this k =1

permutation of components x i , then we can write P j ( x ) = P 0j x, where an n ´ n matrix P 0j is the representation of the permutation P j ( × ) in the group of permutable n ´ n matrices P 0 = ìí P 00 , P10 , K , P 0n ! -1 üý in each of which any row and î þ any column contain exactly one unity and n - 1 zeros, P 00 = I is a unit n ´ n matrix, and the ordering is arbitrary but fixed. We note that an L j -transformation in V n is of the form L j (x ) = - x j t j +

å

k¹ j

( xk - x j ) t k .

LEMMA 1. The transformation Li1 ( Li2 ( Li1 ( x ))) = Li1 ,i2 ,i1 ( x ) permutes x i1 and x i2 : Li1 , i2 , i1 ( x ) = x i1 t i2 + x i2 t i1 +

å

k ¹ i1 k ¹ i2

xk t k ,

i.e., it is a permutation. Without loss of generality, we can put i1 = 1 and i 2 = 2 and obtain L1,2 ,1 ( x ) = L1,2 ( - x1 t1 +

n

å

k=2

( x k - x1 ) t k ) = L1 ( - x 2 t1 +

= x 2 t1 + x1 t 2 +

n

å

k=3

n

å

k =1

( xk - x2 ) t k )

xk t k .

Since any permutation can be obtained after at the most n - 1 transpositions, we obtain the following theorem. 765

THEOREM 1. Any of n ! permutations can be obtained as a composition of definite L j -transformations. Let P 0i , i = 0, n ! - 1 , be a permutable n ´ n matrix in the group P 0 . We fix j = 1, n and replace all the elements of the jth column in P 0i by - 1. We denote the obtained matrix by P ij . Then we have P ij x = L j ( P i0 x ) and, by virtue of Theorem 1, the result of action of P ij on x can be obtained as a result of composition of some different L j -transformations, i.e., as some Lobanov L< j >-transformation. Thus, matrices P ij are representations of L< j >-transformations in the algebra of n ´ n matrices M n ( R ) . The number of different matrices P ij equals n ! ( n + 1) = ( n + 1) ! and coincides with the number of different L< j >-transformations. THEOREM 2. A set

{

}

Ln = L0 , L< j > , < j > = ( j1 , K , j k ), j i = 1, n, k = 1, n , where L0 is the identical transformation, forms a group isomorphic to the permutation group S n+ 1 of order ( n + 1) ! , and the set of matrices Pn = ìí P ij , i = 0, n ! - 1, j = 0, n üý is its exact matrix representation in the group of invertible î þ matrices GLn ( F3 ) . For the multiindex < j > = ( j1 , j 2 , K , j k ) that characterizes the order of successive application of L js -transformations, we put < j > * = ( j k , j k -1 , K , j1 ) . Then, by virtue of assumption (2), we have L< j > ( L< j >* ( x )) = L0 x = x and, hence, the L< j >* -transformation is inverse to the L< j >-transformation. A composition of any L< j >-transformations is also an L< j >-transformation. Hence, Ln is a group whose order is equal to ( n + 1) !. The isomorphism between Ln and Pn is directly established. To complete the proof of the theorem, we note that the module of the determinant of any matrix from Pn equals unity and Ln is isomorphic to S n+ 1 . The latter fact will be stated in the next section in geometric terms. 3. BARYMETRICS IN CENTROAFFINE SPACES AND LOBANOV TRANSFORMATION In an n-dimensional vector space V n whose basis t1 K t n consists of linearly independent vectors, a metric can be introduced by many ways that equips the space with the structure of a metric (normalized) space (for example, Euclidean, cubic, or octahedral metrics [6]). Let a mapping r ( × ) :V n ® R + = [ 0, ¥ ) be a semi-norm, i.e., we have (1) r ( x ) ³ 0 and r ( x ) = 0 Û x = 0, (2) "a ³ 0 r( a x ) ³ a r ( x ) , (3) r ( x + y ) £ r ( x ) + r ( y ) . In addition, we require that, for every L< j >-transformation, we have r ( L< j > ( x )) = r ( x )

(3)

(the validity check of the transformation). Any of well-known norms or semi-norms does not satisfy this requirement. Definition 1 (see [3, 4]). We call the number obtained by the formula n

g ( x ) = å x j - ( n + 1) min ( 0, x1 , K , x n ) j =1

the g-module or Galitsin module of a vector x (a digital message X n ).

766

(4)

Definition 2. We call the number obtained by the formula g d ( x ) = max {0, x1 , x 2 , K , x n } + max {0, - x1 , - x 2 , K , - x n } the symmetric g-module or barycentric norm of x. Definition 3. We call the number obtained by the formula || x||D =

n

å

x 2j -

j =1

2 n

å

I< j

(5)

xi y j

the baryelliptic norm of x. THEOREM 3. Numbers g ( × ) , g s ( × ) , and | | × | |D possess properties 1–3 from the definition of semi-norms and satisfy equality (3). This theorem places Lobanov L< j >-transformations into the class of geometric transformations [7]. In view of its importance, we will prove each introduced characteristic of vectors x (messages X n ). 1. The Galitsin Module g ( x). We assume that x * = min ( 0, x1 , x 2 , K , x n ) . It is obvious that x * £ 0, i.e., we have - x * ³ 0 and x i - x * ³ 0 , i = 1, n. From this, by virtue of (4), we have n

g ( x ) = - x* + å ( xi - x* ) ³ 0 .

(6)

i =1

It follows from the equality g ( x ) = 0 and inequality (6) that we have - x * = x1 - x * = K = x n - x * = 0 , i.e., we obtain x1 = x 2 = K = x n = 0 . The equality g ( a x ) = a g ( x ) is now obvious for a ³ 0 . The triangle inequality follows from the numerical inequality min ( 0, x1 + y1 , K , x n + y n ) ³ min ( 0, x1 , K , x n ) + min ( 0, y1 , K , y n ) . If an L< j >-transformation is a permutation, then condition (3) is obvious by virtue of formula (4). Therefore, without loss of generality, we can consider that x1 £ x 2 £ K £ x n and check whether condition (3) is true on L< j >-transformations. n

We assume that x1 ³ 0 . Then we have g ( x ) = å x j since x * = 0. In this case, we obtain j =1

g ( a j x ) = - nx j +

n

å

k¹ j

x k - ( n + 1) min ( 0, x1 - x j , x 2 - x j , K , x j -1 - x j1 - x j ) ,

x j + 1 - x j , K , x n - x j = - nx j +

n

å

k¹ j

n

x k - ( n + 1)( - x j ) = å x j = g ( x ) . j =1

Let x1 £ x 2 £ K £ x k £ 0 £ x k + 1 £ K £ x n . Then we have g ( x ) = g ( a j x ) = g - nx j +

n

å

m¹ j

x m - ( n + 1) ( x1 - x j ) =

n

å

m =1 n

å

m =1

x m - ( n + 1) x1 . If j = k, then we have

x m - ( n + 1) x1 = g ( x ) .

The case when j > k is similarly considered. 2. The Barycentric Norm g s ( x). Here, we can extend the results being proved. For the number g s ( x ), the equality g s ( ax ) = | a | g s ( x ) , a Î R, is fulfilled, i.e., g s ( x ) is a norm. If an L< j >-transformation is a permutation, then we have g s ( L< j > ( x ) = g s ( x )) and, without loss of generality, we can consider that x1 £ x 2 £ K £ x n . Let us consider the following three possible cases of the concrete definition of g s ( x ) : (a) if we have x n £ 0 , then we obtain g s ( x ) = - x1 and g s ( L j ( x )) = - x j + ( x j - x1 ) = - x1 ; (b) if we have x1 £ 0 and x n ³ 0 , then we obtain g s ( x ) = x n - x1 and g s ( L j ( x )) = ( x n - x j ) + ( x j - x1 ) = g s ( x ) ; (c) if we have x1 ³ 0 , then we obtain g s ( x ) = x n and g s ( L j ( x ) ) = x n - x j + x j = g s ( x ) .

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TABLE 1 j 0 1 2 3 4 5 6 7 8 9 10 11 12 ...

X3 (

6, 3, 5) ¯ (- 6 , - 3 , - 1) ¯ ( - 3, 3, 2) ¯ ( - 5 , 1, - 2 ) ¯ (- 6 , - 1, - 3 ) ¯ ( 6, 5, 3) ¯ ( 1, - 5 , - 2 ) ¯ ( 3, - 3, 2) ¯ (- 3 , - 6 , - 1) ¯ ( 3, 6, 5) ¯ ( - 2 , 1, - 5 ) ¯ ( 2, 3, - 3) ¯ ( 5, 6, 3) ...

3. The Baryelliptic Norm || x|| 2D . By virtue of definition (5), we have | | ax| |D = | a | | | x| |D .

(7)

The well-known proof of the fact that the right side of definition (5) satisfies all the properties that determine a norm is rather cumbersome and is performed by the following scheme. n 2 We first reduce the square-law form || x||2D = å x 2j - å x i x j to the canonical form with the help of the n j =1 i< j transformation proposed in [4]. From this we can conclude that the form is positively defined and, accordingly, we have || x||D ³ 0 and | | x| |D = 0 Þ x = 0 . To prove the triangle inequality, we can use the Cauchy–Buniakowski inequality. The equalities | | L j ( x )| |2D = || x||2D are obtained as a result of simple analytical calculations. We now can complete the proof of Theorem 2. We put t 0 = - t1 - t 2 - K - t n , where t j are basic vectors in V n . For x ÎVrn , we denote g ( x ) = g ( g > 0, x ¹ 0 ) and consider the minimal convex polyhedron with its nodes in gt 0 , gt1 , K , gt n . This polyhedron is an ( n + 1)-hedron (see [8, 9]) Pn+ 1 or a simplex in V n , and the origin of coordinates of the space is in the barycenter of its ( n + 1)-hedron. It is well known (see, for example, [8, 9]) that the group of congruence motions of an ( n + 1)-hedron is isomorphic to the permutation group S n+ 1 . The boundary g Pn+ 1 of this polyhedron is the set of all x ÎV n such that we have g ( x ) = g = const. It follows from Theorem 3 that L j -transformations transform the ( n + 1)-hedron Pn+ 1 into itself, which is proved by the isomorphism between L and S n+ 1 . A set of vectors x ÎV n for which we have g s ( x ) = g = const is the boundary of a convex polyhedron in V n with 2n + 1 - r nodes and n ( n + 1) ( n - 1)-dimensional faces, and the origin of coordinates coincides with the barycenter of this 768

n ( n + 1)-hedron. A remarkable property of this n ( n + 1)-hedron is that, with the help of parallel shifts, we can fill the entire space V n (without holes and intersections) in the same manner as we can fill the entire n-dimensional Euclidean space R n by n-dimensional cubes. The polyhedron mentioned is a Voronoi polyhedron [7] of a lattice in V n , and this lattice is an analog of a hexagonal lattice in R 2 . In [4], for a variety of geometric reasons, these polyhedrons are called rhombohedrons in V n . Let V n = R n ( n ³ 2) be an Euclidean space, and let t1 , K , t n be its orthonormal basis. Then {x Î R n : | | x| |2D = 1} is an ellipsoid in R n in which a convex polyhedron is inscribed with 2n + 1 - r nodes at the ends of the vectors with the following

coordinates: ( x1 , 0, K , 0) + permutations, ( x 2 , x 2 , 0, K , 0) + permutations, ( x n -1 , x n -1 , K , x n -1 , 0) + permutations, and ( x n , x n , K , x n ), where x1 = x n = ± 1 and xk = ±

n , k = 2, n - 1 . k ( n - k + 1)

Such polyhedrons can fill the entire R n . In the example considered below, the digital message X 3 = ( 6, 3, 5) is considered that consists of three figures, a part (half) of the action orbit {L< j > ( X 3 )} of the group L3 is presented, and also the values of the g-module, symmetric g-module, and squared baryelliptic distance are computed. Using different formulas, we obtain the following results for the initial X 3 : g ( X 3 ) = x1 + x 2 + x 3 - 4 min {0, x1 x 2 x 3 } = 6 + 3 + 5 - 4 × 0 = 14 ; g s ( X 3 ) = max {0, x1 x 2 x 3 } + max {0, - x1 - x 2 - x 3 } = 6 ; || X 3 ||2D = x12 + x 22 + x 32 = 36 + 9 + 25 -

2 ( x1 x 2 + x1 x 3 + x 2 x 3 ) 3

2 (18 + 30 + 15) = 28 . 3

Successively transforming L j ( × ), we obtain the results presented in Table 1 for the initial message X 3 = ( 6, 3, 5) (an arrow corresponds to a chosen arbitrary number j in L j ( × )-transformations). Here, we have g ( × ) = 14, g s ( × ) = 6, and || × ||2D = 28 . The fact that we have gs ( x ) = gs ( - x ) and | | x| |D = | |- x| |D allows us to introduce a 2-extension of the group L of ~ Lobanov transformations. We define a set of transformations L = h e L< j >, L< j > Î L h, where e Î { - 1, 1}. ~ THEOREM 4. The set of transformations L is a group isomorphic to S n+ 1 ´ C 2 , where C 2 is a cyclic group of the ~ ~ second order, the order of the group L equals 2 × ( n ! + 1), and, for any L< j > Î L, we have ~ ì g s ( L< j >x ) = g s ( x ),ü ï ï í ý. ï | | L~ ( x )| | = | | x| | ï D D þ < j> î In the above example, the orbit circumference of X 3 = ( 6, 3, 5) is doubled and becomes equal to 2 × 4 ! = 48 .

4. DIGITAL MESSAGES AND ALGEBRAS OF BARYCENTRIC NUMBERS The interpretation of a message as a vector x of an n-dimensional vector space V n allowed one to obtain a number of informal geometric results for Lobanov transformations. In this section, the vector space V n is equipped with a new internal operation (along with the composition of vectors), namely, the multiplication of arbitrary vectors from V n that are equipped with the structure of the algebra of hypercomplex numbers.

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During the proof of Theorem 3, a vector t 0 = - t1 - t 2 - K - t n is added to basic vectors t1 , t 2 , K , t n , which allows one to analytically define a barycentral ( n + 1)-hedron Pn+ 1 . We introduce the notation ì i + j, if i + j < k , (i + j )k = í î k - ( i + j ), if i + j ³ k. n

n

i =1

j =1

For any vectors x = å x i t i and y = å y j t j from V n , we put by definition n æ x×y:= å ç ç k =1 è

n ö xi y j - å xi yn - i ÷ t k ÷ j )= k i =1 ø

å

(i +

(8)

(in the first internal sum, the summation is taken over all the i and j for which we have i + j º k ( mod n )). The vector on the right side of (8) is correctly and uniquely determined by vectors x and y. We call it the product of the vectors x and y and operation (8) the operation of barycentric multiplication. It is easy to make sure that x × y = y × x (the multiplication is commutative), n

x × å ( - t k ) = ( S ( - t k )) × x = x, k =1

n

i.e., t 0 = - å t k possesses the properties of unity, ( x + y ) × z = x × z + y × z (the distributive law), and ( x × y ) × z = k =1

x ( y × z )(the associative law). The vectors {t 0 , t1 , K , t n } form a group with respect to the introduced multiplication operation, and this group is isomorphic to the cyclic group C n+ 1 of order n + 1. We call the constructed algebraic structure the algebra of barycentric (hypercomplex) numbers and denote it by {V n , + , × , C n + 1 } and call vectors x Î {V n , + , × , C n + 1 } barycentric numbers. (See the elements of analysis in algebras of hypercomplex numbers in [5, 10, 11].) In the algebra {V n , + , × , C n + 1 }, vectors x (and, hence, digital messages X 4 ) can be raised to any integer power k, i. e., we have xk = x × x × K x . 14243 k times

Combining algebraic properties of vectors (barycentric numbers) and Lobanov transformations, we obtain the following series of unidirectional transformations (see Sec. 1): y = L< j > x k + b , y = ( L< j > x ) k + b ,

(9)

y = x k + L< j > b , y = L< j > ( x k + b) , where k is a fixed number and b is a fixed vector (for example, a directly transformed vector x). Our reasoning can be continued in an obvious manner. If we understand transformations (9) as ciphering transformations of numerical messages, then these transformations are based on Lobanov transformations L< j >, powers of k, and affine shifts of b. The authors agree to make independent computational experiments (9) for small n and k and also for more sophisticated algebras of dimension n = 4 r ( n = 4, 16, 64 ) that are tensor powers of quaternion algebras. The results and conclusions (which are yet unpublished) completely coincide.

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5. MULTISECTIONING OF DIGITAL MESSAGES Let the length n of a digital message be a multiple of some number, n = ms, m, s Î N + . A message X n can be divided into m sections of length s as follows: X n = ( x1 , x 2 , K , x s ; x s+ 1 , x s+ 2 , K , x 2 s ; K ; x (V n -1)

s+ 1

, x (V n -1)

s+ 2

, K , x ms ) .

(10)

Now, the methods of transformation of digital information that are considered in Secs. 2 and 3 can be applied independently to each of m sections of partition (10). We consider that such a multisectioning and subsequent coding is promising in coding continuous signals. A signal S ( t ) , t Î[ 0, T ] , that is continuous on [ 0, T ] can be quantized at moments t i , i = 0, n - 1 , n = mr, that form a T T uniform partition of the time interval into n - 1 subintervals of length W = = = t i + 1 to obtain numerical values n mr S 1 , S 2 , K , S n , where S i = S ( t i -1 ) , i = 1, n. As a result, a numerical message S n = ( S 1 , K , S n ) is formed, which is naturally multisectionalized as follows: X n = ( S 1 , S 2 , K , S r ; S r + 1 , S r + 2 ; K ; S 2 r ; S ( m -1) r + 1 , S ( m -1) r + r , K , S mr ) .

(11)

The theorem of readings [7, Chapter 3], which is of great importance in communication theory, claims that if an insignificant restrictive constraint on the frequency structure of a signal S ( t ) is imposed, then the signal is completely determined by the collection of its values S i (see (11), i.e., by a discrete digital message X n . Applying the methods stated above (possibly, in a combination) and the theorem of readings, we arrive at transformed signals S$ ( t ) whose frequency spectrum can substantially differ from the frequency spectrum of the initial signal S ( t ) .

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