J. Comp. & Math. Sci. Vol. 1(2), 189-192 (2010).
On Einstein’s Summation Convention based Discrete Logarithm Problem under Exterior Algebra and Cryptography SUNIL KUMAR KASHYAP, BIRENDRA KUMAR SHARMA 1 and AMITABH BANERJEE2 Department of Mathematics Rungta College of Engineering and Technology, Chhattisgarh Swami Vivekanand Technical University, Bhilai, Chhattisgarh, 491024 (INDIA) 7sunilkumarkashyap@gmail.com 1 School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, Chhattisgarh, 492010 (INDIA) 2 Department of Mathematics, Govt. DBGA Post Graduate College, Pt. Ravishankar Shukla University, Raipur Raipur, Chhattisgarh 492001, (INDIA) ABSTRACT Albert Einstein’s Theory on “The Summation Convention” is very popular in the Research World of Physics. But the motivation of Einstein was “The Fourth Dimension” (Based on the Non-Euclidian Geometry) and he saw the dream that, “All the Existing Sciences can be defined as the Unified Science. We study the whole literature and motivated by his motivation and try to give an idea for the information security under the Euclidian Phenomenon of Computer Sciences. This paper is centrally defined the Einstein’s Summation Convention based Discrete Logarithm Problem under Exterior Algebra. If we analyze this theory as a computer scientist, then we find that “The Unified Theory” of Einstein is not any other object as you think but in front of you; that is your computer. Although you don’t believe this hypothesis but after study of this paper and application of this proposed theory in the properly in the computer then we understand the real and practical idea of this thinking. But I try to give the theory of cryptography which is based on the proposed theory. AMS 2000 Mathematics Subject Classification Number: 94A60. Key Words: Einstein’s Summation Convention, Discrete Logarithm Problem, Exterior Algebra, Cryptography. Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
190 Sunil Kumar Kashyap et al., J.Comp.&Math.Sci. Vol.1(2), 189-192 (2010). 1. INTRODUCTION
: T0p (V ) p (V ), given by,,
A symmetric linear form g on
V is a symmetric tensor of type (0, 2). If {e j } Nj1 is a basis of V and { i }iN1 is i
j
()( 1 , 2 ,... p ) 1 A( (1) , p!
,
( 2)
,...
( p)
.
its dual basis, then g g ij (Einstein’s Summation Convention), because i j i j j i from a basis of 2 (V ). For any vector v V , we can write,
The sum is over all permutations of (12…p), and (1)... ( p ) is +1 if the permutation is even and -1 if the permutation is odd. () is sometimes denoted by a .
g (v ) g ij i j (v) g ij i j (v k ek )
2.1.2. Definition.
k
i
j
j
i
v g ij (ek ) g ij v . kj
We analyzed the above Einstein’s Summation Convention and found the new results. First, we found the new Discrete Logarithm Problem under the Exterior Algebra. Second, we motivated to design a new Public Key Cryptography based on our new Discrete Logarithm Problem. 2. Preliminaries. 2.1. Exterior Algebra. The following discussion will focus on tensors of type (r, 0). The set of all skew-symmetric tensors of type (p,0) forms a subspaces of p (V ). This subspace is denoted by p (V ). It is not, however an algebra unless we define a skew-symmetric product analogous to that for the symmetric case. The following definitions will be specifying more about this: 2.1.1. Definition. An anti-symmetrizer is a linear operator
The exterior product (also called the wedge, Grassmann, alternating, or veck product) of two skew-symmetric tensors A p (V ) and B p (V ) is a skew-symmetric tensor belonging to p q (V ) and given by the following: A B
( r s )! ( r s )! ( A B ) ( A B ) a . r! s! r! s!
More explicitly,
AB( 1 ,..., r s )
1 A( (1) ,..., r! s!
( r ) ) B( ( r 1) ,..., ( r s ) ). 2.1.3. Definition. An oriented basis of an N-dimensional vector space is an ordered collection of N linearly independent vecotors. If {v i } iN1 is one oriented basis and
{u i }iN1 is a second one, then u1 u 2 ... u N (det R ) v1 v 2 ... v N , Where R is the transformation matrix and det R is a nonzero number (R is
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
Sunil Kumar Kashyap et al., J.Comp.&Math.Sci. Vol.1(2), 189-192 (2010). 191 invertible), which can be positive or negative. 2.1.4. Definition. Let V be a vector space. Let V * have ve i N the oriented basis { }i 1 . The oriented volume N (V * ) of V is defined as follows;
1 2 ... N . Note that if {ei } is ordered as { j },
i
j
its dual basis, then g g ij (Einstein’s Summation Convention), because i j i j j i from a basis of 2 (V ). For any vector v V , we can write,,
g (v) g ij i j (v ) g ij i j (v k e k ) v k g ij i j (ek ) g ij v j i . kj
Thus, g ( v) V * . We can observe this equation under the convention form of g, then summation of this convention can be defined under the following proposition as;
t h e n (e1 , e2 ,..., e N ) 1 / N !, a n d w e say that {ei } is positively oriented with respect to . In general, {vi } is positively This shows that g can be thought of oriented with respect to µ if (v1 , v 2 ,..., as a mapping, g : V V * ,
,..., v N ) 0.
The volume element of V is defined in terms of a basis for V*. The reason for this will become apparent later, when we shall see that dx, dy, and, dz form a basis for (R 3) * and dxdydz dxdydz. 3. The Proposed Mathematics. 3.1. Theorem. Going from a vector in V to its unique image in V* is the Einstein’s Summation Convention based Discrete Logarithm Problem under Exterior Algebra. Proof: A symmetric linear form g on V is a symmetric tensor of type (0, 2). N
If {e j } j 1 is a basis of V and { i }iN1 is
For this equation to make sense, it should not matter which factor in the symmetric product v contracts with. But this is a trivial consequence i j of the symmetries g ij g ji , and ,
j i . The components g ij v j of g (v ) in the basis { i } Ni1 of V* are denoted by vi , so g ( v ) vi j . We have thus lowered the index of v j by the use of the symmetric bilinear form g. In applications vi is uniquely defined; furthermore, there is a one-toone correspondence between vi , and , v i . This can happen if and only if the mapping g : V V * is invertible, in which case there must exist a unique g 1 : V * V , or, g 1 2 (V * ) 2 (V ), such that,
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
192 Sunil Kumar Kashyap et al., J.Comp.&Math.Sci. Vol.1(2), 189-192 (2010).
v j e j v g 1 g (v) g 1 (v i i ) v i g 1 ( i ) vi [( g 1 ) ij e j e k ]( i ) 1
jk
i
1
OR, ji
vi ( g ) e j ek ( ) v i ( g ) e j . Comparison of the L.H.S. and the R.H.S. yields v j v i ( g 1 ) ji . It is compulsory to omit the -1 and simply write, v j g ji vi , The above equation represents the convention form of v and g in the term of ji under the Exterior Algebra. The discrete logarithm problem is conversely find in this equation with the different variate of v and g. This completes the proof. 3. The Proposed Cryptography. Now, we can implement the proposed theory in the structure of cryptology as the following cryptosystem, which is really an idea of some unified frame of Einstein’s Dream. 4.1. Key Generation. 4.1.1. Select Theorem 3.1. 4.1.2. Select, V, V*and the unique rule of unique image for V V * under 4.1.1. 4.1.3. Select the public key (under, V V * , OR, V * V ). 4.1.4. Select the private key (under,
V V * , OR, V * V ). 4.1.5. Select the message (M). 4.2. Encryption.
C ( M )(V V * ) C ( M )(V * V ) 4.3. Decryption
M (C )(V * V ) OR , M (C )(V V * ) 5. CONCLUSION The main objectives of this new research works are followings; 1. To redefine Einstein’s Summation Convention. 2. To redefine the Discrete Logarithm Problem on (1). 3. To redefine The Exterior Algebra for (1) and (2). 4. To redefine the mathematical relations among (1), (2) and (3). 5. To redesign cryptography under (1), (2), (3) and (4). 6. REFERENCES 1. Abraham, R., Marsden, J., Ratiu, T., Manifolds, Tensor Analysis and Applications, 2nd Edition, SpringerVerlag, Page No. 287-292 (1988). 2. Fulton, W., harris, J., Representaion Theory, Springer-Verlag, Page No. 84-94 (1991). 3. Halmos, P., Finite Dimensional Vector Spaces, 2nd Edition, Van Nostrand, Page No. 201-212 (1958). 4. Zeidler, E., Applied Functional Analysis, Springer-Verlag, Page No. 176-185 (1995).
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)