J. Comp. & Math. Sci. Vol. 1(2), 201-203 (2010).
A New Discrete Logarithm Problem in m-Dimensional Manifolds 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 In this paper, we present a new discrete logarithm problem in m-dimensional manifolds. This problem is very difficult to solve, therefore we motivated to give the new theory of cryptography, and obviously the security of this cryptography is based on the proposed problem. 2000 AMS Mathematics Subject Classification Number: 94A60. Key words: Discrete Logarithm Problem, m-Dimensional Manifolds, Cryptography.
1. INTRODUCTION The discrete logarithm based cryptography is very popular in the modern world of cryptography. The first theory of public key cryptography
was based on the discrete logarithm problem, given by Diffie, Hellman1 of his revolutionary paper entitled “New directions in cryptography” in the year 1976. After this invention, there are several public key cryptography are in
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
202 Sunil Kumar Kashyap et al., J.Comp.&Math.Sci. Vol.1(2), 201-203 (2010). the existence in the real world although the percentage of the successful and professionally useful has been only 1%, because the format of any public key cryptography essential the very depth knowledge of the subject in many areas of science, engineering and technology and some areas of arts also. Mathematic and computer science plays the lead role.
basis of P (M ). In particular, P (M ) is m-dimensional. An arbitrary vector,
We are giving again a new scheme of public key cryptography, whose security is based on our proposed new discrete logarithm problem in m-dimensional manifolds. Our claim and challenge in single line that is “This problem is tougher to solve in one direction but toughest to reverse direction”. Therefore our cryptography will be secure in future, because there are no algorithm exists in present for solve the proposed problem.
given by,
2. T he Proposed M athematical Problem.
t P (M ), can be written as the abovee discrete logarithm problem. Proof:
The coordinate frame i
m
p i 1
at P is
a set of operators i ( P ) : F ( P ) R
f x i
( i p ) f
P
d f ( 1 ( x1 ( P),..., x i 1 ( P ), u, x i 1 ( P ),..., du ),..., x m ( p)) u c .
Let, M V , a vector space. Choose a basis {ei } in V with its dual considered as coordinate functions. Then, at every v V , there is a natural isomorphism : V (V ) mapping a vector
u i ei V , onto, i i v v (V ). In this section, we give the proposed Discrete Logarithm Problem in m-dimensional manifolds. 2.1. Theorem:
t ii
p
Suppose we have two coordinate system at P, {x i } with tangents i
P
and
i { y j } with tangents j P : t i
, where, i t ( x i ).
j j P.
Finding the value of the index i is the discrete logarithm problem under the m-dimensional manifold M
and P M . The set i
We can use relation to obtain terms of
m
p i 1
P
forms a
j:
We have
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
i
in
Sunil Kumar Kashyap et al., J.Comp.&Math.Sci. Vol.1(2), 201-203 (2010). 203 3.2. Encryption.
i t( x i ) ( j j P ) ( x i ) ) j j y
i i j x ( x ) P y j
k The Cipher Text= C [C1 m( i
)i ,
, C 2 t ]. P
.
This is the required discrete logarithm problem. This completes the proof.
3. The Proposed Cryptography. The working process of the proposed cryptography is based on the below subsections; 3.1.
p
3.3. Decryption. The Plain Text =
Pm
C1 . (C 2 ) i
4. CONCLUSION The main objectives of this research paper are the followings; 4.1. To give the new hard discrete logarithm problem. 4.2. To give the new secure cryptography.
Key Generation.
Select, the following keys;
We strongly feel that this paper fulfills the above objectives.
The original message = m,
5. REFERRENCES
k , t , , i, , P, x, M , p ( M ) under the Theorem 2.1 and k is addidional private key parallel to i.
1 Diffie W., Hellman M.E., New directions in cryptography, Transactions on information theory, 22, 644-654 (1976).
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)