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Companion Matrix Encryption Using the Knapsack Problem Based on Elgamal

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Companion Matrix Encryption Using the Knapsack Problem Based on Elgamal Journal of Computer and Mathematical Sciences, Vol.6(8),416-422, August 2015 ISSN 0976-5727 (Print) (An International Research Journal), www.compmath-journal.org ISSN 2319-8133 (Online) Companion Matrix Encryption Using the Knapsack Problem Based on Elgamal Khushboo Thakur1, B. P. Tripathi2 and B. K. Sharma3 1,3School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, C.G., INDIA. email: [email protected] 2Department of Mathematics, Govt. N.P.G. College of Science, Raipur, Chhattisgarh, INDIA. (Received on: August 14, 2015) ABSTRACT The Merkle-Hellman invented in 1978 is based on the super increasing subset problem. Ralph Merkle and Martin Hellman used the subset problem to create a cryptosystem to encrypt data. In this paper we have verified a companion matrix of message which have been encrypted through the Merkle-Hellman encryption scheme and sent by use of Elgamal. So that only the proposed receiver is efficient to decode the message. Our result also illustrated with the help of a mathematical example. Keywords: Merkle-Hellman encryption scheme, elgamal, Ralph Merkle and Martin Hellman. 1. INTRODUCTION The knapsack problem is a NP complete problem in combinatorial optimization. The knapsack problem selects the most useful items from a number of items given that the knapsack has a certain capacity. Knapsack problems are widely used to design solutions to the industrial problems using Public-key cryptography. The 0-1 knapsack problem states that there is a knapsack with a given capacity and a certain number of items that need to be filled inside the knapsack. Each item has a value and a weight associated with it. The knapsack problem selects the items that can be put in the knapsack so that the value of all the items maximizes the weight and does not increase the total capacity of the knapsack. This can be denoted as Maximize n ∑bi xi i=0 August, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org 417 Khushboo Thakur, et al., J. Comp. & Math. Sci. Vol.6(8), 416-422 (2015) n ≤ W Subject to ∑ wi xi i=0 where ‘b’ is the value associated with each item i. ‘w’ is the weight associated with each item i. ‘W’ is the maximum capacity of the knapsack. ‘n’ is the number of items. The subset sum problem is a special case of the knapsack problem3 .This problem finds a group of integers from a list vector V, where V = ( , , , ) with the subset of v1 v2 v3 vn elements in the vector V gives a sum of S. It also determines if a vector X = ( , , , ) exists where element of {0,1} so that V ∗ X = S3. Then A super x1 x2 x3 xn xi increasing knapsack vector s is created and the super-increasing property is hidden by creating a second vector M by modular multiplication and permutation. The vector M is the public key of the cryptosystem and is used to decrypt the message5. The Merkle Hellman system is based on the subset sum problem1. The subset sum problem is a special case of the knapsack problem. The subset sum problem is hard, its decision problem was shown to be NP-complete by Karp6. T A super-increasing knapsack vector s is created and the super- increasing property is hidden by creating a second vector M by modular multiplication and permutation. The vector M is the public key of the cryptosystem and s is used to decrypt the message2. 2. ENCRYPTING MESSAGES This cryptosystem performs encryption in two steps. First the polynomials are converted to their binary equivalent. These polynomials are encrypted through the Merkle-Hellman encryption scheme whose main idea is to create a subset problem which can be solved easily and then to hide the super- increasing nature by modular multiplication and permutation. Secondly, these encrypted polynomial are further encrypted though the use of EL- GAMAL concepts. We choose a prime number p and a primitive root g mod p and also choose a random exponent a ∈{}0............p − 2 and calculate `A' such that a A = g mod p We choose another random integer b ∈{0............p − 2} using these details we find out the value of `B' such that b B = g mod p August, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org Khushboo Thakur, et al., J. Comp. & Math. Sci. Vol.6(8), 416-422 (2015) 418 Here (p,g, A) is a public key and private key is a. Thus the formula to encrypt the message using Elgamal cryptosystem is b c = A m mod p 3. MATHEMATICAL EXPLANATION Step 1: The first step is to choose key of length 7 bits. These are used to perform the first encryption process. Step 2: The second step is to convert the polynomial of the message into binary. The binary sequence is represented by the variable y. Step 3: Choose a super increasing sequence of number of positive integers. A super increasing sequence is one where every number is greater than the sum of all preceding numbers. S = ( , , , ) s1 s2 s3 .........sn Step 4: The forth step is to choose a random integer (z) such that n z f ∑ si i=0 and a random integer r, such that gcd (z, r) = 1 where r and z are co prime. The sequence s and the numbers z and r be the private key of the cryptosystem. All the elements ( , , ,......... ) of the sequence s are multiplied with the number r and the modulus s1 s2 s3 sn of the multiple is taken by dividing with the number z. Now calculate the sequence k = ( , , ,......... ) k1 k 2 k 3 k n Where = r ∗ mod z k i si The public key is k and private key is (r , z , s). Step 5: The message is encrypted by multiplying all the elements of sequence with the k i corresponding elements of sequence where is the i- th bit of the message and yi yi yi ∈{}0,1 . The numbers are then added to create the encrypted message forms the cipher M i text of the cryptosystem. Therefore, n = M i ∑ k i yi i=0 August, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org 419 Khushboo Thakur, et al., J. Comp. & Math. Sci. Vol.6(8), 416-422 (2015) 3.1. Example. Encrypting the message ⎡0 0 0 0 0 0 0 ⎤ ⎢ ⎥ ⎢1 0 0 0 0 0 0 ⎥ ⎢0 1 0 0 0 0 0 ⎥ ⎢ ⎥ A= ⎢0 0 1 0 0 0 0 ⎥ ⎢0 0 0 1 0 0 −1⎥ ⎢ ⎥ ⎢0 0 0 0 1 0 −1⎥ ⎢ ⎥ ⎣0 0 0 0 0 1 −1⎦ Step 1: The first step is to convert the Companion matrix in binary equivalent- A= 1 1 1 0 0 0 0 Step 2: The second step is to choose a super-increasing sequence s is created s = (1, 3, 11, 23, 72, 114, 258) This problem is easy because s is a super-increasing sequence. Step 3: The binary sequence is y = ( , , ,......... ) , choose a number z that is y1 y2 y3 yn greater than the sum of all super increasing sequence then z = 457 and choose a number r that is in the range [1; z) where r = 15. The private key consists of z, s and r. To calculate a public key, generate the sequence k by multiplying each element in s by r mod z k = (15, 45, 165, 345, 166, 339, 214) because k1 = 1 ∗ 15 mod 457 = 15 k2 = 3 ∗ 15 mod 457 = 45 k3 = 11 ∗ 15 mod 457 = 165 k4 = 23 ∗ 15 mod 457 =345 k5 = 72 ∗ 15 mod 457 = 166 k6 = 114 ∗ 15 mod 457 = 339 k7 = 258 ∗ 15 mod 457 = 214 where the sequence k is a public key. Step 4: The message is encrypted by multiplying all the elements of sequence k with the corresponding elements of sequence y and adding the resulting sum. Therefore, the encrypted message is n M= ∑k i yi i=0 August, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org Khushboo Thakur, et al., J. Comp. & Math. Sci. Vol.6(8), 416-422 (2015) 420 Encrypting the message ⎡0 0 0 0 0 0 0 ⎤ ⎢ ⎥ ⎢1 0 0 0 0 0 0 ⎥ ⎢0 1 0 0 0 0 0 ⎥ ⎢ ⎥ A == ⎢0 0 1 0 0 0 0 ⎥ ⎢0 0 0 1 0 0 −1⎥ ⎢ ⎥ ⎢0 0 0 0 1 0 −1⎥ ⎢ ⎥ ⎣0 0 0 0 0 1 −1⎦ Its binary equivalent is 1 1 1 0 0 0 0 Where k = (15, 45, 165, 345, 166, 339, 214) and y = (1 1 1 0 0 0 0) Then, M = 15 + 45 + 165 = 225 Step 5: This is encrypted using elgamal concepts. Here we choose a prime number p=283, g=179 and choose random number b=113. The value of `a' is choosen is 106. Therefore, A = 179 106 mod 283 A = 74 Thus the message is encrypted according to the formula C = A b M mod p Thus the encrypted code for the first character becomes C = 74 113 ∗ 225 mod 283 = 81 These codes are sent to the receiver. Thus the message to be transmitted to the receiver is 0081. 4. DECRYPTING MESSAGES During the decryption process, the blocks of encrypted code are separated. On these blocks decryption is performed using the ELGAMAL concepts. The formula used is p−1−a M = B C mod p The output of it is decrypted using Merkle-Hellman Knapsack cryptosystem decryption process. 4.1. Mathematical Explanation. To decrypt the message M, the recipient of the message would have to find the bitstream which satisfies the Equation1.
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