Project Report. Title: Finding and Implementing Auto Parallelization in RSA Encryption and Decryption Algorithm

Transcription

1 Project Report Title: Finding and Implementing Auto Parallelization in RSA Encryption and Decryption Algorithm Satyam Mishra Undergraduate Student, LNM Institute of Information Technology )

2 Abstract RSA encryption and decryption algorithm talks about secrecy of messages between two users. It involves a public key and a private key generation. The public key can be known to everyone and is used for encrypting messages. Messages encrypted with the public key can only be decrypted using the private key. The private key is not publicly known, it is only known to receiver so that she/he can decrypt the encrypted message. These keys (public and private keys) for the RSA algorithm are generated using some mathematical operations. I have designed a program code which implements RSA algorithm. The program code generates the encrypted message by taking motivation from a well known padding scheme Secure Hash Algorithm 1, or SHA-1. Generally, implementation of this algorithm involves huge calculations, thus if execution is performed sequentially, it may take large time and may test patience of the user for a large input. I have implemented some parallel operations in the program code thereby, reducing the total time. I have used POSIX threads threading library and tried to find best possible quick implementation of RSA encryption and decryption algorithm. Problem Definition What is RSA Algorithm and mathematics behind it? RSA Algorithm: Some idea of RSA algorithm has been given in abstract above. You may refer that again for more information. The encryption and decryption in the RSA algorithm as follows: First, I had to generate the key pair and then those keys were to be used for encryption and decryption. Key generation RSA involves a public key and a private key. The mathematics behind RSA algorithm is as follows: Mathematics behind RSA

3 Firstly two distinct prime numbers 'p' and 'q' need to be generated between the range 64 to 127. The reason for choosing prime numbers between this range is : (i) Both primes have to be of similar bit-length. (ii) Since these primes are converted into binary while doing encryption of the message, and then some general operations are performed on these binary numbers, selection of large range of primes was not done because their binary conversions may get out of range and thus may create problems in further calculations. 1. After this a number n is computed which is equal to pq. ( n = pq) o This n is used as the modulus for both the public and private keys. 2. Then I computed phi(n) = (p 1)(q 1). 3. I choose an integer public exponent e such that 1 < e < phi(n) and greatest common divisor of (e, phi(n)) = 1; i.e e and phi(n) are co-prime. We use here some functions such gcd ( int, int ) to return exponent e and gcd2( int, int ) to find out gcd of two numbers. (gcd() and gcd2() are the functions used in the original program code of the algorithm, code can be made available on request.) 4. Now, in parallel to above 4 steps, I simultaneously calculate plain- text, m, of the message to be encrypted i.e a integer value of encoded form of message. Program reads this message from a file. So here I apply the first instance of parallelism in the algorithm. 5. Next, I apply another instance of parallelism in algorithm in finding private-key and cipher text. 6. Private-key is calculated using function priv ( int, int ) in the program. While calculating cipher text c (say), it involves function c= m^e (mod n). (m raised to power e, then I take remainder after dividing by n(which is equal to pq, from step 2). In these threads, there are several steps such as, calculation of m raised to power e i.e. m^e., I have used here Modular exponentiation method and have applied Right to left binary method to calculate the value of m ( a two digit number in our program) raised to power e (single or two digit integer in our program). In this calculation of m^e, since it might be a large calculation, I have implemented threads inside this sub-thread too. (explained below) 7. After that we can simply determine d as: mod phi(n). (image source:wikipedia) i.e d is the multiplicative inverse of e 9. Next, receiver gets the cipher-text and uses the formula m=c^d(mod n) [ here c is ciphertext, d is private-key exponent ] to decrypt the encrypted message, and thus able to read the original message.

4 Main Implementation: First we need to understand how encryption is done: Message to be encrypted or plain text Encryption Algorithm Encrypted message or cipher text Private key known only to sender and receiver This diagram is a simple representation of how algorithm actually works. Mathematical part of algorithm has been explained above. With calculating the public key exponent, I have parallely applied SHA-1 to find out the encrypted message. Appendix (at last of this report) gives a brief information about SHA-1. Reader may refer that for more info about SHA-1. A Flowchart explanation of how the program works, how threads are generated and how parallelism is used is shown on next page:

5 Flowchart Explanation: Start Main Thread This thread finds out prime numbers, their product and public key exponent, e This thread calculates the message digest and plain-text Return to main ^ thread This thread calculates the private key exponent, d. This thread finds out cipher-text which will be send to receiver. This calculates one step of Modular Expo. Method (Right to left binary method) This calculates square of number (plaintext) Return to the thread called by main function Return to Main thread End

6 Outcomes of the Project A padded or encrypted message of a readable message is obtained thereby, securing the readable message between the communicating path of sender and receiver. Parallelism has been applied appropriately at some instances and program-code generates padded message successfully. The difficulty level of generation of cipher-text of the message has also been learnt by me as a implementer, thereby demonstrating the importance of the algorithm which is dominating the field of cryptography for last 35 years. Insights and Motivation Currently, algorithm generates 7 bits of primes (64 to 127). I had to confine myself to generate primes upto only 7 bits because of the integer range available in GCC library. And thus, I am motivated to extend the project to 256 or 512 bits of primes using GNU Multiple Precision Arithmetic Library and continue it. References 1. A study on parallel RSA factorization by Yi-Shiung Yeh, Ting-Yu Huang, Han-Yu Lin and Yu-Hao Chang- JOURNAL OF COMPUTERS, VOL. 4, NO. 2, FEBRUARY Analysis of the RSA algorithm by Betty Huang, April 8, A Parallel Implementaition of RSA by David Pearson, July 22, Paper by Enterprise Security Solutions- Cryptography Using Compaq MultiPrime Technology in a Parallel Processing Environment 5. RSA algorithm by DI management. 6. RSA(algorithm) on Wikipedia. 7. Marcello de Sales Blog: RSA algorithm: a step by step process.

7 Appendix Brief Information about Secure Hash Algorithm 1 Introduction: For a message of length < 2^64 bits, the SHA-1 produces a 160-bit condensed representation of the message called a message digest. The SHA-1 is designed to have the following properties: it is computationally NOT feasible to find a message which corresponds to a given message digest, or to find two different messages which produce the same message digest. The length of the message is the number of bits in the message (the empty message has length 0). If the number of bits in a message is a multiple of 8, for compactness we can represent the message in hex. The purpose of message padding is to make the total length of a padded message a multiple of 512. The SHA-1 sequentially processes blocks of 512 bits when computing the message digest. The following specifies how this padding shall be performed. As a summary, a "1" followed by m "0"s followed by a 64-bit integer are appended to the end of the message to produce a padded message of length 512 * n. The 64-bit integer is l, the length of the original message. The padded message is then processed by the SHA-1 as n 512-bit blocks. How the message digest is produced: The message digest is computed using the final padded message. The computation uses two buffers, each consisting of five 32-bit words, and a sequence of eighty 32-bit words. The words of the first 5-word buffer are labeled A,B,C,D,E. The words of the second 5-word buffer are labeled H0, H1, H2, H3, H4. The words of the 80-word sequence are labeled W0, W1,..., W79. A single word buffer TEMP is also employed in the program-code. To generate the message digest, the 16-word blocks M1, M2,..., Mn and are processed in order. The processing of each Mi involves 80 steps. Before processing any blocks, the {Hi} are initialized as follows: in hex, H0 = H1 = EFCDAB89 H2 = 98BADCFE H3 = H4 = C3D2E1F0. Now there are some steps in which SHA-1 actually works. The more info can be found at URL: Although I am not able to implement all steps but I am successful in implementing following steps in calculation of padded message: 1. Convert each character of message in ASCII and then convert in binary. 2. Take 4 bits of binary conversion from left and convert in into hexadecimal.

8 3. As I have to produce a padded message like H0 H1 H2 H3 H4, I divide the hexadecimal conversion in 5 parts and calculate for each part its H(i) by summing up bits of each part and adding them to defined H(i). After processing Mn, the message digest is the 160-bit string represented by the 5 words H0 H1 H2 H3 H4. Finally this algorithm turns the message into an integer (m) which is used for generating the cipher text (c) in encryption. This cipher text and the decryption key(d) are used to decrypt the original message at the receiver end. ********