ANALYSIS OF RSA ALGORITHM USING GPU · PDF fileRSA is an algorithm for public-key cryptography [1] and is considered as one of the great advances in the field of public key cryptography.

International Journal of Network Security & Its Applications (IJNSA), Vol.6, No.4, July 2014

DOI : 10.5121/ijnsa.2014.6402 13

ANALYSIS OF RSA ALGORITHM USING GPU

PROGRAMMING

Sonam Mahajan1 and Maninder Singh

2

1Department of Computer Science Engineering, Thapar University, Patiala, India

2 Department of Computer Science Engineering, Thapar University, Patiala, India

ABSTRACT Modern-day computer security relies heavily on cryptography as a means to protect the data that we have

become increasingly reliant on. The main research in computer security domain is how to enhance the

speed of RSA algorithm. The computing capability of Graphic Processing Unit as a co-processor of the

CPU can leverage massive-parallelism. This paper presents a novel algorithm for calculating modulo

value that can process large power of numbers which otherwise are not supported by built-in data types.

First the traditional algorithm is studied. Secondly, the parallelized RSA algorithm is designed using

CUDA framework. Thirdly, the designed algorithm is realized for small prime numbers and large prime

number . As a result the main fundamental problem of RSA algorithm such as speed and use of poor or

small prime numbers that has led to significant security holes, despite the RSA algorithm's mathematical soundness can be alleviated by this algorithm.

KEYWORDS CPU, GPU, CUDA, RSA, Cryptographic Algorithm.

1. INTRODUCTION

RSA (named for its inventors, Ron Rivest, Adi Shamir, and Leonard Adleman [1] ) is a public

key encryption scheme. This algorithm relies on the difficulty of factoring large numbers which

has seriously affected its performance and so restricts its use in wider applications. Therefore, the

rapid realization and parallelism of RSA encryption algorithm has been a prevalent research

focus. With the advent of CUDA technology, it is now possible to perform general-purpose

computation on GPU [2]. The primary goal of our work is to speed up the most computationally

intensive part of their process by implementing the GCD comparisons of RSA keys using

NVIDIA's CUDA platform.

The reminder of this paper is organized as follows. In section 2,3,4, we study the traditional RSA

algorithm. In section 5, we explained our system hardware. In section 6,7,8, 9 we explained the

design and implementation of parallelized algorithm. Section 10 gives the result of our

parallelized algorithm and section 12 concludes the paper.

2. TRADITIONAL RSA ALGORITHM[1]

RSA is an algorithm for public-key cryptography [1] and is considered as one of the great

advances in the field of public key cryptography. It is suitable for both signing and encryption.

Electronic commerce protocols mostly rely on RSA for security. Sufficiently long keys and up-to-

date implementation of RSA is considered more secure to use.

International Journal of Network Security & Its Applications (IJNSA), Vol.6, No.4, July 2014

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RSA is an asymmetric key encryption scheme which makes use of two different keys for

encryption and decryption. The public key that is known to everyone is used for encryption. The

messages encrypted using the public key can only be decrypted by using private key. The key

generation process of RSA algorithm is as follows:

The public key is comprised of a modulus n of specified length (the product of primes p and q),

and an exponent e. The length of n is given in terms of bits, thus the term 8-bit RSA key" refers

to the number of bits which make up this value. The associated private key uses the same n, and

another value d such that d*e = 1 mod (n) where (n) = (p - 1)*(q - 1) [3]. For a plaintext M

and cipher text C, the encryption and decryption is done as follows:

C = Me mod n, M = Cd mod n.

For example, the public key (e, n) is (131,17947), the private key (d, n) is (137,17947), and let

suppose the plaintext M to be sent is: parallel encryption.

Firstly, the sender will partition the plaintext into packets as: pa ra ll el en cr yp ti on. We

suppose a is 00, b is 01, c is 02, ..... z is 25.

Then further digitalize the plaintext packets as: 1500 1700 1111 0411 0413 0217 2415

1908 1413.

After that using the encryption and decryption transformation given above calculate the

cipher text and the plaintext in digitalized form.

Convert the plaintext into alphabets, which is the original: parallel encryption.

3. OPERATION [1]

The three steps of RSA algorithm Key Generation, Encryption and Decryption are explained as

follows:

3.1 Key generation

As explained above RSA is an asymmetric key encryption scheme. It makes use of two different

keys for encryption and decryption. The public key that is known to everyone is used for

encryption. The messages encrypted using the public key can only be decrypted by using private

key. The keys for RSA algorithm are generated as follows:

Figure 1. RSA Key Generation

International Journal of Network Security & Its Applications (IJNSA), Vol.6, No.4, July 2014

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3.2 Encryption process

Alice transmits its public key (n,e) to the Bob and keeps the private key secret. Suppose Bob

wishes to send message M to Alice. He first convert the message M to an integer m , such that

m is between 0 and n using padding scheme [1][3]. Finally cipher text c is calculated as: C = me

mod n. This cipher text is then transmitted to Alice.

3.3 Decryption process

Alice can decode the cipher text using her private key component d as follows:

m = Cd mod n

Given m, original message M can be recovered.

3.4 Example

Figure 2. RSA Example

3.6 Modular exponential

3.6.1 Modular arithmetic

Public key cryptography is computationally expensive as it mostly includes raising a large power

to a base and then reducing the result using modulo function. This process is known as modular

exponential. In practical, RSA algorithm should be fast , i.e., modular exponential function should

be fast and along with it should be efficient. The simple modular operations are given in the

Figure 3.

Figure 3. Modular Arithmetic

3.6.2 Naive modular exponentiation

In this method modular multiplications are applied repeatedly. For example: with g=4, e=13 and

m= 497, the naive modular exponential will solve ge mod m as shown in figure and gives the

result as 445. This method is not efficient as it performs e-1 modular multiplications. In

cryptography the security depends on the larger value of e and efficiency depends how efficiently

these modular multiplications and modular exponential functions are solved. The plaintext ,

International Journal of Network Security & Its Applications (IJNSA), Vol.6, No.4, July 2014

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cipher-text or even partial cipher text is supposed to be of large in value and it will require a large

amount of modular multiplications if we rely on this naive algorithm.

Figure 4. Naive Modular Exponentiation Example

3.6.3 Repeated square-and-multiply methods

It makes use of the fact that if the e value is even, then the modular exponential is calculated as

ge mod m =( ge/2 * ge/2) mod m ,hence by reducing the amount of modular multiplications to 2z

where z is the number of bits when converting e to binary form. The algorithm comes with two

forms as shown below.

Right-to-left binary modular exponential

Left-to-right binary modular exponential

This algorithm works efficiently for large value of e.

International Journal of Network Security & Its Applications (IJNSA), Vol.6, No.4, July 2014

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Figure 5. Repeated Square-and-Multiply Methods

3.6.4 Left-to-right k-ary exponentiation

This algorithm is the generalization of the previous left-to-right binary modular exponentiation

described previously. In this algorithm each iteration processes more than one bit of the exponent

and works efficiently when pre-computations are done in advance and is used again and again.

Figure 6. Left-To-Right K-ary Modular Exponentiation

3.6.5 Sliding window exponential

The main idea behind this algorithm is the reduction of pre-computations as compared to the

above discussed algorithm and hence ultimately reduction in the average number of

multiplications done.

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Figure 7. Sliding Window Exponential

3.7 ADVANTAGES AND DISADVANTAGES OF RSA ALGORITHM

The RSA algorithm is known for its increased security and convenience. In RSA algorithm

private keys are neither transmitted or nor revealed to anyone. By contrast in secret-key system

keys are exchanged either manually or with the help of communication channel. This is because

in secret-key system same key is used for encryption and decryption and there exist a chance that

an enemy can retrieve this secret ke