# Modification of Traditional RSA into Symmetric-RSA Cryptosystems

Prerna Mohit (Indian Institute of Technology (Indian School of Mines), India) and G. P. Biswas (Indian Institute of Technology (Indian School of Mines), India)
DOI: 10.4018/978-1-7998-1763-5.ch007
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## Abstract

This paper addresses the modification of RSA cryptography namely Symmetric-RSA, which seem to be equally useful for different cryptographic applications such as encryption, digital signature, etc. In order to design Symmetric-RSA, two prime numbers are negotiated using Diffie-Hellman key exchange protocol followed by RSA algorithm. As the new scheme uses Diffie-Hellman and RSA algorithm, the security of the overall system depends on discrete logarithm as well as factorization problem and thus, its security is more than public-key RSA. Finally, some new cryptographic applications of the proposed modifications are described that certainly extend the applications of the existing RSA.
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## 2. Preliminaries

Since the modifications of RSA are proposed using Diffie-Hellman (DH) key exchange protocol, thus DH and RSA techniques are introduced below.

### 2.1. Diffie-Hellman (DH) Protocol

In (Diffie et al., 1976), Whitfield Diffie and Martin Hellman published an elementary article for secure exchange of a contributory common key between two remote participants over public channels. It does not require any prior information and is known to be the first public-key cryptosystem. In DH protocol, a finite multiplicative group <ZP, ×> with a generator g are publicly assumed, and two public messages are exchanged for negotiation of a secret key. Let A and B are two participants, who exchange the following two public messages, where A→ B: C means A sends message C to B:

A→B: X = gx (mod P), where 1< x < P and x is a random secret chosen by ABA: Y = gy (mod P), where 1< y < P and y is a random secret chosen by B

The common contributory secret key K (say) is calculated by the participants independently as

K =Yx(mod P) = (gy)x (mod P) =Xy (mod p) =(gx)y (mod P)

This key K is secure, because the DHP (DH problem) and the underlying DLP (discrete logarithm problem) as given below are intractable in polynomial time for a large prime modulus P.

• DHP: Given public values (X = gx, Y = gy), the computation of K = gxy (mod P)

• DLP: Given either of the public values (X = gx or Y = gy), the computation of random secrets x or y

This protocol has huge applications in designing other useful cryptosystems and some of them are Secure Socket Layer (SSL), Transport Layer Security (TLS), Internet Protocol Security (IPSec), Public Key Infrastructure (PKI), Digital Signature Standard (DSS), PGP (Pretty Good Privacy), SET (Secure Electronic Transaction) etc. Although the DH protocol has numerous applications, it is vulnerable to several attacks like man-in-the-middle, Denial-of-Service (DoS) etc.

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