International Journal of Computer Applications (0975 – 8887) Volume 74– No.13, July 2013 31 Three Novel Theorems for Applied Cryptography Rahul Yadav Scholar (Master of Technology) Department of CS, SunRise University, Alwar Rajasthan, India Deepak Chaudhary Assistant Professor Department of CS, IET, Alwar Rajasthan, India ABSTRACT With advancements in computing capabilities public key cryptosystems are going to be more complex yet vulnerable over the modern day‟s computer networks and associated security mechanism, especially those based on novel approaches of applied mathematics. This paper explores three novel theorems derived while studying and implementing RSA algorithm, one of the strongest public key cryptosystem. The proposed Theorems are best suited and adequate for RSA algorithm yet being applicable to some of other existing algorithms and theorems of applied mathematics. The first theorem deals with concept of ambiguity while calculating multiplicative inverse of encryption key which in some of instances returns undesirable negative numbers not useful as decryption key . Second theorem deals with unconcealed multiplicative inverses, unconcealed are values which remain unchanged after any mathematical transformations. Concept of unconcealed multiplicative inverses is useful in key generation for RSA cryptosystem. Third theorem deals with the concept of unconcealed exponentiation modulo quite useful in finding unconcealed signature and messages to form UM Matrix for RSA. General Terms Cryptography, Public Key Cryptosystem, RSA Algorithm, Novel Theorems, & Applied Mathematics for Computers Keywords Quarter division of Ring, Unconcealed Transformation, Ambiguous Values, Bézout's identity, Extended Euclid algorithm 1. INTRODUCTION Most of the modern day‟s Public Key Cryptosystems are applications of mathematical approaches designed for data security, especially public key cryptosystem. While dealing with implementation of RSA we found many applied algorithms as conceptual base of RSA approach of data security, which includes Prime Number generation & their testing (like Miller-Robin Test [1] ), Multiplication algorithms (like Booth multiplication [2] ), Totient Function, Multiplicative Inverses of the form Bézout's identity. Euclid‟s Extended Algorithm is one which is capable of easing major computation part of RSA Algorithm‟s Key generation part. Additionally it‟s a fact that Fermat‟s Little Theorem, Euler‟s Theorem and Totient Function lies on the heart of this cryptosystem. 1.1 RSA ENCRYPTION & DECRYPTION APPROACHES To encrypt a message M with RSA approach, using a public encryption key (e,n), proceed as follows. (Here e & n are a pair of integers positive in nature.) In very first step, represent the message as an integer between 0 & n — 1. (Break a long message into a series of blocks, & represent each block as such an integer.) Users are advised to use any standard notation. Main purpose here is nothing to do with encryption of the message but solitary to get it into the numeric form necessary for enciphering. Then, encrypt the message by raising it to the eth power modulo n. That is, the result (the cipher text C) is the remainder when M is divided by n. e To decrypt the cipher text, raise it to another power d, further modulo n. The encryption & decryption algorithms E & D are thus: = () = ( ), . ……. (1) () = ( ), . …….. (2) Encryption does not result into increased size of a message; both the message & the cipher text are integers in the range 0 to n — 1. The encryption key is thus the pair of positive integers (e,n). Similarly, the decryption key is the pair of positive integers (d, n). Each user makes his encryption key public, & keeps the corresponding decryption key private. (These integers should properly be subscripted as in, U A , e A , & d A , since each user has his own set. nevertheless, we will solitary consider a typical set, & A will omit the subscripts.) How should you choose your encryption & decryption keys, if you want to use RSA approach? Users first compute n as the product of two prime numbers p & q: = ∗ . These prime numbers are very large, “random” prime numbers. Although you will make n public, the factors p & q will be effectively hidden from everyone else due to the enormous difficulty of finding factors of n. It also enables to hide the way d can be derived from e. You then pick the integer d to be a large (in terms of digit / bit count), randomly chosen integer which is relatively prime to totient function of n i.e. (p — 1) * (q — 1). That is, check that d satisfies: (,( — 1) ∗ ( — 1)) = 1 The integer e is calculated using p, q, & d to be the “multiplicative inverse” of d, modulo (p — 1) • (q — 1). Thus we have ∗ ≡ 1 ( ( — 1) ∗ ( — 1)). We prove in the next section that this guarantees that (1) & (2) hold, i.e. that E & D are in verse permutations. The aforementioned approach should not be confused with the “exponentiation” technique presented by Diffie & Hellman to solve the key distribution issue. Their technique permits two users to determine a key in common to be used in a normal