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Discrete Log Problem

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1. Introduction
Initially, the encryption of message was based on symmetric key cryptography where sender and receiver of message use the same key for encryption and decryption.But, to use the same key, sender and receiver must share the key in advance. And if their locations are different than there is risk in transmission of the key. Later in 1976,a cryprosystem,which is known as Diffie-hellman key-exchange, was published by Whitefield Diffie and Martin Hellman and concept behind the cryptosystem is known as public key encryption. In public key cryptosystem, each one gets a pair of keys, public key and private key. The pubic key is freely available to everyone while the private key remains secret. The sender, who wants to send a message securely to someone, use public key of receiver to encrypt the message and receiver use his private key to decrypt the message.This system doesn’t require secure key transmission.So, it resolves the one of the problem faced by symmetric key cryptosystem. If someone is able to compute respective private key from a given public key, then this system is no more secure. So, Public key cryptosystem requires that calculation of respective private key is computationally impossible from given public key. In most of the Public key cryptosystem, private key is related to public key via Discrete Logarithm. Examples are Diffie-Hellman Key Exchange, Digital Signature Algorithm (DSA), Elgamal which are based on DLP in finite multiplicative group.

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2. Discrete logarithm problem
The Discrete Logarithm Problem (DLP)is the problem of finding an exponent x such that g x ≡ h (mod p) where, g is a primitive root for Fp and h is a non-zero element of Fp . Let, n be the order of g. Then solution x is unique up to multiples of n and x is called discrete logarithm of h to the base g (i.e.) x = logg h. In cryptosystem based on Discrete Logarithm , x is used…...

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