The ElGamal Cryptosystem is an entire public-key cryptosystem like RSA, but based on discrete logs p large so secure and > m = message 1 Bob chooses prime p, primitive root a, integer a Bob computes b ≡ aa (mod p) Bob publishes (a, p, b) and holds a secret Alice chooses secret k, computes and sends to Bob the pair (r,t) where r ≡ ak (mod p) t ≡ bkm (mod p) Bob calculates: tr-a ≡ m (mod p) Why does this … ElGamal cryptosystems and Discrete logarithms De nition Let Gbe a cyclic group of order nand let be a generator of G. For each A2Gthere exists an unique 0 a n 1 such that A= a. The ElGamal cryptosystem is used in some form in a number of standards including the digital signature standard (DSS) and the S/MIME email standard. The ElGamal system is a public-key system is that message expansion by a factor of two takes prime p and an integer g, whose powers modulo p Another potential disadvantage of the ElGamal The ElGamal system is a public-key cryptosystem based on the discrete logarithm problem. * * In 1984, T. Elgamal announced a public-key scheme based on discrete logarithms, closely related to the Diffie-Hellman technique [ELGA84, ELGA85]. The encryption algorithm is similar in nature to the Diffie-Hellman key agreement protocol (see Question 24). not the same as signature verification, nor is decryption the DSA (see Question 26) This cryptosystem is based on the difficulty of finding discrete logarithm in a cyclic group that is even if we know g a and g k, it is extremely difficult to compute g ak. The Note, it resembles the order of xand ypowers on the curve (i.e x3 and y2). As with Diffie-Hellman, the global elements of ElGamal are a prime number q and a, which is a primitive root of q. The ElGamal signature algorithm is He then computes. However, such message expansion is negligible if the cryptosystem is used only for exchange of The ElGamal PKC • Based on the difficulty of discrete logarithm, was invented by Tahir ElGamal in 1985. Upon receiving the ciphertext, Alice computes. Suppose Bob wishes to send a message m to Alice. It derives the strength from the assumption that the discrete logarithms cannot be found in practical time frame for a given number, while the inverse operation of the power can be computed efficiently. Bob sends (y1 ,y2) The ElGamal Cryptosystem is an entire public-key cryptosystem like RSA, but based on discrete logs p large so secure and > m = message 1 Bob chooses prime p, primitive root a, integer a Bob computes b ≡ aa (mod p) Bob publishes (a, p, b) and holds a secret Alice chooses secret k, computes and sends to Bob the pair (r,t) where r ≡ ak (mod p) t ≡ bkm (mod p) Bob calculates: tr-a ≡ m (mod p) Why does this … • Alice wants to send a message m to Bob. place during encryption. In 1984 aherT ElGamal introduced a cryptosystem which depends on the Discrete Logarithm Problem.The ElGamal encryption system is an asymmet- ric key encryption algorithm for public-key cryptography which is based on the Die-Hellman key exchange.ElGamal depends on the one way function, means that the encryption and decryption are done in separate functions.It depends on the …

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