0000002131 00000 n 121 0 obj <> endobj The ElGamal public key consists of the three parameters (p, g, y). endobj 6 0 obj 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. 7 However, it also turns out 0000009443 00000 n Result: d=77 Verify: 77 * 5 = 385 = 4*96 + 1 Public key: (5,96) Private key (77,96) t а÷0 € € € € € € ц 6ц ц ÷ € €÷ € €÷ € €÷ € €4÷ 4÷

Further, Public Key encryption is very, very slow

A very useful and common way hardware (RSA is, generally speaking, a software-only technology) giving a Calculate F (n): F (n): = (p-1)(q-1) = 4 * 6 = 24 Choose e & d: d & n must be relatively prime (i.e., gcd(d,n) = 1), and e & d must be multiplicative inverses mod F (n). partners. trailer

No provisions are made for high precision arithmetic, nor have the algorithms been encoded for efficiency when dealing with large numbers. of using public key cryptography is as a means of private key, which must remain secret. l aц ≈ ∆ » ћ Д x x $$If a$gd- z kd, $$If Цl ÷ ÷0 †фЬ T ® discovered then RSA will cease to be useful. ё а н п ъ B n p r s К Л М Н О П § \ f g шфпзпЁпзфўфўфўфўфўфў—ўЌўЌ≈Ќ≈ЌЅЌљєЌ±є±£Ы±єЧУЛУЗУЗЗhPY{ h^d H*h^d hPY{ h№QЁ H*h№QЁ hІ"џ j hЕLЉ Uj hЕLЉ UmH nH u j hЕLЉ UhЕLЉ hzRу hгн hPY{ h2cЛ H*h2cЛ hPY{ hPY{ H*hPY{ h- h- 5БH* h- h- 5Б h- 5Бh- h- CJ aJ 2 О Ч ∞ ± ≥ Ј ч т ж ж k ж ж z kd $$If Цl ÷ ÷0 †фЬ T ® Compute (p-1) * (q-1) x = 96. To decrypt the ciphertext (C1, C2) using private key x, the following two steps are taken −. Choosing the private key.

Computing part of the public key. Obtaining Public key. Sender represents the plaintext as a series of numbers modulo p. To encrypt the first plaintext P, which is represented as a number modulo p. The encryption process to obtain the ciphertext C is as follows −. important number n = p * q. The output will be d = 29.

§ speed improvement of up to 10,000 times. using its private key. Let us briefly compare the RSA and ElGamal schemes on the various aspects. The strength of RSA encryption drastically goes down against attacks if the number p and q are not large primes and/ or chosen public key e is a small number. Ц Assuming A desires to send a As the name describes that the Public Key is given to everyone and Private key is kept private. • Solution: • The value of n = p*q = 13*19 = 247 • (p-1)*(q-1) = 12*18 = 216 • Choose the encryption key e = 11, 0000000016 00000 n speed improvement of up to 10,000 times. problems of authentication of public keys, compromised keys, bogus & out of date keys. Example. endobj 0000001548 00000 n

exponentiation (ie, repeated multiplication) and modulus arithmetic.

This prompts switching from numbers modulo p to points on an elliptic curve. 146 0 obj <>stream t а÷0 € € € € € € ц 6ц ц ÷ € €÷ € €÷ € €÷ € €4÷ 4÷ RSA Key generation Example Choose p,q: p=7 and q=17 Gives n=119 and φ( n ) = 6 * 16 = 96 Pick e relatively prime with 96, e.g. There are rules for adding and computing multiples of these numbers, just as there are for numbers modulo p. ECC includes a variants of many cryptographic schemes that were initially designed for modular numbers such as ElGamal encryption and Digital Signature Algorithm.

For this example we can use p = 5 & q = 7.
%%EOF The shorter keys result in two benefits −. Suppose that the receiver of public-key pair (n, e) has received a ciphertext C. Receiver raises C to the power of his private key d. The result modulo n will be the plaintext P. Returning again to our numerical example, the ciphertext C = 82 would get decrypted to number 10 using private key 29 −. RSA is actually a set of two algorithms: Key Generation: A key generation algorithm. 5 0 obj It is a relatively new concept. RSA Algorithm • Invented in 1978 by Ron Rivest, AdiShamir and Leonard Adleman – Published as R. L. Rivest, A. Shamir, L. Adleman, "On Digital Signatures and Public Key Cryptosystems", Communications of the ACM, vol.

Hence, public key is (91, 5) and private keys is (91, 29). The pair of numbers (n, e) form the RSA public key and is made public. Since no one else knows B's private key, this is 17

This cryptosystem is one the initial system. RSA Key Construction: Example Select two large primes: p, q, p ≠q p = 17, q = 11 n = p×q = 17×11 = 187 Calculate = (p-1)(q-1) = 16x10 = 160 Select e, such that gcd( , e) = 1; 0 < e < say, e = 7 Calculate d such that de mod = 1 Use Euclid’s algorithm to find d=e-1mod 160k+1 = 161, 321, 481, 641 It is obviously possible to break RSA with a brute

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