Rachana Y. Patil

91 Views

Download Presentation
## Rachana Y. Patil

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Symmetric and asymmetric-key cryptography will exist in**parallel and continue to serve the community. We actually believe that they are complements of each other; the advantages of one can compensate for the disadvantages of the other. Symmetric-key cryptography is based on sharing secrecy; asymmetric-key cryptography is based on personal secrecy. 10.2**Asymmetric key cryptography uses two separate keys: one**private and one public. Locking and unlocking in asymmetric-key cryptosystem 10.3**Plaintext/Ciphertext**Unlike in symmetric-key cryptography, plaintext and ciphertext are treated as integers in asymmetric-key cryptography. Encryption/Decryption C = f (Kpublic , P) P = g(Kprivate , C) 10.5**RSA CRYPTOSYSTEM**The most common public-key algorithm is the RSA cryptosystem, named for its inventors (Rivest, Shamir, and Adleman).**Procedure**Encryption, decryption, and key generation in RSA**Example**Bob chooses 7 and 11 as p and q and calculates n = 77. The value of f(n) = (7 − 1)(11 − 1) or 60. Now he chooses two exponents, e and d, from Z60∗. If he chooses e to be 13, then d is 37. Note that e × d mod 60 = 1 (they are inverses of each Now imagine that Alice wants to send the plaintext 5 to Bob. She uses the public exponent 13 to encrypt 5. Bob receives the ciphertext 26 and uses the private key 37 to decipher the ciphertext:**Example**Now assume that another person, John, wants to send a message to Bob. John can use the same public key announced by Bob (probably on his website), 13; John’s plaintext is 63. John calculates the following: Bob receives the ciphertext 28 and uses his private key 37 to decipher the ciphertext:**Example**n=221 e=5 find d p=19 q=23 e=3 find Ø(n) and d e=17 n=187 find d n=19519 e=17 find d**Bob chooses p = 11 and e1 = 2. and d = 3 e2 = e1d = 8. So**the public keys are (2, 8, 11) and the private key is 3. Alice chooses r = 4 and calculates C1 and C2 for the plaintext 7. Bob receives the ciphertexts (5 and 6) and calculates the plaintext.**Example**In ElGamal,given the prime p=31 Choose an appropriate e1 and d,then calculate e2 Encrypt the plaintext message 5 Decrypt the ciphertext to obtain the plaintext**SYMMETRIC-KEY AGREEMENT**Alice and Bob can create a session key between themselves. This method of session-key creation is referred to as the symmetric-key agreement.**Note**The symmetric (shared) key in the Diffie-Hellman method is K = gxy mod p.**Example**Assume that g = 7 and p = 23. The steps are as follows: • Alice chooses x = 3 and calculates R1 = 73 mod 23 = 21. • Bob chooses y = 6 and calculates R2 = 76 mod 23 = 4. • Alice sends the number 21 to Bob. • Bob sends the number 4 to Alice. • Alice calculates the symmetric key K = 43 mod 23 = 18. • Bob calculates the symmetric key K = 216 mod 23 = 18. • The value of K is the same for both Alice and Bob; gxy mod p = 718 mod 35 = 18.**Example**Alice and Bob decide to use diffie hellman key exchange protocol To agree upon a common key, they choose p=13 and g=2.Each chooses his own secret number and exchange the numbers 6 and 11. • What will be the common secret key they derived? • What are their secret numbers? • Can intruder M gain any knowledge from the protocol run if he sees P,g and the two public key 6 and 11? If yes show how**Digital Signature**The digital signature process. • The sender uses a signing algorithm to sign the message. • The message and the signature are sent to the receiver. • The receiver receives the message and the signature and applies the verifying algorithm to the combination. • If the result is true, the message is accepted; otherwise, it is rejected.**Need for Keys**Note A digital signature needs a public-key system. The signer signs with her private key; the verifier verifies with the signer’s public key. 13.29**Note**A cryptosystem uses the private and public keys of the receiver: a digital signature uses the private and public keys of the sender. 13.30**DIGITAL SIGNATURE SCHEMES**Several digital signature schemes have evolved during the last few decades. Some of them have been implemented. 13.31**Note**Key Generation Key generation in the RSA digital signature scheme is exactly the same as key generation in the RSA In the RSA digital signature scheme, d is private; e and n are public. 13.33**Signing and Verifying**RSA digital signature scheme 13.34**Example**Alice selects n=221 and e=15.Find Private key of Alice. If Alice wants to send message M=11 to Bob. Calculate The Signature and show Bob can Verify the message.**ElGamal Digital Signature Scheme**General idea behind the ElGamal digital signature scheme 13.36**Note**Key Generation The key generation procedure here is exactly the same as the one used in the cryptosystem. In ElGamal digital signature scheme, (e1, e2, p) is Alice’s public key; d is her private key. 13.37**Verifying and Signing**ElGamal digital signature scheme 13.38**Example**• Bob chooses p=11,e1=2,r=9,d=8 and sign message M=5 using Elgamal digital signature scheme. Calculate s1 and s2 and show how Alice can verify the signature • Alice chooses p=23,e1=5,d=3 a random number 9 and sign message M=7 before sending it to bob. Calculate s1 and s2 and show how bob can verify the signature.