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**Asymmetric-Key Cryptography**

GROUP MEMBER :- SOURAV SHASHANK SURAJ YADAV SONAL RATHI SUBHAM SINGHAL

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**Asymmetric key cryptography uses two separate keys: one private and one public.**

Locking and unlocking in asymmetric-key cryptosystem

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**General idea of asymmetric-key cryptosystem**

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**C = f (Kpublic , P) P = g(Kprivate , C)**

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)

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**C M Public Key (asymmetric): Requirements:**

1. For every message M, encrypting with public key and then decrypting resulting ciphertext with matching private key results in M. 2. Encryption and Decryption can be efficiently applied to M 3. It is impractical to derive decryption key from encryption key. Sender Receiver (encryption) (public key of Receiver) C M (decryption) (private key of Receiver)

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**Combining Public/Private Key Systems**

Public key encryption is more expensive than symmetric key encryption For efficiency, combine the two approaches (1) A B (2) Use public key encryption for authentication; once authenticated, transfer a shared secret symmetric key (2) Use symmetric key for encrypting subsequent data transmissions

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**RivestShamirAdelman (RSA) Method**

Named after the designers: Rivest, Shamir, and Adleman Public-key cryptosystem and digital signature scheme. Based on difficulty of factoring large integers For large primes p & q, n = pq Public key e and private key d calculated

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RSA Key Generation Every participant must generate a Public and Private key: 1. Let p and q be large prime numbers, randomly chosen from the set of all large prime numbers. 2. Compute n = pq. 3. Choose any large integer, d, so that: GCD( d, ϕ(n)) = (where ϕ(n) = (p1)(q1) ) 4. Compute e = d-1 (mod ϕ(n)). 5. Publish n and e. Keep p, q and d secret. Note: Step 4 can be written as: Find e so that: e x d = 1 (modulo ϕ(n)) If we can obtain p and q, and we have (n, e), we can find d

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**RivestShamirAdelman (RSA) Method**

Assume A wants to send something confidentially to B: A takes M, computes C = Me mod n, where (e, n) is B’s public key. Sends C to B B takes C, finds M = Cd mod n, where (d, n) is B’s private key + Confidentiality B A M M C Me mod n Cd mod n (e, n) (d, n) Encryption Key for user B (B’s Public Key) Decryption Key for user B (B’s PrivateKey)

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**RSA Method Example: 1. p = 5, q = 11 and n = 55.**

(p1)x(q1) = 4 x 10 = 40 2. A valid d is 23 since GCD(40, 23) = 1 3. Then e = 7 since: 23 x 7 = 161 modulo 40 = 1 in other words e = 23-1 (mod 40) = 7

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**Topics discussed in this section:**

ELGAMAL CRYPTOSYSTEM Besides RSA another public-key cryptosystem is ElGamal. ElGamal is based on the discrete logarithm problem discussed. Topics discussed in this section: ElGamal Cryptosystem Procedure Proof Analysis

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**Key generation, encryption, and decryption in ElGamal**

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Key Generation

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Note The bit-operation complexity of encryption or decryption in ElGamal cryptosystem is polynomial.

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**Proof of ElGamal Cryptosystem**

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**Here is a trivial example. Bob chooses p = 11 and e1 = 2**

Here is a trivial example. 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.

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Questions?

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