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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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RSA Public Key Crypto System. About RSA Announced in 1977 by Ronald Rivest, Adi Shamir, and Leonard Adleman Relies on the relative ease of finding large.

RSA Public Key Crypto System. About RSA Announced in 1977 by Ronald Rivest, Adi Shamir, and Leonard Adleman Relies on the relative ease of finding large.

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