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Public Key Cryptography and the RSA Algorithm

Public Key Cryptography and the RSA Algorithm. Prepared By : Diwan. Private-Key Cryptography. traditional private/secret/single key cryptography uses one key Key is shared by both sender and receiver if the key is disclosed communications are compromised

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Public Key Cryptography and the RSA Algorithm

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  1. Public Key Cryptography and theRSA Algorithm Prepared By : Diwan

  2. Private-Key Cryptography • traditional private/secret/single key cryptography uses one key • Key is shared by both sender and receiver • if the key is disclosed communications are compromised • also known as symmetric, both parties are equal • hence does not protect sender from receiver forging a message & claiming is sent by sender

  3. Public-Key Cryptography • probably most significant advance in the 3000 year history of cryptography • uses two keys – a public key and a private key • asymmetric since parties are not equal • uses clever application of number theory concepts to function • complements rather than replaces private key cryptography

  4. Public-Key Cryptography • public-key/two-key/asymmetric cryptography involves the use of two keys: • a public-key, which may be known by anybody, and can be used to encrypt messages, and verify signatures • a private-key, known only to the recipient, used to decrypt messages, and sign (create) signatures • is asymmetric because • those who encrypt messages or verify signatures cannot decrypt messages or create signatures

  5. Public-Key Cryptography

  6. Why Public-Key Cryptography? • developed to address two key issues: • key distribution – how to have secure communications in general without having to trust a KDC with your key • digital signatures – how to verify a message comes intact from the claimed sender • public invention due to Whitfield Diffie & Martin Hellman at Stanford U. in 1976 • known earlier in classified community

  7. Public-Key Characteristics • Public-Key algorithms rely on two keys with the characteristics that it is: • computationally infeasible to find decryption key knowing only algorithm & encryption key • computationally easy to en/decrypt messages when the relevant (en/decrypt) key is known • either of the two related keys can be used for encryption, with the other used for decryption (in some schemes)

  8. Public-Key Cryptosystems

  9. Public-Key Applications • can classify uses into 3 categories: • encryption/decryption (provide secrecy) • digital signatures (provide authentication) • key exchange (of session keys) • some algorithms are suitable for all uses, others are specific to one

  10. RSA • by Rivest, Shamir & Adleman in 1977 • best known & widely used public-key scheme • uses large integers (eg. 1024 bits) • security due to cost of factoring large numbers • nb. factorization takes O(e log n log log n) operations (hard)

  11. RSA Key Setup • each user generates a public/private key pair by: • selecting two large primes at random - p, q • Calculate N=P X Q • Select The Public Key (encryption Key) E such that it is not a factor of (P-1) and (Q-1). • Select The Private key (decryption key) D such that Following Equation is true: • (D X E) mod (P-1) X (Q - 1) = 1 • For Encryption Find Cipher Text CT From PT: • CT= PT^E Mod N • For Decryption Find PT from CT • PT= CT^D Mod N

  12. RSA Use • to encrypt a message M the sender: • obtains public key • computes: C=Me mod N, where 0≤M<N • to decrypt the ciphertext C the owner: • uses their private key • computes: M=Cd mod N • note that the message M must be smaller than the modulus N (block if needed)

  13. RSA Example • Select primes: p=7 & q=17 • ComputeN = P X Q =7×17= 119 • Compute (P–1)(Q-1)=6 X 16 = 96 • Factor of 96 are 2,2,2,2,2 & 3. • Thus We have to choose E such that none of the factor of E is 2 and 3. • Let Choose E as 5 • (D X 5) Mod (7 - 1) X (17 -1) =1 • (D X 5) Mod (96) =1 • Possibilities are (97,193,289,385,…) but only 385 is divisible by 5 so D X 5 = 385, D = 77 • If Plain text = 10 then CT = 10 ^ 5 mod 119 = 40 • For Decryption PT = 40 ^ 77 Mod 119 = 10 which is original plain text.

  14. RSA Key Generation • users of RSA must: • determine two primes at random - p, q • select either e or d and compute the other • primes p,qmust not be easily derived from modulus N=p.q • means must be sufficiently large • typically guess and use probabilistic test • exponents e, d are inverses, so use Inverse algorithm to compute the other

  15. Summary • have considered: • principles of public-key cryptography • RSA algorithm, implementation, security

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