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Computer and Network Security

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oscar-workman
28 diapositives · 423 Views · Oct 28, 2014

This lecture delves into RSA (Rivest-Shamir-Adleman) and other public key cryptosystems, covering their unique features and key management strategies. Learn about the core principles of public key cryptography, which uses a pair of keys—a public key for encryption and a private key for decryption. Explore its benefits in secure communication and the crucial aspect of digital signatures. This session emphasizes the significance of public key distribution and practical RSA encryption examples, shedding light on its role in modern security practices.

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  1. Computer and Network Security Rabie A. Ramadan Lecture 6

  2. RSA Other Public Key Cryptosystems Key management Table of Contents

  3. Public Key Cryptography and Rivest-Shamir-Adleman (RSA)

  4. Every Egyptian received two names, which were known respectively as the true name and the good name, or the great name and the little name; and while the good or little name was made public, the true or great name appears to have been carefully concealed. —The Golden Bough, Sir James George Frazer

  5. Traditional private/secret/single key cryptography uses one key Shared by both sender and receiver If this key is disclosed communications are compromised Also is symmetric, parties are equal Hence does not protect sender from receiver forging a message & claiming is sent by sender Private-Key Cryptography

  6. Uses two keys – a public & a private key Asymmetric since parties are not equal Uses clever application of number theoretic concepts to function Complements rather than replaces private key cryptography Public-Key Cryptography

  7. 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 It is asymmetric because those who encrypt messages or verify signatures cannot decrypt messages or create signatures Public-Key Cryptography

  8. Public-Key Cryptography – Encryption

  9. Public-Key Cryptography -Authentication

  10. Public-Key Cryptosystems Secrecy and Authentication

  11. Developed to address two key issues: key distribution – how to have secure communications digital signatures – how to verify a message comes intact from the claimed sender encryption/decryption (provide secrecy) Public invention due to Whitfield Diffie & Martin Hellman at Stanford Uni in 1976 Public-Key Cryptography

  12. 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 keyscan be used for encryption, with the other used for decryption (in some schemes) Public-Key Characteristics

  13. By Rivest, Shamir & Adleman of MIT in 1977 Best known & widely used public-key scheme Based on exponentiation in a finite (Galois) field over integers modulo a prime Uses large integers (eg. 1024 bits) Security due to cost of factoring large numbers factorization takes O(e log n log n log n) operations (hard) RSA

  14. Each user generates a public/private key pair by: Selecting two large primes at random - p, q Computing their system modulus N=p.q note ø(N)=(p-1)(q-1) Selecting at random the encryption key e where 1<e<ø(N), gcd(e,ø(N))=1 Solve following equation to find decryption key d e.d=1 mod ø(N) and 0≤d≤N publish their public encryption key: KU={e,N} keep secret private decryption key: KR={d,p,q} RSA

  15. to encrypt a message M the sender: obtains public key of recipient KU={e,N} computes: C=Me mod N, where 0≤M<N to decrypt the ciphertext C the owner: uses their private key KR={d,p,q} computes: M=Cd mod N note that the message M must be smaller than the modulus N (block if needed) RSA Use

  16. RSA Summary

  17. Select primes: p=17 & q=11 Compute n = pq =17×11=187 Compute ø(n)=(p–1)(q-1)=16×10=160 Select e : gcd(e,160)=1; choose e=7 Determine d: de=1 mod 160 and d < 160 d=23 Publish public key KU={7,187} Keep secret private key KR={23,17,11} RSA Example

  18. X = Y mod m Dividing X and Y by m must give the same number 7 = 23 mod 8  7/8 = 23/8 = 7 22 = 13 mod 9  22/9 = 13/9 = 4 Now  d.e = 1 mod 160  e =7  7d = 1 mod 160 7d /160 = 1/160  7d has to be something related to 160 +1 If d =23 then 23*7 /160 = 161/160 = 1/160 =1 Then e =7 and d =23 a+ kp = a mod p If d.e = (a+kp) that is what I need Mod Operations

  19. X = Y mod m Dividing X and Y by m must give the same number 7 = 23 mod 8  7/8 = 23/8 = 7 22 = 13 mod 9  22/9 = 13/9 = 4 Now  d.e = 1 mod 160  e =7  7d = 1 mod 160 7d /160 = 1/160  7d has to be something related to 160 +1 If d =23 then 23*7 /160 = 161/160 = 1/160 =1 Then e =7 and d =23 a+ kp = a mod p If d.e = (a+kp) that is what I need Primarily and Coprima

  20. sample RSA encryption/decryption is: given message M = 88 (nb. 88<187) encryption: C = 887 mod 187 = 11 decryption: M = 1123 mod 187 = 88 RSA Example (Cont.)

  21. Three approaches to attacking RSA: brute force key search (infeasible given size of numbers) mathematical attacks (based on difficulty of computing ø(N), by factoring modulus N timing attacks (on running of decryption) RSA Security

  22. Key Management

  23. Public-key encryption helps address key distribution problems Have two aspects of this: Distribution of public keys Use of public-key encryption to distribute secret keys KeyManagement

  24. Can be considered as using one of: Public announcement Publicly available directory Public-key authority Public-key certificates Distributionof Public Keys

  25. Users distribute public keys to recipients or broadcast to community at large eg. append PGP keys to email messages or post to news groups or email list Major weakness is forgery Anyone can create a key claiming to be someone else and broadcast it Until forgery is discovered can masquerade as claimed user PublicAnnouncement

  26. Can obtain greater security by registering keys with a public directory Directory must be trusted with properties: contains {name,public-key} entries participants register securely with directory participants can replace key at any time directory is periodically published directory can be accessed electronically Still vulnerable to tampering or forgery PubliclyAvailableDirectory

  27. Improve security by tightening control over distribution of keys from directory Has properties of directory And requires users to know public key for the directory Then users interact with directory to obtain any desired public key securely Does require real-time access to directory when keys are needed Could be a bottleneck Public-KeyAuthority

  28. Public-KeyAuthority