Public Key Cryptosystems and RSA
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The document discusses public key cryptography and RSA. It begins by defining encryption and distinguishing between private and public key cryptosystems. It then provides an overview of RSA, including how key generation works by selecting prime numbers, calculating the public and private keys, and how encryption and decryption are performed. The document concludes by discussing some potential attacks against RSA, such as factoring challenges and risks of non-random number generation.
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Public Key Cryptosystems and RSA
- 1. PUBLIC KEY CRYPTOSYSTEMS AND RSA Christopher Theisen http://theisencr.github.io/whitewater_pkc/
- 2. AGENDA ◦ Encryption ◦ Private Key Cryptosystems ◦ Public Key Cryptosystems ◦ Introduction to RSA ◦ Simple RSA Example ◦ Exercise: Key Generation and Message ◦ Attacks against RSA
- 3. ENCRYPTION ◦ Definition: “The process of converting information or data into a code, with the goal of preventing unauthorized access.” ◦ Important for protecting data you want to keep private ◦ Credit cards, personal information, etc.
- 4. PRIVATE KEY CRYPTOSYSTEMS ◦ Use of a single, shared key that can encrypt and decrypt information ◦ Messages are encrypted using the shared key, then the encrypted message is sent to the other party ◦ Use Case: sustained messages between two known parties
- 5. PRIVATE KEY CRYPTOSYSTEMS Public Key Cryptography | RSA | Example | Exercise | Attacks
- 6. PUBLIC KEY CRYPTOSYSTEMS Public Key Cryptography | RSA | Example | Exercise | Attacks Distinguished from private key: ◦ Private Key: A secret, exclusive key for encryption and decryption ◦ Public Key: Separate, public key for encryption and decryption. ◦ Use Case: authentication step and exchange of shared secret key for further communication
- 7. PUBLIC KEY CRYPTOSYSTEMS Public Key Cryptography | RSA | Example | Exercise | Attacks
- 8. PUBLIC KEY - CONFIDENTIALITY Public Key Cryptography | RSA | Example | Exercise | Attacks
- 9. PUBLIC KEY - AUTHENTICATION Public Key Cryptography | RSA | Example | Exercise | Attacks
- 10. PUBLIC KEY – CONFIDENTIALITY AND AUTHENTICATION Public Key Cryptography | RSA | Example | Exercise | Attacks Bob Private Key -> Alice Public Key -----> Alice Private Key -> Bob Public Key
- 11. RIVEST-SHAMIR-ADLEMAN (RSA) ◦ Developed by Ron Rivest, Adi Shamir, and Leonard Adleman ◦ Based on the difficulty of factoring large prime numbers ◦ Someone with the product of two primes can encrypt, but only someone who knows both primes can decrypt. Public Key Cryptography | RSA | Example | Exercise | Attacks
- 12. TRAPDOOR FUNCTION ◦ Easy to compute in one direction, hard to compute in the other without special information (the trapdoor) Public Key Cryptography | RSA | Example | Exercise | Attacks
- 13. RSA – KEY GENERATION 1. Pick two large primes, p and q p = 11 q = 3 2. Calculate n = pq n = 11 * 3 n = 33 Public Key Cryptography | RSA | Example | Exercise | Attacks
- 14. RSA – KEY GENERATION 3. Calculate λ(n) = (p-1)(q-1) λ(n) = (11-1)(3-1) λ(n) = 10*2 λ(n) = 20 4. Choose a small number e, coprime to λ(n) e = 3 Alternate: Fix e first (e=3, e=17, e=65,537) Public Key Cryptography | RSA | Example | Exercise | Attacks
- 15. RSA – KEY GENERATION 5. Find d, satisfying de mod λ(n) = 1 Isolating d: d = (1 + x* λ(n)) / e, where x is any integer. x = 0 => d = (1 + 0) / 3 (no) x = 1 => d = (1 + 20) / 3 = 7 (yes!) d = 7 Public Key Cryptography | RSA | Example | Exercise | Attacks
- 16. RSA – KEY GENERATION p = 11 q = 3 n = 33 λ(n) = 20 e = 3 d = 7 Private Key = (n, d) (33, 7) Public Key = (n, e) (33, 3) Public Key Cryptography | RSA | Example | Exercise | Attacks
- 17. RSA – MESSAGES Private Key = <n, d> <33, 7> Public Key = <n, e> <33, 3> We want to send the integer “m” as a message. Sending Messages: Encryption: c = me mod n Decryption: m = cd mod n Public Key Cryptography | RSA | Example | Exercise | Attacks
- 18. RSA – ENCRYPTION Private Key = <n, d> <33, 7> Public Key = <n, e> <33, 3> Encryption: c = me mod n m = 4 c = 43 mod 33 c = 64 mod 33 c = 31 Public Key Cryptography | RSA | Example | Exercise | Attacks
- 19. RSA – DECRYPTION Private Key = <n, d> <33, 7> Public Key = <n, e> <33, 3> Decryption: m = cd mod n c = 31 m = 317 mod 33 m = 27,512,614,111 mod 33 m = 4 Public Key Cryptography | RSA | Example | Exercise | Attacks
- 20. RSA - ALGORITHM Key Generation 1. Pick two large primes, p and q 2. n = pq 3. λ(n) = (p-1)(q-1) 4. Choose a small number e, coprime to λ(n) 5. Find d, satisfying d*e mod λ(n) = 1 Public Key: <n, e> Encryption: c = me mod n Secret Key: <n, d> Decryption: m = cd mod n Public Key Cryptography | RSA | Example | Exercise | Attacks
- 21. EXERCISE: RSA KEY SIZE Links to tools for RSA Demo (work in pairs or more): theisencr.github.io/whitewater_pkc/ Pink generates a public key – “Packed public key” field Blue copies public key, unpacks, encodes message “OpenPGP Multi Precision Integer (MPI) of Public Key (base64)” Pink copies encoded message, decrypts. Spend time checking out performance of each step. Public Key Cryptography | RSA | Example | Exercise | Attacks
- 22. EXERCISE: ATTACKS Imagine Eve wanted to intercept communications that Bob (Blue) sends to Alice (Pink). Open an additional Pink tab. Experiment with “tricking” Bob (Blue) into communicating with Eve (new Pink). What’s the key step? Public Key Cryptography | RSA | Example | Exercise | Attacks
- 23. AUTHENTICATING PUBLIC KEYS ◦ You need to trust that Alice’s public key is *really* her key! ◦ Three approaches: ▫ Certificate Authorities – central repository of validated keys ▫ Web of Trust – get people to “vote” that your key is accurate (Distributed Ledger) ▫ Meet in real life and exchange keys Public Key Cryptography | RSA | Example | Exercise | Attacks
- 24. ATTACKING RSA Public Key Cryptography | RSA | Example | Exercise | Attacks
- 25. ATTACKING RSA: FACTORING CHALLENGE ◦ Brute forcing RSA requires prime factorization ◦ Monetary reward for cracking large RSA values ◦ RSA-XXX: XXX = number of bits ◦ RSA-768: Factored in December 2009 – 2 ½ years ◦ “On a single core 2.2 GHz AMD Opteron processor with 2 GB RAM, sieving would have taken about fifteen hundred years” Public Key Cryptography | RSA | Example | Exercise | Attacks
- 26. ATTACKING RSA - RANDOMNESS p = Randomly Chosen q = Randomly Chosen n = p and q λ(n) = p and q e = chosen from p, q d = found from e, λ(n) Public Key Cryptography | RSA | Example | Exercise | Attacks
- 27. ATTACKING RSA - RANDOMNESS ◦ “Random numbers” are actually pseudo-random ◦ Ways to generate “random” numbers ▫ Seed by time ▫ Seed by execution history - /dev/random ▫ Seed by atmospheric noise Public Key Cryptography | RSA | Example | Exercise | Attacks
- 28. Public Key Cryptography | RSA | Example | Exercise | Attacks
- 29. ATTACKING RSA - RANDOMNESS ◦ 2012 paper by Heninger et al. at USENIX ▫ “Mining your P’s and Q’s: Detection of Widespread Weak Keys in Network Devices” ◦ Plain terms: if the P/Q of two keys are the same, you can determine the other factor of both. ◦ How rare is this? Public Key Cryptography | RSA | Example | Exercise | Attacks
- 30. ATTACKING RSA - RANDOMNESS ◦ If you use poor randomness, common factors (somewhat) common! ◦ Heninger et al. harvested 5 million SSL keys ◦ Found high common factors in 0.5% of the keys (25,000 keys) ◦ Result: can compute the private keys of those 25,000! Public Key Cryptography | RSA | Example | Exercise | Attacks
- 31. ATTACKING RSA - RANDOMNESS ◦ /dev/urandom: supplies random bytes based on disk activity, non-blocking ◦ Why would disk activity be not-so-random on devices like these? Public Key Cryptography | RSA | Example | Exercise | Attacks
- 32. SUMMARY SLIDE Public Key Cryptography | RSA | Example | Exercise | Attacks ◦ Differences between Private and Public Key Cryptosystems ◦ Introduction to RSA ◦ Walkthrough of RSA at scale ◦ Attacks against RSA
Editor's Notes
- #12: Factoring = breaking a number into smaller numbers