Enhancing RSA Security: Attacks, Vulnerabilities, and the OAEP Function

1. Attacks on RSA. Safe modes.

2. From the previous lecture… Secret B p, q, n:=pq

3. Textbook RSA (V) • Security: • Is encryption secure? • Can we recover the secret key ? Key recovery as hard as factorizing • Can we recover in anyotherway ? Values are long-term Deterministic Each maps to unique

4. Textbook RSA (VI) • Security: • IND-CPA/IND-CCA: can’t say anything about Encryption is deterministic: Can always distinguish m from m’ • Plaintext recovery: can’t find from Not guaranteed if few possible messages Try out all alternatives – find plaintext OK if chosen at random from large set • Not very secure; but we can improve it

5. Textbook RSA ++ • Improving Textbook RSA: pre-processing RSA encryption Secret pre-processing Security will depend on this step

6. PKCS and Bleichenbacher • Preprocessing with PKCS1, mode 2 • Pad with random number (make it probabilistic) 00 02 random pad FF message 1024 bits • Bleichenbacher ’98: use the regularity of the ciphertext (they must start with “00|02”) to recover plaintext!

7. PKCS and Bleichenbacher (II) • Core idea Ciphertext Continue ERROR! Decrypt Does m start with “00|02”? • Attacker starts with ciphertext • Re-randomize it: • Is it PKCS? Repeat until you know rM starts with 00|02 • Move to next part of message ciphertexts

8. Contents • Pre-processing • How OAEP works • Hash Functions; Random Oracles (brief) • Improvements on OAEP • Attacks on factoring – generic • Pollard’s • Pollard- • Unsafe modes for RSA • Small pk and related ciphertexts • Small sk: Wiener’s attack • Some physical attacks

9. The OAEP Function • A new pre-processing function: OAEP • OAEP = Optimal Asymmetric Encryption Padding • By Bellare & Rogaway, 1994; in RFC 2437 bits bits bits K = size of n=pq m pad r = parameters (to be set) G,H = hash functions G = bit XOR H X Y

10. The OAEP Function • In detail: OAEP m pad r G • Hash functions • A box with input of any size, and output of fixed size In this case: input is bits, output is • Collision-resistance: can’t find with • Random oracles: always outputs new string Outputs consistently: consistent

11. The OAEP Function • In detail: OAEP m pad r G • How it works: r G random bits bits = m pad

12. The OAEP Function • In detail: OAEP r H • How it works: H random bits bits = r

13. RSA-OAEP Decryption • are random oracles Hard to invert • How do we decrypt? • Go in reverse: receive Decrypt: m pad r G H X Y

14. RSA-OAEP Decryption • are random oracles Hard to invert • How do we decrypt? • Go in reverse: receive Decrypt: H = r

15. RSA-OAEP Decryption • are random oracles Hard to invert • How do we decrypt? • Go in reverse: receive Decrypt: Recover: m pad r G H X Y

16. RSA-OAEP Decryption • are random oracles Hard to invert • How do we decrypt? • Go in reverse: receive Decrypt: Recover: r G = m pad

17. RSA-OAEP Decryption • are random oracles Hard to invert • How do we decrypt? • Go in reverse: receive Decrypt: Recover: Retrieve: Check: pad has the right format

18. The OAEP Function • In detail: OAEP • Functions are random oracles: thatis, theygiverandom output. In practice: use SHA-1 • Randomness chosenfreshlyevery time • How about the padding? m pad r • Original OAEP: ([BR94]) • OAEP+: with W a random oracle ([S01])

19. Improving OAEP: SAEP bits bits bits m W(m,r) r H X Y • No need for function • Function israndom oracle. Input size: bits. Output size: bits

20. Contents • Pre-processing • How OAEP works • Hash Functions; Random Oracles (brief) • Improvements on OAEP • Generic attacks on factoring • Small Small or • Pollard- • Unsafe modes for RSA • Small pk and related ciphertexts • Small sk: Wiener’s attack • Some physical attacks

21. Attacks on RSA • For the remainder of this lecture = We • 1st goal: • Given something of the form , find • Strategies: • Generic: factor . Given, easy to recover • Specific: retrieve plaintext without factoring

22. Small • Easy case: we are given and • If are prime, then • Given and Calculate: This gives: Also: So: and: Factorization: and

23. Small • Hard case: we are given only Try to guess Use: Then: Note: are odd. Thus: and are even Algorithm SmallDiff: Input Complexity parameter Write Let . While DO IF is a square (itisequal to for a positive integer) THEN: if and are prime, Output and ELSE:

24. Small or : Pollard’s • Attack on factoring – bad (p-1) • Vulnerability: with one small prime • Pollard’s-(p-1) factors in steps if smallest factor If is small, then this method is fast • Idea: if is prime, thenis not Since all are odd (impair), iseven Obviously will have 2 as a factor We are hoping has onlysmallfactors and wewilltry to retrievethem all All in the same set

25. Small or : Pollard’s • Attack on factoring – bad (p-1) • Vulnerability: with one small prime • Supposition: • How large can be for each? Well, for any , so • Start with definite upper bound: As , anydivides. So divides So

26. Small or : Pollard’s • Attack on factoring – bad (p-1) • Vulnerability: with one small prime As , anydivides. So divides So Pick random Check that • If : then. Hooray! • If and • With high probability • Then • Else, pick a new a

27. Exercise time! • Write pseudocode for Pollard’s

28. So far • Small • Given and : calculate Take: Factorization: and • Given : verify values of for integer For each check if isinteger If so, if are prime then: Output Else, next and repeatprocedure

29. So far • Small Pick random Check that • If : then. Hooray! • If and • With high probability • Then • Else, pick a new a and repeat

30. Pollard’s • General factorization attack (are we lucky?) • Strategy: find specific small suchthat Most likely then, • Imagine we could calculate Say wehad: • Suppose we find suchthat, then: divides Then with high probability • But, we don’t know . We do this.

31. Pollard’s • Strategy: we compute: • Find: suchthat • With high probability • Choice: speed vs. storage • Storage: method as above. Need to store all • Speed: Floyd’s cycle finding algorithm: • and • Mod n: Only checking pairs at a time

32. Floyd’s Cycle-Finding Alg. Source:http://home.online.no/~vlaenen/

33. Exercise time! • Put the method (with Floyd’s cycle-finding algorithm) in pseudocode/algorithm form!

34. Contents • Pre-processing • How OAEP works • Hash Functions; Random Oracles (brief) • Improvements on OAEP • Generic attacks on factoring • Small Small or • Pollard- • Unsafe modes for RSA • Small pk and related ciphertexts • Small sk: Wiener’s attack • Some physical attacks

35. Unsafe Modes for RSA • Small public key • More receivers with same small (different) • Same plaintext is sent to users

36. Unsafe Modes for RSA • Small public key • One receiver with small (different) • Two related plaintexts: and • If knows the relationship of the messages, she can use polynomial multiplication to find • Recommended • e = • This leads to fast encryption

37. More Unsafe Modes • Small secret key • Better for decryption: makes it more efficient • Use: least common multiple LCM Math “magic” Divide by dpq

38. More Unsafe Modes • Small secret key • If issmall, then. • If issmall, then. Tend to 0 • This means that converges towards • Continued fractions and some trial and error gives d

39. Physical Attacks • Implementation: Square and Multiply • Standard way to do exponentiation • Write in binary []. Set • For DO: Square AND Multiply • If then set Square • Else, set • Example:

40. Physical Attacks • Implementation: Square and Multiply • Timing attack: multiply takes longer than square • Time the operation and write out the order of ops M, Sq, Sq, M, Sq, Sq, M, Sq, M, Sq, Sq, M • Retrieve key from inverse Square and Multiply • Power attack: multiply burns more than square • Retrieve for smartcards Source: http://www.dbs.com.hk/

41. Thanks!