1 / 34

Hellman’s TMTO Attack

Hellman’s TMTO Attack. Popcnt. Before we consider Hellman’s attack, consider simpler T ime- M emory T rade- O ff “Population count” or popcnt Let x be a 32-bit integer Define popcnt( x ) = number of 1 ’s in binary expansion of x How to compute popcnt( x ) efficiently?.

kin
Download Presentation

Hellman’s TMTO Attack

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Hellman’s TMTO Attack Hellman’s TMTO 1

  2. Popcnt • Before we consider Hellman’s attack, consider simpler Time-Memory Trade-Off • “Population count” or popcnt • Let x be a 32-bit integer • Define popcnt(x) = number of 1’s in binary expansion of x • How to compute popcnt(x) efficiently? Hellman’s TMTO 2

  3. Simple Popcnt • Most obvious thing to do is popcnt(x) // assuming x is 32-bit value t = 0 for i = 0 to 31 t = t + ((x >> i) & 1) next i return t end popcnt • Is this the most efficient method? Hellman’s TMTO 3

  4. More Efficient Popcnt • Pre-compute popcnt for all 256 bytes • Store pre-computed values in a table • Given x, lookup its bytes in this table • Sum these values to find popcnt(x) • See next slide for pseudo-code… Hellman’s TMTO 4

  5. More Efficient Popcnt Initialize: table[i] = popcnt(i) for i = 0,1,…,255 popcnt(x) // assuming x is 32-bit word p = table[x & 0xff] + table[(x >> 8) & 0xff] + table[(x >> 16) & 0xff] + table[(x >> 24) & 0xff] return p end popcnt Hellman’s TMTO 5

  6. Popcnt TMTO • Advantage(s)? • Each popcnt now requires 4 steps, not 32 • Disadvantage? • Pre-computation (one time work) • Additional storage • Balance “time” vs “memory” where… • Time == popcnt computation(s) • Memory == storage/pre-computation Hellman’s TMTO 6

  7. TMTO Basics • Pre-computation • One-time work • Results stored in a table • Pre-computation results used to make each subsequent computation faster • In general, larger pre-computation requires more initial work and larger “memory” • But then each computation takes less “time” • Try to balance “memory” and “time” Hellman’s TMTO 7

  8. Block Cipher Notation • Consider a block cipher C = E(P, K) where P is plaintext block of size n C is ciphertext block of size n K is key of size k Hellman’s TMTO 8

  9. Block Cipher as Black Box • For TMTO, treat block cipher as black box • Details of crypto algorithm not relevant Hellman’s TMTO 9

  10. Hellman’s TMTO Attack • This is a chosen plaintext attack • We choose P • Then we are given corresponding C • That is, C = E(P, K) • We want to find the key K Hellman’s TMTO 10

  11. TMTO Attack • Given: P and C = E(P, K) • Find: key K • Two “obvious” approaches • Exhaustive key search “Memory” is 0, but “time” of 2k-1 for each attack • Pre-computeC = E(P, K)for all keysK Then given any C, simply look up key K in table “Memory” of 2k but “time” of 0 per attack • TMTO lies between 1. and2. Hellman’s TMTO 11

  12. Chain of Encryptions • For simplicity, we assume block length n and key length k are equal, that is, n = k • Attack still works if this does not hold • Define a chain of encryptions as SP = K0 =Starting Point K1 = E(P, SP) K2 = E(P, K1) : : EP = Kt = E(P, Kt1) =End Point Hellman’s TMTO 12

  13. Encryption Chain • Note that ciphertext from one iteration is used as key at next iteration • Note also that the same (chosen) plaintextP used at each iteration Hellman’s TMTO 13

  14. Pre-computation • Pre-compute m encryption chains, each of length t +1 • Save only the start and end points (SP0, EP0) (SP1, EP1) : (SPm-1, EPm-1) EP0 SP0 EP1 SP1 EPm-1 SPm-1 Hellman’s TMTO 14

  15. TMTO Attack • Memory: Pre-compute encryption chains and save (SPi, EPi) fori = 0,1,…,m1 • This is one-time work • Sort based on EPi • For each attack for unknown key K do… • We have P and C = E(P, K) where P is same plaintext used to compute chains • Time: Compute a chain (max of t steps) X0 = C, X1 = E(P, X0), X2 = E(P, X1),… Hellman’s TMTO 15

  16. TMTO Attack • Consider the computed chain X0 = C, X1 = E(P, X0), X2 = E(P, X1), … • Suppose for some i we find Xi= EPj EPj C SPj K • Since C= E(P, K) key K appears to lie just before ciphertext C in this chain! Hellman’s TMTO 16

  17. TMTO Attack • Summary of attack phase: compute X0 = C, X1 = E(P, X0), X2 = E(P, X1),… • If for some i we find Xi= EPj then reconstruct chain from SPj Y0 = SPj, Y1 = E(P,Y0), Y2 = E(P,Y1),… • We find C = Yti = E(P, Yti1) (always?) • So that K =Yti1 (always?) Hellman’s TMTO 17

  18. Trudy’s Perfect World • Suppose block cipher has k = 56 • That is, the key length is 56 bits • Suppose we can find m = 228 chains each of length t = 228 with no overlap • Is this realistic? • Then all 256 possible keys covered by exactly one element of a chain • This looks promising… Hellman’s TMTO 18

  19. Trudy’s Perfect World • If we can find m = 228 chains each of length t = 228 and no intersection, then… • Memory:228 pairs (SPj, EPi) • Time: about 228 (per attack) • Start at C, find some EPj in about 227 steps • Find K with about 227 more steps • So, work per attack is about 228 • Trivial work and attack never fails! • For Trudy, life doesn’t get any better… Hellman’s TMTO 19

  20. Trudy’s Perfect World • No chains intersect • Every ciphertext C is in one chain SP0 EP0 C EP1 SP1 K EP2 SP2 Hellman’s TMTO 20

  21. The Real World • Real chains are not so well-behaved • Chains can cycle and merge K C EP SP • Chain beginning at C goes to EP • But chain from SP to EP does not give K • This looks like Trudy’s nightmare… Hellman’s TMTO 21

  22. The Real World • Chains can cycle and merge K C EP SP • This adversely affects attacks, as mentioned on previous slide • It also, adversely affects pre-computation and probability of success • Why? Hellman’s TMTO 22

  23. Real-World TMTO Issues • Merging chains, cycles, false alarms, … • Pre-computation is lots of work • Must attack many times to amortize cost • Success is not assured • Depends on initial work, merging, cycles, … • What if block size not equal key length? • This is easy to deal with • What is the probability of success? • This is not so easy to compute… Hellman’s TMTO 23

  24. To Reduce Merging • Compute chain asF(E(P, Ki1)) where F permutes the bits • Then we have F(E(P, K)) =F(C) • Chains computed using different functions can intersect, but will not merge SP0 F0 chain EP1 SP1 F1 chain EP0 Hellman’s TMTO 24

  25. Hellman’s TMTO in Practice • Let m = random starting points for each F t = encryptions in each chain r = number of “tables”, i.e., functions F • Then mtr pre-computed chain elements • Pre-computation is about mtr work • Each TMTO attack requires • About mr “memory” (storage), and tr “time” • If we choose m = t = r = 2k/3 then probability of success is at least 0.55 Hellman’s TMTO 25

  26. Success Probability? • Throw n balls into m urns • Expected number of urns that have at least one ball? • This is classic “occupancy problem” • See Feller, Intro. to Probability Theory • Why is this relevant to TMTO ? • “Urns” correspond to keys • “Balls” correspond to constructing chains Hellman’s TMTO 26

  27. Success Probability • Using occupancy problem approach… • Assuming k-bit key and m,t,r as defined on previous slide • Can choose m = t = r but not necessary • Then, approximately, P(success) = 1 emtr/k • An upper bound can be given that is slightly “better” Hellman’s TMTO 27

  28. Success Probability • Success probability P(success) = 1 emtr/k Hellman’s TMTO 28

  29. Distributed TMTO • Employ “distinguished points” • Do not use fixed-length chains • Instead, compute chain until some distinguished point is found • Example of distinguished point: Hellman’s TMTO 29

  30. Distributed TMTO • Similar pre-computation as before, except we have triples: (SPi, EPi,li) fori = 0,1,…,rm • Where li is the length of the chain • And r is number of tables • And m is number of random starting points • Let Mi be the maximum lj for the ith table • Each table has a fixed random function F Hellman’s TMTO 30

  31. Distributed TMTO • Suppose r computers are available • Each computer deals with one table • So, only one random function F per computer • “Master” gives computer i values Fi, Mi, C • And definition of distinguished point • Computer i computes chain beginning with C and using Fi of (at most) length Mi Hellman’s TMTO 31

  32. Distributed TMTO • If computer i finds a distinguished point within Mi steps • Returns result to “master” for secondary test • Master searches for K on corresponding chain, using same approach as in non-distributed TMTO • False alarms can occur (e.g., distinguished points) • If no distinguished point found in Mi steps • Computer i gives up, since key not on any Fi chains • Note that computer i does not need to know any SP or EP or l • Master knows (SPi, EPi,li) fori = 0,1,…,rm Hellman’s TMTO 32

  33. TMTO for DES • Attack is feasible against DES • Recall that for DES, we have k = 56 • Assume that we use m= t= r= 2k/3 ≈ 218.67 • Then pre-computation is about 256work • And we store about 237 starting/end point pairs • So, each time attack is conducted… • …use about237“memory” and requires 237 “time” Hellman’s TMTO 33

  34. TMTO: The Bottom Line • Attack is not specific to DES • Applies to any block cipher • Inner working of cipher is not relevant • But key length is… • No fancy math required • Just a clever algorithm • Lesson: Clever algorithms can break crypto! Hellman’s TMTO 34

More Related