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1 CS 255 Lecture 6 Hash Functions Brent Waters

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2 Recap-Notions of Security What attacker can do Random plaintext attack Chosen plaintext attack Chosen ciphertext attack Attacker’s Goal Discover secret key Decrypt a ciphertext, C * Distinguish two messages

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3 Recap- Notions of Security 3x3=9 possible notions of security Strongest system =Semantic security against CCA weakest adversary goal + most adversary power

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4 Recap- Semantic Security of Counter Mode 1) Defined notion of security for block cipher --Indistinguishable from PRP --Formal definition game --Believe this is true for AES…

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5 Recap- 2) Prove that if cipher is indist. from Random Permutation then counter mode is semantically secure against CPA attack --Assume counter mode is not ) A breaks it --Build algorithm B that uses algorithm A --Want to show that A’s answer gives B information to play his game

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6 Why do we do this? Aren’t we assuming AES, 3DES secure anyway? Why not just make same assumption for mode X? Reduce to simplest assumptions possible

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7 Hash Functions Hash function- h: {0,1} * \rightarrow {0,1} n typically n ¼ 160 bits (will see why soon) Hi, I recently….. …should be used h(x) 01100100…1

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8 Properties Compression Pre-image resistanc: Given y=h(x) difficult to determine x’ s.t. h(x’)=y 2 nd preimage resistance: Given x find x’ x s.t. h(x) = h(x’) Collision resistance: Find x’ x s.t. h(x)=h(x’)

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9 Relations If h is collision resistant then h is 2 nd order pre- image resistant How do we show this? Reduction—simple here

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10 Applications Show three applications and do one together For each one keep in mind what properties we need

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11 Password protection pword=jeitlse Password file U1=… U2=… What should we put in there? What if backup tape stolen? What property do we need

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12 Virus protection Worried virus might modify an application Small amount of trusted storage on USB token What properties do we need? Mirror sites distributing software

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13 Digital Signatures One party can sign a message M, many parties can verify Contract signing, code signing Raw signature scheme only signs messages ~160 bits What properties do we need?

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14 Birthday Attack for Collisions Let r 1, … r j 2 [0,1…B] When n=1.2 sqrt(B) then Pr[ 9 i j: r i =r j ] Pr[ 9 i j: r i =r j ] =1-Pr[ 8 i j:r i r j ] =1-(1-1/B)(1-2/B)...(1-(n-1)/B) =1- n-1 (1-i/B) ¼ 1- n-1 e -i/B =1-e 1/2n 2 /B =1-1/e.7 for n=1.2 sqrt(B) =1/2

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15 Lesson 80 bit hash implies 40 bit security (for collisions) Need 160 bit hash output For n integers have ¼ n^2 pairs each is a possibility for a collision

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16 Iterated Construction (Merkle-Damgard) M1M2M3M4pad IV ffff H0H0 H1H1 H2H2 H3H3 1.f – Compression function 2.H i – chaining variables 3.IV – Initial Value

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17 Iterated Construction (Merkle-Damgard) M1M2M3M4pad IV ffff H0H0 H1H1 H2H2 H3H3 Padding: 100000 | length Pad out last message block Add one block with message length

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18 Collision resistance If compression function resistant then so is iterated construction Way we prove this is to show if we have M M’ and hash(M)=hash(M’) then we can find two different inputs to compression function (x,y) and (x’,y’) such that f(x,y)=f(x,y) -Note (x,y) (x’,y’) if x x’ or y y’

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19 Collision Resistance Suppose h(M)=h(M’) IV=H 0, H 1,H 2....H t IV=H 0 ’, H 1 ’, H 2 ’...H r ’ Collision means H t = H r ’ Case I: Suppose t r then H t =H r ’ =f(H t-1, t)=f(H r-1 ’, r) ) collision!

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20 Collision Resistance Suppose h(M)=h(M’) M=M 0, M 1...M t-1, M’=M 0, M 1,... M r-1 IV=H 0, H 1, H 2....H t IV=H 0 ’, H 1 ’, H 2 ’...H r ’ Case 2: t r (Messages same # of blocks) Look at ith chaining variable Have H i =H i ’ so f(H i,M i )=f(H i ’,M i ’) if M i M i ’ or if H i H i ’ then have a collision otherwise repeat observation for i-1 chaining var. However, 9 j: M j M j ’ so must have a collision at some point

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21 Block cipher construction Matyas-Meyer f(M,H)=E(M,g(h)) © M E HiHi g MiMi © H i+1... Thm: Suppose E k (x) =E(X,K) is a collection of random permutations. Then finding a collision take 2 n/2 evaluations of E. Best possible.

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22 Customized Hash functions Merkle-Damgard types—compression function faster than block ciphers MD4128 Collisions found MD5128 28.5MB/s Collisons found SHA-1160 15.2MB/s SHA-2 160,256 RIPEMD 160 12.6 Collisions found

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23 “Provable” hash functions Discrete log problem: Given g a mod p Output a f(a,b)=g a h b mod p Slow

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24 Paper submission project Professors/grad students submit papers to conferences electronically Strict deadlines: 9pm Jan. 29 th People always wait to last minute – get flood of papers at end Graphics people send in videos – potentially GBs of data– no way server can handle them all

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25 Solutions? Attacks? Properties

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