Hashes and Message Digests Haipeng Dai haipengdai@nju.edu.cn 313 CS Building Department of Computer Science and Technology Nanjing University
Hashes and Message Digests Haipeng Dai haipengdai@nju.edu.cn 313 CS Building Department of Computer Science and Technology Nanjing University
Integrity vs.Secrecy Integrity:attacker cannot change message without being detected Encryption:attacker cannot read message correctly without key -Note:encryption does not guarantee integrity. Reason:attacker can change ciphertext arbitrarily and any ciphertext can be decrypted to get the corresponding plaintext(although possibly garbage). M network Alice Bob Learns M-attack on secrecy Changes M-attack on integrity 2
2 Integrity vs. Secrecy Integrity: attacker cannot change message without being detected Encryption: attacker cannot read message correctly without key ─ Note: encryption does not guarantee integrity. ─ Reason: attacker can change ciphertext arbitrarily and any ciphertext can be decrypted to get the corresponding plaintext (although possibly garbage). network Learns M – attack on secrecy Alice Bob M M Changes M – attack on integrity M’
Integrity Protection VIRUS badFile goodFile ①小e①ites BigFirmTM hash(goodFile) User Software manufacturer wants to ensure that the executable file is received by users without modification... Sends out the file to users and publishes its hash in NY Times The goal is integrity,not secrecy Idea:given goodFile and hash(goodFile), very hard to find badFile such that hash(goodFile)-hash(badFile) 3
3 Integrity Protection Software manufacturer wants to ensure that the executable file is received by users without modification… Sends out the file to users and publishes its hash in NY Times The goal is integrity, not secrecy Idea: given goodFile and hash(goodFile), very hard to find badFile such that hash(goodFile)=hash(badFile) goodFile BigFirm™ User VIRUS badFile The Times hash(goodFile)
Authentication Authenticity is identification and assurance of origin of information -We'll see many specific examples in different scenarios network 第 4
4 Authentication Authenticity is identification and assurance of origin of information ─ We’ll see many specific examples in different scenarios network
Authentication with Shared Secrets SECRET SECRET msg,H(SECRET,msg) Alice Bob Alice wants to ensure that nobody modifies message in transit (both integrity and authentication) Idea:given msg, very hard to compute H(SECRET,msg)without SECRET; easy with SECRET 5
5 Authentication with Shared Secrets Alice wants to ensure that nobody modifies message in transit (both integrity and authentication) Idea: given msg, very hard to compute H(SECRET, msg) without SECRET; easy with SECRET msg, H( SECRET, msg ) Alice Bob SECRET SECRET
Hash Functions:Main Idea Hash is also called message digest Arbitrary-length message to fixed-length digest message hash function H message digest bit strings of any length n-bit strings H is a lossy compression function -Collisions:h(x)=h(x)for some inputs x,x' -Result of hashing should "look random"(make this precise later) ●Intuition:half of digest bits are“I',any bit in digest is“l”half the time Cryptographic hash function needs a few properties... 6
6 Hash Functions: Main Idea Hash is also called message digest Arbitrary-length message to fixed-length digest H is a lossy compression function ─ Collisions: h(x)=h(x’) for some inputs x, x’ ─ Result of hashing should “look random” (make this precise later) ● Intuition: half of digest bits are “1”; any bit in digest is “1” half the time Cryptographic hash function needs a few properties… bit strings of any length n-bit strings . . . . . x’ x’’ x y’ y hash function H message digest message
Hashes and Message Digest One-wayness:given h(x),hard to find x -Also called“pre-image resistance” Collision resistance:hard to find x,x'such that h(x)=h(x') -Also called "strong collision resistance" -Brute-force collision search is only O(2n/2),not 2n -Birthday Paradox -n needs to be at least 128 ■ Weak collision resistance:given x,hard to find x'such that h(x)=h(x') -Also called“second pre-image resistance” -Brute-force attack requires O(2m)time 7
7 Hashes and Message Digest One-wayness: given h(x), hard to find x ─ Also called “pre-image resistance” Collision resistance: hard to find x, x’ such that h(x)=h(x’) ─ Also called “strong collision resistance” ─ Brute-force collision search is only O(2n/2), not 2n ─ Birthday Paradox ─ n needs to be at least 128. Weak collision resistance: given x, hard to find x’ such that h(x)=h(x’) ─ Also called “second pre-image resistance” ─ Brute-force attack requires O(2n) time
One-Wayness Intuition:hash should be hard to invert -"Preimage resistance" -Leth(x)=y∈{0，l}for a random x' -Given y,it should be hard to find x such that h(x)= ▣How hard? -Brute-force:try every possible x,see if h(x)=y -SHA-1 (common hash function)has 160-bit output .Assuming 264 trials per second,can do 289 trials per year Will take 271 years to invert SHA-1 on a random image 8
8 One-Wayness Intuition: hash should be hard to invert ─ “Preimage resistance” ─ Let h(x’) = y ∈{0,1}n for a random x’ ─ Given y, it should be hard to find x such that h(x)=y How hard? ─ Brute-force: try every possible x, see if h(x)=y ─ SHA-1 (common hash function) has 160-bit output ● Assuming 264 trials per second, can do 289 trials per year ● Will take 271 years to invert SHA-1 on a random image
Collision Resistance (CR) Should be hard to find x,x'such that h(x)h(x) -Brute-force collision search is only O(22),not O(2") -For SHA-1,this means O(280)vs.O(2160) 9
9 Collision Resistance (CR) Should be hard to find x, x’ such that h(x)=h(x’) ─ Brute-force collision search is only O(2n/2), not O(2n) ─ For SHA-1, this means O(280) vs. O(2160)
Birthday Paradox Problem Birthday Problem:What is the minimum number of persons in a room so that the odds are better than 50%that two of them will have the same birthday?23 persons. General Problem: Given a random variable that is an integer with uniform distribution between 1 and n and a selection of k instances(ksn)of the random variable,what is the probability,P(n,k),that there is at least one duplicate? How many possibilities? -nk(samples with duplicate) n(n-1)...(n-k+1)(samples without duplicate) Probability of no repetition? nx(n-1)x.×(n-k+l) n ·msw。三卧】 Because 1-x 0.5? P(n,k)-1-e-k(k-by2n>1/2 →e-2mln2Note:for large k,kk-l)≈k2 →k2/2>ln2→k>√2ln2)n≈l.18vn 0.2 Ifn=365,we get k>1.18x√365=22.54 02 04 0.6 0.8 10
10 Birthday Paradox Problem Birthday Problem: What is the minimum number of persons in a room so that the odds are better than 50% that two of them will have the same birthday? 23 persons. General Problem: Given a random variable that is an integer with uniform distribution between 1 and n and a selection of k instances (k≤n) of the random variable, what is the probability, P(n, k), that there is at least one duplicate? How many possibilities? ─ nk (samples with duplicate) ─ n(n-1)…(n-k+1) (samples without duplicate) Probability of no repetition? Thus Because 1-x ≤e-x (x≥0), we have What value of k is required such that P(n, k)>0.5? P(n, k)= ⇒ ⇒ ⇒ Note: for large k, k(k-1) ≈k2 ⇒ ⇒ If n=365, we get k n n×(n−1)×...×(n−k+1) − × × − × − = − − − + × × − × − = − × − × × − + = − n k n n n n k n n n n n n n n k P n k k 1 ... 1 2 1 1 1 1 1 ... 1 2 1 ( 1) ... ( 1) ( , ) 1 1 1/2 ( 1)/2 − > −k k− n e 1/2 ( 1)/2 ln2 /2 ln2 2 k n> k> 2(ln2)n≈1.18 n k>1.18× 365=22.54
