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Hashing

David Evans

http://www.cs.virginia.edu/evans

Note: only 3

people (out of 4) have

voted that notes are

useful. I won’t make notes (regularly) until at least 10 people do.

CS588: Security and Privacy

University of Virginia

Computer Science

Remote Coin Flipping (Ch 1)

Picks random x

Bob

Alice

f (x)

Picks “odd”

or “even”

Alice wins

if x does

not match

Bob’s pick

“odd” or “even”

Checks

f (x) matches

value received

in step 1

x

University of Virginia CS 588

Magic Function f

- One Way:
- For every integer x, easy to compute f(x)
- Given f (x), hard to find any information about x

- Collision Resistant:
- “Impossible” to find pair (x, y) where x y and f (x)=f (y)

University of Virginia CS 588

Normal CS Hashing

“dog”

“neanderthal”

“horse”

H (char s[]) = (s[0] – ‘a’) mod 10

University of Virginia CS 588

Regular Hash Functions

- Many-to-one: maps a large number of values to a small number of hash values
- Even distribution: for typical data sets, P(H(x) = n) = 1/N where N is the number of hash values and n = 0 .. N – 1.
- Efficient: H(x) is easy to compute.

How well does

H (char s[]) = (s[0] – ‘a’) mod 10

satisfy these properties?

University of Virginia CS 588

Cryptographic Hash Functions

- One-way: for given h, it is hard to find x such that H(x) = h.
- Collision resistance:
Weak collision resistance: given x, it is hard to find y x such that H(y) = H(x).

Strong collision resistance: it is hard to find any x and y x such that H(y) = H(x).

University of Virginia CS 588

Fair Remote Coin Flipping?

What goes wrong if f is

not one-way?

What goes wrong if f is not weak collision resistant?

What goes wrong if f is not strong collision resistant?

Picks random x

Bob

Alice

f (x)

Picks “odd”

or “even”

Alice wins

if x does

not match

Bob’s pick

“odd” or “even”

Checks

f (x) matches

value received

in step 1

x

University of Virginia CS 588

Using Hashes

- Alice wants to send Bob and “I owe you” message.
- Bob should be able to show the message to a judge to compel Alice to pay up.
- Bob should not be able to make his own “I owe you” from Alice, or change the contents of the one she sent him.

University of Virginia CS 588

IOU Protocol (Attempt 1)

M

H(M)

Bob

Alice

M

H(M)

Hmmm...Bob can just make up M and H(M)!

Judge

University of Virginia CS 588

IOU Protocol (Attempt 2)

M

EKA[H(M)]

Bob

Alice

secret key KA

M

EKA[H(M)]

Can Bob cheat?

Shared secret KA

Can Alice cheat?

Yes, send Bob: M, junk.

Judge will think Bob cheated!

Judge

knows KA

University of Virginia CS 588

IOU Protocol (Attempt 3)

M

EKRA[H(M)]

Bob

Alice

knows KUA

{KUA, KRA}

M

EKRA[H(M)]

Why not just

use EKRA[M]?

Bob can verify H(M) by decrypting, but cannot forge M, EKRA[H(M)] pair without knowing KRA.

Known public-key encyrption algorithms are slow

Judge

knows KUA

University of Virginia CS 588

No Collision Resistance

- Suppose we use: H (char s[]) = (s[0] – ‘a’) mod 10
- Alice sends Bob:
“I, Alice, owe Bob $2.”, EKRA[H (M)]

- Bob sends Judge:
“I, Alice, owe Bob $2000000.”, EKRA[H (M)]

- Judge validates
EKUA[EKRA[H (M)]] = H(“I, Alice, owe Bob $2000000.”)

and makes Alice pay.

University of Virginia CS 588

Weak Collision Resistance

- Given x, it should be hard to find y x such that H(y) = H(x).
- Similar to a block cipher except no need for secret key:
- Changing any bit of x should change most of H(x).
- The mapping between x and H(x) should be confusing (complex and non-linear).

University of Virginia CS 588

A Better Hash Function?

- H(x) = DES (x, 0)
- Weak collision resistance?
- Given x, it should be hard to find y x such that H(y) = H(x).
- Yes – DES is one-to-one. (These is no such y.)

- A good hash function?
- No, its output is as big as the message!

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What we need:

- Produce small number of bits (say 64) that depend on the whole message in a confusing, non-linear way.
- Have we seen anything like this?

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Cipher Block Chaining

Pn

P2

P1

IV

...

DES

K

DES

DES

K

K

Cn

C2

C1

Use last ciphertext block as hash. Depends on all plaintext blocks.

University of Virginia CS 588

Actual Hashing Algorithms

- Based on cipher block chaining
- No need for secret key or IV (just use 0)
- Don’t use DES
- Performance
- Better to use bigger blocks

- MD5 [Rivest92] – 512 bit blocks, produces 128-bit hash
- SHA [NIST95] – 512 bit blocks, 160-bit hash

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Why big hashes?

- 3DES is (probably) secure with 64-bit blocks, why do secure hash functions need at least 128 bit digests?
- 64 bits is fine for weak collision resistance, but we need strong collision resistance too.

University of Virginia CS 588

Strong Collision Resistance

- It is hard to find any x and y x such that H(y) = H(x).
- Difference from weak:
- Attacker gets to choose both x and y, not just y.

- Scenario:
- Suppose Bob gets to write IOU message, send it to Alice, and she signs it.

University of Virginia CS 588

Cryptographic Hash Functions

- Many-to-one: compresses
- Even distribution: P(H(x) = n) = 1/N
- Efficient: H(x) is easy to compute.
- One-way: given H(x), hard to find x
- Collision resistance:
Weak collision resistance: given x, it is hard to find y x such that H(y) = H(x).

Strong collision resistance: it is hard to find any x and y x such that H(y) = H(x).

University of Virginia CS 588

IOU Request Protocol

x

EKRA[H(x)]

Bob

Alice

knows KUA

{KUA, KRA}

y

EKRA[H(x)]

Bob picks x and y such that H(x) = H(y).

Judge

knows KUA

University of Virginia CS 588

Finding x and y

Bob generates 210 different agreeable (to Alice) xi messages:

I, { Alice | Alice Hacker | Alice P. Hacker | Ms. A. Hacker }, { owe | agree to pay } Bob { the sum of | the amount of } { $2 | $2.00 | 2 dollars | two dollars } { by | before } { January 1st | 1 Jan | 1/1 | 1-1 } { 2006 | 2006 AD}.

University of Virginia CS 588

Finding x and y

Bob generates 210 different agreeable (to Bob) yi messages:

I, { Alice | Alice Hacker | Alice P. Hacker | Ms. A. Hacker }, { owe | agree to pay } Bob { the sum of | the amount of } { $2 quadrillion | $2000000000000000 | 2 quadrillion dollars | two quadrillion dollars } { by | before } { January 1st | 1 Jan | 1/1 | 1-1 } { 2006 | 2006 AD}.

University of Virginia CS 588

Bob the Quadrillionaire!?

- For each message xi and yi, Bob computes hxi = H(xi) and hyi = H(yi).
- If hxi = hyjfor some i and j, Bob sends Alice xi, gets EKRA[H(x)]back.
- Bob sends the judge yjand EKRA[H(xi)].
- Is this different from when Alice chooses x?

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Chances of Success

- Hash function generate 64-bit digest (n = 264)
- Hash function is good (randomly distributed and diffuse)
- Chance a randomly chosen message maps to a given hash value: 1 in n = 2-64
- By hashing m good messages, chance that a randomly chosen bad message maps to one of the m different hash values: m * 2-64
- By hashing m good messages and m bad messages: m * m * 2-64 (approximation)

University of Virginia CS 588

Is Bob a Quadrillionaire?

- m = 210
- 210 * 210 * 2-64 = 2-44 (still a pauper)
- Try m= 232
- 232 * 232 * 2-64 = 20 = 1 (yippee!)
- Flaw: some of the messages might hash to the same value, might need more than 232 to find match.

University of Virginia CS 588

Birthday “Paradox”

What is the probability that two people in this room have the same birthday?

Text, Chapter 3.6

University of Virginia CS 588

Birthday Paradox

Ways to assign k different birthdays without duplicates:

N = 365 * 364 * ... * (365 – k + 1)

= 365! / (365 – k)!

Ways to assign k different birthdays with possible duplicates:

D = 365 * 365 * ... * 365 = 365k

University of Virginia CS 588

Birthday “Paradox”

Assuming real birthdays assigned randomly:

N/D = probability there are no duplicates

1 - N/D = probability there is a duplicate

= 1 – 365! / ((365 – k)!(365)k)

University of Virginia CS 588

Generalizing Birthdays

n!

(n – k)! nk

P(n, k) = 1 –

Given k random selections from n possible values, P(n, k) gives the probability that there is at least 1 duplicate.

University of Virginia CS 588

Birthday Probabilities

P(no two match) = 1 – P(all are different)

P(2 chosen from N are different)

= 1 – 1/N

P(3 are all different)

= (1 – 1/N)(1 – 2/N)

P(n trials are all different)

= (1 – 1/N)(1 – 2/N) ... (1 – (n – 1)/N)

ln (P)

= ln (1 – 1/N) + ln (1 – 2/N) + ... ln (1 – (k – 1)/N)

University of Virginia CS 588

Happy Birthday Bob!

ln (P) = ln (1 – 1/N) + ... + ln (1 – (k – 1)/N)

For 0 < x < 1: ln (1 – x) x

ln (P) – (1/N + 2/N + ... + (n – 1)/N)

Gauss says:

1 + 2 + 3 + 4 + ... + (n – 1) + n = ½ n (n + 1)

So,

ln (P) ½ (k-1) k/N

Pe½ (k-1)k / N

Probability of match 1 –e½ (k-1)k / N

University of Virginia CS 588

Applying Birthdays

P(n, k) > 1 – e-k*(k-1)/2n

- For n = 365, k = 20:
P(365, 20) > 1 – e-20*(19)/2*365

P(365, 20) > .4058

- For n = 264, k = 232: P (264, 232) > .39
- For n = 264, k = 233: P (264, 233) > .86
- For n = 264, k = 234: P (264, 234) > .9996

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Is 128 bits enough?

- For n = 2128, k = 240: P (2128, 240) > 10-15
- If your guesses are random, need to try 240 inputs to have a 10-15 chance of finding a collision
- Assumes you hash function is perfect

University of Virginia CS 588

A Most Disturbing Program!

From http://www.freedom-to-tinker.com/archives/000664.html

#!/usr/bin/perl -w

use strict;

use Digest::MD5 qw(md5_hex);

# Create a stream of bytes from hex.

my @bytes1 = map {chr(hex($_))} qw(d1 31 dd 02 c5 e6 ee c4 69 3d 9a 06 98 af f9 5c 2f ca b5 87 12 46 7e ab 40 04 58 3e b8 fb 7f 89 55 ad 34 06 09 f4 b3 02 83 e4 88 83 25 71 41 5a 08 51 25 e8 f7 cd c9 9f d9 1d bd f2 80 37 3c 5b d8 82 3e 31 56 34 8f 5b ae 6d ac d4 36 c9 19 c6 dd 53 e2 b4 87 da 03 fd 02 39 63 06 d2 48 cd a0 e9 9f 33 42 0f 57 7e e8 ce 54 b6 70 80 a8 0d 1e c6 98 21 bc b6 a8 83 93 96 f9 65 2b 6f f7 2a 70);

my @bytes2 = map {chr(hex($_))} qw(d1 31 dd 02 c5 e6 ee c4 69 3d 9a 06 98 af f9 5c 2f ca b5 07 12 46 7e ab 40 04 58 3e b8 fb 7f 89 55 ad 34 06 09 f4 b3 02 83 e4 88 83 25 f1 41 5a 08 51 25 e8 f7 cd c9 9f d9 1d bd 72 80 37 3c 5b d8 82 3e 31 56 34 8f 5b ae 6d ac d4 36 c9 19 c6 dd 53 e2 34 87 da 03 fd 02 39 63 06 d2 48 cd a0 e9 9f 33 42 0f 57 7e e8 ce 54 b6 70 80 28 0d 1e c6 98 21 bc b6 a8 83 93 96 f9 65 ab 6f f7 2a 70);

# Print MD5 hashes

print md5_hex(@bytes1), "\n", md5_hex(@bytes2), "\n";

79054025255fb1a26e4bc422aef54eb4

79054025255fb1a26e4bc422aef54eb4

University of Virginia CS 588

Hash Collisions

- Collisions announced in SHA-0 at Crypto 2004
- No collisions yet found in SHA-1 (which replaced SHA-0 as a standard in 1994)
- NIST is nervous http://csrc.nist.gov/hash_standards_comments.pdf

University of Virginia CS 588

NIST Comments

- “At the recent Crypto2004 conference, researchers announced that they had discovered a way to "break" a number of hash algorithms, including MD4, MD5, HAVAL-128, RIPEMD and the long superseded Federal Standard SHA-0 algorithm. The current Federal Information Processing Standard SHA-1 algorithm, which has been in effect since it replaced SHA-0 in 1994, was also analyzed, and a weakened variant was broken, but the full SHA-1 function was not broken and no collisions were found in SHA-1. The results presented so far on SHA-1 do not call its security into question. However, due to advances in technology, NIST plans to phase out of SHA-1 in favor of the larger and stronger hash functions (SHA-224, SHA-256, SHA-384 and SHA-512) by 2010.”

University of Virginia CS 588

Charge

We’ll cover SSL

after Spring Break…

but, this should make you nervous…

Wednesday 3:30

Chenxi Wang Seminar

“Defending against Large Scale Attacks on the Internet”

Thursday 9:30 (please arrive on time for class, not like usual!)

Chenxi Wang guest lecture

Using hashes to provide censorship-resistant publishing

University of Virginia CS 588

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