Cryptography: Understanding Hash Functions and Their Role in Security, Slides of Computer Science

Download Cryptography: Understanding Hash Functions and Their Role in Security and more Slides Computer Science in PDF only on Docsity!

Part 1  Cryptography 1

Hash Functions

Part 1  Cryptography 2

Hash Function Motivation

 Suppose Alice signs M

o Alice sends M and S = [M]

Alice

to Bob

o Bob verifies that M = {S}

Alice

o Is it OK to just send S?

 If M is big, [M]

Alice

is costly to compute

 Suppose instead, Alice signs h(M), where

h(M) is much smaller than M

o Alice sends M and S = [h(M)]

Alice

to Bob

o Bob verifies that h(M) = {S}

Alice

Part 1  Cryptography 4

Pre-Birthday Problem

 Suppose N people in a room

 How large must N be before the

probability someone has same

birthday as me is  1/

o Solve: 1/2 = 1  (364/365)

N

for N

o Find N = 253

Part 1  Cryptography 5

Birthday Problem

 How many people must be in a room before

probability is  1/2 that two or more have

same birthday?

o 1  365/365  364/365   (365N+1)/

o Set equal to 1/2 and solve: N = 23

Part 1  Cryptography 7

Non-crypto Hash (1)

 Data X = (X

0

,X

1

,X

2

,…,X

n-

), each X

i

is a byte

 Spse hash(X) = X

0

+X

1

+X

2

+…+X

n-

 Is this secure?

 Example: X = (10101010,00001111)

 Hash is 10111001

 But so is hash of Y = (00001111,10101010)

 Easy to find collisions, so not secure…

Part 1  Cryptography 8

Non-crypto Hash (2)

 Data X = (X

0

,X

1

,X

2

,…,X

n-

 Suppose hash is

o h(X) = nX

0

+(n-1)X

1

+(n-2)X

2

+…+1X

n-

 Is this hash secure?

 At least

o h(10101010,00001111)h(00001111,10101010)

 But hash of (00000001,00001111) is same

as hash of (00000000,00010001)

 Not one-way, but this hash is used in the

(non-crypto) application rsync

Part 1  Cryptography 10

Popular Crypto Hashes

 MD5  invented by Rivest

o 128 bit output

o Note: MD5 collision recently found

 SHA-1  A US government standard

(similar to MD5)

o 160 bit output

 Many others hashes, but MD5 and SHA-

most widely used

 Hashes work by hashing message in blocks

Part 1  Cryptography 11

Crypto Hash Design

 Desired property: avalanche effect

o Change to 1 bit of input should affect about

half of output bits

 Crypto hash functions consist of some

number of rounds

 Want security and speed

o Avalanche effect after few rounds

o But simple rounds

 Analogous to design of block ciphers

Part 1  Cryptography 13

Hash Uses

 Authentication (HMAC)

 Message integrity (HMAC)

 Message fingerprint

 Data corruption detection

 Digital signature efficiency

 Anything you can do with symmetric crypto

Part 1  Cryptography 14

Online Auction

 Suppose Alice, Bob and Charlie are bidders

 Alice plans to bid A, Bob B and Charlie C

 They don’t trust that bids will stay secret

 Solution?

o Alice, Bob, Charlie submit hashes h(A), h(B), h(C)

o All hashes received and posted online

o Then bids A, B and C revealed

 Hashes don’t reveal bids (one way)

 Can’t change bid after hash sent (collision)

Part 1  Cryptography 16

Shamir’s Secret Sharing

(X

0

,Y

0

(X )

1

,Y

1

)

(0,S)

 Two points determine a line

 Give (X

0

,Y

0

) to Alice

 Give (X

1

,Y

1

) to Bob

 Then Alice and Bob must

cooperate to find secret S

 Also works in discrete case

 Easy to make “m out of n”

scheme for any m  n X

Y

2 out of 2

Part 1  Cryptography 17

Shamir’s Secret Sharing

(X

0

,Y

0

)

(X

1

,Y

1

)

(0,S)

 Give (X

0

,Y

0

) to Alice

 Give (X

1

,Y

1

) to Bob

 Give (X

2

,Y

2

) to Charlie

 Then any two of Alice, Bob

and Charlie can cooperate to

find secret S

 But no one can find secret S

 A “2 out of 3” scheme X

Y

(X

2

,Y

2

)

2 out of 3

Part 1  Cryptography 19

Secret Sharing Example

 Key escrow  required that your key be

stored somewhere

 Key can be used with court order

 But you don’t trust FBI to store keys

 We can use secret sharing

o Say, three different government agencies

o Two must cooperate to recover the key

Part 1  Cryptography 20

Secret Sharing Example

(X

0

,Y

0

)

(X

1

,Y

1

)

(0,K)

 Your symmetric key is K

 Point (X

0

,Y

0

) to FBI

 Point (X

1

,Y

1

) to DoJ

 Point (X

2

,Y

2

) to DoC

 To recover your key K,

two of the three agencies

must cooperate

 No one agency can get K

X

Y

(X

2

,Y

2

)