Session 5

Hash functions and digital signatures

Contents

• Hash functions– Definition– Requirements– Construction– Security– Applications

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Contents

• Digital signatures– Definition– Digital signatures – procedure– Digital signature with RSA– Signing enciphered messages– Signing and hashing

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Hash functions - definition• Let k, n be positive integers• A function f with n bit output and k

bit key is called a hash function if1. f is a deterministic function2. f takes 2 inputs, the first is of arbitrary

length and the second is of length k3. f outputs a binary string of length n

• Formally:

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nk* ,,,:f 101010

Hash functions - definition• The key k is assumed to be known/fixed,

unlike in cipher systems• If k is known/fixed, the hash function is

unkeyed• If k is secret the hash function is keyed• k is known/fixed in most of the

applications (e.g. digital signature schemes)

• k is kept secret in Message Authentication Codes (MACs)

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Hash functions – security requirements• In order to be useful for

cryptographic applications, any hash function must satisfy at least 3 properties (3 “levels of security”) (1)1. One-wayness (or preimage resistance):

a hash function f is one-way if, for a random key k and an n -bit output string w, it is difficult for the attacker presented with k and w to find x such that fk (x )=w.

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Hash functions – security requirements• Security requirements (2)

2. Second preimage resistance (or weak collision resistance): a hash function f is second preimage resistant if it is difficult for an attacker presented with a random key k and a random input string x to find y x such that fk

(x )=fk (y ).

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Hash functions – security requirements• Security requirements (3):

3. (Strong) collision resistance: a hash function f is collision resistant if it is difficult for an attacker presented with a random key k to find x and y x such that fk (x )=fk (y ).

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Hash functions – security requirements• The collision resistance implies the

second preimage resistance.• The second preimage resistance and

one-wayness are incomparable– The properties do not follow from one

another– Still, a hash function that would be one-

way but not second preimage resistant would be quite artificial

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Hash functions – security requirements• In practice, collision resistance is the

strongest security requirement of all the three requirements– the most difficult to satisfy– the easiest to breach

• Breaking the collision resistance property is the goal of most attacks on hash functions.

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Hash functions – other requirements• Certificational weakness– A good hash function should possess

avalanche property• changing a bit of input would approximately

change a half of the output bits– No input bits can be reliably guessed

based on the hash function’s local output (local one-wayness)

– Failure to satisfy these (and some other) properties is called certificational weakness.

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Hash functions – other requirements• It is also required that a hash

function is feasible to compute, given x (and k ).

• This is the reason why some theoretically strong constructions of hash functions are not used extensively in practice.

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Hash functions – other requirements• Example: so called algebraic hash

functions, based on the same difficult mathematical problems that are used in public key cryptography– Shamir’s function (factoring)– Chaum-vanHeijst-Pfitzmann’s function

(discrete log)– Newer designs: VSH (factoring), LASH

(lattice), Dakota (modular arithmetic and symmetric ciphers)

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Hash functions - construction

• The Merkle-Damgård construction– A classical hash function design– Iterates a compression function– A compression function• takes a fixed length input• outputs a fixed length (shorter) output.

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Hash functions - construction

• In practice, symmetric cipher systems are used as compression functions (usually block ciphers).

• Let g =(x,k ) be a block cipher, where x is the plaintext message, and k is the key.

• The length of the block x is n bits and the length of the key k is m bits, m >n.

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Hash functions - construction

• The hash function f to be constructed– has the (theoretically) unlimited input

length– has the output bit length n

• The input string to the hash function f is y.

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Hash functions - construction

• Hash function iterations– Pad y such that the length of the padded

input y ’ is the least possible multiple of m.

– Let where yi {0,1}m .– Let f0 be a fixed initialization vector of

length n (in bits).– Then, for i =1,..., r, fi =g (fi -1, ).

– Finally, f =fr .17/44

'r

''' y||||y||yy 21

'iy

Hash functions - construction

• Remark:– The padding algorithm and f0 depend on

the particular hash function.• Schematic of the Merkle-Damgård

design

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Hash functions - construction• Advantages of using block ciphers as

compression functions– Efficient, i.e. fast– Usually already implemented

• Disadvantage– Employing a strong block cipher in hash

function design does not guarantee a good hash function.

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Hash functions - construction

• Examples of Merkle-Damgård designs– The MD (Message Digest) family of hash

functions (MD4, MD5), n =128.– The NIST SHA (Secure Hash Algorithm)

family of hash functions (SHA-1 (n =160), SHA-2 (i.e. SHA-256, SHA-512)).

• They all use custom block cipher rounds.

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Hash functions - construction

• The speed of such a design depends on the number of rounds of the block cipher involved.

• Example–MD4 – 3 rounds–MD5 – 4 rounds – more secure– But MD5 is 30% slower than MD4.

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Hash functions - security

• Security of the most often used hash functions, MD5 and SHA-1 has been recently compromised – collisions were found.

• They are now considered insecure.• Consequence: the SHA-3 contest, the

proposals are due October 2008.22/44

Hash functions - applications

• Data integrity protection– Digital signature schemes

• Authentication–Message authentication codes (MACs)– If MAC uses a hash function it is called

HMAC– HMAC standard RFC2104 (Bellare-

Canetti-Krawczyk, 1996).23/44

Digital signatures - definition

• Digital signature– A number dependent on some secret

known only to the signer and on the contents of the signed message

–Must be verifiable in case of• a signer repudiating a signature• a fraudulent claimant

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Digital signatures - definition

• Applications– Authentication– Data integrity protection and non-

repudiation– Certification of public keys in large

networks.

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Digital signatures - procedure

• Basic elements (1)–M – the set of messages that can be

signed– S – the set of signatures, e.g. binary

strings of fixed length– SA – signing transformation for the entity

A

• SA is kept secret by A• Used to create signatures from M

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SM:SA

Digital signatures - procedure

• Basic elements (2)– VA – verification transformation for the

A’s signatures

• Publicly known• Used by other entities to verify signatures

created by A

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false,trueSM:VA

Digital signatures - procedure

• Both SA and VA should be feasible to compute

• It should not be computationally feasible to forge a digital signature y on a message x– Given x, only A (i.e. Alice) should be

able to compute the signature y such that VA(x,y )=true. 28/44

Digital signatures - procedure

• Signing a message x– Alice uses the algorithm SA to compute

the signature over the message x– Alice publishes (or sends to some

recipient) the message x, together with the signature y =SA(x )

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Digital signatures - procedure

• Verifying a signature of a message published/sent by Alice– Upon receiving the pair (x,y ), the verifier

uses the algorithm VA (publicly known) to verify the integrity of the received message x

– If VA (x,y )=true, the signature is verified.

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Digital signatures - procedure

• It can be shown that asymmetric ciphers can be used for digital signature purposes

• To prevent forgery, it should be infeasible for an attacker to retrieve the secret information used for signing – the transformation SA.

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Digital signature with RSA

• Alice signs the message x by using the deciphering transformation

• Alice is the only one that can sign, since dA is kept secret.

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Ad nxy A mod

Digital signature with RSA

• Bob verifies the signature y received from Alice by employing encipherment of y using Alice’s public key (eA,nA), i.e.

• If c =x, then the signature y is verified.

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Ae nyc A mod

Digital signature with RSA - security

• Suppose Eve wants to sign her own message x ’ with Alice’s signature y (i.e. to forge Alice’s signature).

• Eve does not know dA, she only knows Alice’s public key (eA,nA ).

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Digital signature with RSA - security

• Direct verification, if Eve’s signed document (x ’,y ) is to be verified

– This will fail, since c ≠x ’.• Thus, what Eve needs is another

signature, y ’, such that• Getting y ’ is a difficult problem.

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Ae nyc A mod

'mod' xny AeA

Digital signature with RSA - security

• Another possibility for Eve – she can choose y ’ first and then generate the message

• y ’ will then be easily verified, i.e. such a forgery is successful.

• But then the probability that x ’ is meaningful is very small.

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Ae nyx A mod''

Signing enciphered messages

• Suppose Alice wants to send a signed enciphered message x to Bob.– Alice computes her signature y =SA (x )– Then Alice enciphers both x and y by

means of Bob’s public key– The ciphertext z is transmitted to Bob.

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Signing enciphered messages

• Deciphering and verification– Bob deciphers z by means of his private

key and thus obtains (x,y )– Then Bob uses Alice’s public verification

function VA to verify the Alice’s signature y.

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Signing and hashing

• Usually, public key ciphers are used in digital signature schemes

• If the original message is signed, the signature is at least as long as the message – inefficient

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Signing and hashing

• Another problem is that of Eve’s ability to generate the signature and then get the corresponding message that may be meaningful, although with small probability.

• Solution: sign hashed message.

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Signing and hashing

• The hash function f is made public• Starting with a message x, Alice first

computes f (x ), which is significantly smaller than x

• Alice then computes y =SA(f (x ))• Alice then sends (x,y ) to Bob.

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• Verification process– Bob computes f (x )– Bob also computes VA (f (x ),y )– If VA (f (x ),y ) =true, then Alice’s

signature is verified.

Signing and hashing

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• Suppose Eve has (x,y =SA(f (x ))• Eve would like to sign her own message

x ’ with Alice’s signature (i.e. to forge it)• So she needs SA(f (x ’))=SA(f (x )), which

means she needs f (x ’)=f (x ). This is difficult if f (x ) is second preimage resistant.

Signing and hashing - security

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• Moreover, it is highly unlikely that Eve would be able to find two messages, x’ and x ’’ with the same hashes and consequently signatures, if f is collision resistant.

• So it is difficult for Eve to choose the signature first and then get the corresponding message.

Signing and hashing - security

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