DIGITAL SIGNATURES
Fred Piper
Codes & Ciphers Ltd
12 Duncan Road
Richmond
Surrey
TW9 2JD
Information Security Group
Royal Holloway, University of London
Egham, Surrey
TW20 0EX
Digital Signatures
2
Outline
1.
Brief Introduction to Cryptography
2.
Public Key Systems
3.
Basic Principles of Digital Signatures
4.
Public Key Algorithms
5.
Signing Processes
6.
Arbitrated Signatures
7.
Odds and Ends
NOTE:
We will not cover all the sections
Digital Signatures
3
The Essence of Security
–
Recognition of those you know
–
Introduction to those you don’t
know
–
Written signature
–
Private conversation
Digital Signatures
4
The Challenge
•
Transplant these basic
social mechanisms to the
telecommunications
and/or business
environment.
Digital Signatures
5
•
Sender
–
Am I happy that the whole world sees this ?
–
Am I prepared to pay to stop them ?
–
Am I allowed to stop them ?
•
Recipient
–
Do I have confidence in :
–
the originator
–
the message contents and message stream
–
no future repudiation.
•
Network Manager
–
Do I allow this user on to the network ?
–
How do I control their privileges ?
The Security Issues
Digital Signatures
6
Cryptography is used to provide:
1.
Secrecy
2.
Data Integrity
3.
User Verification
4.
Non

Repudiation
Digital Signatures
7
Cipher System
cryptogram
c
Enciphering
Algorithm
Deciphering
Algorithm
Key
k(E)
Key
k(D)
message
m
message
m
Interceptor
Digital Signatures
8
The Attacker’s Perspective
Deciphering
Algorithm
Unknown Key
k(D)
Known
c
Wants
m
Note
:
k(E)
is not needed unless
it helps determine
k(D)
Digital Signatures
9
Two Types of Cipher System
•
Conventional or Symmetric
–
k(D)
easily obtained from
k(E)
•
Public or Asymmetric
–
Computationally infeasible to
determine
k(D)
from
k(E)
Digital Signatures
10
•
THE SECURITY OF THE SYSTEM IS
DEPENDENT ON THE SECURITY OF
THE KEYS
Digital Signatures
11
Public Key Systems
•
Original Concept
•
For a public key system an enciphering
algorithm is
agreed and each would

be
receiver publishes the key
which
anyone
may use to send a message to him.
•
Thus for a public key system to be secure it must not be
possible to deduce the message from a knowledge of the
cryptogram and the enciphering key. Once such a system
is set up, a directory of all receivers plus their enciphering
keys is published. However, the only person to know any
given receiver’s deciphering key is the receiver himself.
Digital Signatures
12
Public Key Systems
•
For a public key system, encipherment
must be a ‘one

way function’ which has a
‘trapdoor’. The trapdoor must be a secret
known only to the receiver.
•
A ‘one

way function’ is one which is easy
to perform but very difficult to reverse. A
‘trapdoor’ is a trick or another function
which makes it easy to reverse the
function
Digital Signatures
13
Some Mathematical One

Way
Functions
1.
Multiplication of two large primes.
2.
Exponentiation modulo
n ( n = pq )
.
3.
x
a
x
in
GF(2
n
)
or
GF(p)
.
4.
k
E
k
(m)
for fixed
m
where
E
k
is encryption
in a symmetric key system which is secure
against known plaintext attacks.
5.
x
a
.
x
where
x
is an
n

bit binary vector and
a
is a fixed
n

tuple of integers. Thus
a
.
x
is an
integer.
Digital Signatures
14
Public Key Cryptosystems
–
Enable secure communications without
exchanging secret keys
–
Enable 3rd party authentication ( digital
signature )
–
Use number theoretic techniques
–
Introduce a whole new set of problems
–
Are extremely ingenious.
Digital Signatures
15
Digital Signatures
•
According to ISO, the term Digital
Signature is used: ‘to indicate a
particular authentication technique
used to establish the origin of a
message in order to settle disputes
of what message (if any) was sent’.
Digital Signatures
16
Digital Signatures
A signature on a message is some data that
•
validates a message and verifies its origin
•
a receiver can keep as evidence
•
a third party can use to resolve disputes.
It depends on
•
the message
•
a secret parameter only
available to the sender
It should be
easy to compute
(by one person only)
easy to verify
difficult to forge
Digital Signatures
17
Digital Signature
•
Cryptographic checksum
•
Identifies sender
•
Provides integrity check for data
•
Can be checked by third party
Digital Signatures
18
Hand

Written Signatures
•
Intrinsic to signer
•
Same on all documents
•
Physically attached to message
•
Beware plastic cards.
Digital Signatures
•
Use of secret parameter
•
Message dependent.
Digital Signatures
19
Principle of Digital Signatures
•
There is a (secret) number which:
•
Only one person can use
•
Is used to identify that person
•
‘Anyone’ can verify that it has been
used
NB:
Anyone who knows the value of a
number can use that number.
Digital Signatures
20
Attacks on Digital Signature
Schemes
To impersonate A, I must either
•
obtain A’s private key
•
substitute my public key for A’s
NB
: Similar attacks if A is receiving secret
data encrypted with A’s public key
Digital Signatures
21
Obtaining a Private Key
Mathematical attacks
Physical attacks
NB
: It may be sufficient to obtain a
device which contains the key.
Knowledge of actual value is not
needed.
Digital Signatures
22
Certification Authority
AIM :
To guarantee the authenticity of public keys.
METHOD :
The Certification Authority guarantees the
authenticity by signing a certificate containing
user’s identity and public key with its secret key.
REQUIREMENT :
All users must have an authentic copy of the
Certification Authority’s public key.
Digital Signatures
23
Certification Process
Verifies
credentials
Creates
Certificate
Receives
(and checks)
Certificate
Presents Public
Key and
credentials
Generates
Key Set
Distribution
Centre
Owner
Digital Signatures
24
How Does it Work?
•
The Certificate can accompany all Fred’s
messages
•
The recipient must directly or indirectly:
•
Trust the CA
•
Validate the certificate
The CA certifies
that Fred Piper’s
public key
is………..
Electronically
signed by
the CA
Digital Signatures
25
User Authentication Certificates
•
Ownership of certificate does not
establish identity
•
Need protocols establishing use of
corresponding secret keys
Digital Signatures
26
WARNING
•
Identity Theft
•
You
‘
are
’
your private key
•
You ‘are’ the private key
corresponding to the public key in
your certificiate
Digital Signatures
27
Certification Authorities
•
Problems/Questions
•
Who generates users’ keys?
•
How is identity established?
•
How can certificates be cancelled?
•
Any others?
Digital Signatures
28
Fundamental Requirement
Internal infrastructure to support
secure technological implementation
Digital Signatures
29
Is everything OK?
Announcement in Microsoft Security
Bulletin
MS01

017
“
Ve
ri
Sign Inc recently advised
Microsoft that on January 29

30 2001
it issued two Ve
ri
Sign Class 3 code

signing digital
certificates
to an
individual who fraudulently claimed to
be a Microsoft employee.”
Digital Signatures
30
RSA System
•
Publish integers n and e where n = pq (p and q large
primes) and e is chosen so that (e,(p

1)(q

1)) = 1.
•
If message is an integer m with 0 <
m < n then the
cryptogram c = m
e
(mod n).
•
The primes p and q are ‘Secret’ (i.e. known only to the
receiver) and the system’s security depends on the
fact that knowledge of n will not enable the interceptor
to work out p and q.
Digital Signatures
31
RSA System
Since (e,(p

1)(q

1)) = 1 there is an integer d such that
ed = 1(mod(p

1)(q

1)).
[NOTE: without knowing p and q it is ‘impossible’ to
determine d.]
To decipher raise c to the power d.
Then m=c
d
(=m
ed
) (mod n).
System works because if n=pq,
a
k(p

1)(q

1) + 1
= a (mod n)
for all a, k.
Digital Signatures
32
RSA Summary and Example
Theory
Choice
n = p.q
2773 = 47.59
p=47 q=59
e.d 1(mod(p

1) (q

1))
17.157
≡
1(mod 2668)
e=17 d=157
Public key is (e,
n)
(17,2773)
Private
key is (d,n)
(157,2773)
Message M (0 < M < n)
M = 31
NB :
Knowledge of p and q is required to compute d.
Encryption using P
rivate
Key :
C
≡
M
e
(mod n)
587
≡
31
17
(mod 2773)
Decryption using
Private
Key :
M
≡
C
d
(mod n)
3
1
≡
587
157
(mod 2773)
Digital Signatures
33
E
l
G
amal
C
ipher
–
Work in GF(q)
–
For practical systems
•
q = large prime
•
q = 2
n
–
Note:
We will
not
define GF(2
n
). For a
prime q arithmetic in GF(q) is
arithmetic modulo q.
Digital Signatures
34
E
l
G
amal
C
ipher
System wide parameters : integers g,p
NB:
p is a large prime and g is a primitive element
mod p.
A chooses
private
key x such that 1 < x < p

1
A’s public key is y = g
x
mod p.
Note:
x is called the discrete logarithm of y modulo p
to the base g.
Digital Signatures
35
El Gamal Encryption
If B wants to send secret message m to A then
1.
B obtains A’s public key y plus g and p
2.
B generates random integer k.
3.
B sends g
k
(mod p) and
c
= my
k
(mod p) to A.
A
uses x
to compute y
k
from g
k
and then
evaluates m.
Digital Signatures
36
El Gamal Cipher
Important facts from last slide
•
g is special type of number
•
sender needs random number
generator
•
cryptogram is twice as long as
message
Digital Signatures
37
El Gamal

Encryption

Worked Example
Prime
p
= 23
Primitive element
a
= 11
Private key
x
= 6
Public key
y
= 11
6
(mod 23) = 9
To encipher
m
= 10
Assume random value
k
= 3
a
k
=
11
3
mod 23
=
20
y
k
=
11
18
mod 23
=
16
my
k
=
10.16 mod 23
=
22
Thus transmit (20, 22)
Digital Signatures
38
El Gamal

Worked Example
To decrypt
20, 22
y
k
= (a
k
)
x
= 20
6
= 16 mod 23
To find
m
: solve
c
=
my
k
mod
p
i.e. solve 22 =
m
16 mod 23
Solution
m
= 10
Digital Signatures
39
Modular Exponentiation
•
Both RSA and El Gamal involve computing
x
a
(mod
N
) for large
x, a
and
N
•
To speed up process need:
•
Fast multiplication algorithm
•
Avoid intermediate values becoming too
large
•
Limit number of modular multiplications
Digital Signatures
40
How to Create a Digital Signature
Using RSA
MESSAGE
HASHING
FUNCTION
HASH OF MESSAGE
Sign using Private Key
SIGNATURE

SIGNED HASH OF MESSAGE
Digital Signatures
41
How to Verify a Digital Signature Using
RSA
HASH OF MESSAGE
Verify the
Received Signature
Re

hash the
Received Message
Verify using
Public Key
Message
Hashing
Function
HASH OF MESSAGE
Message
Signature
Signature
Message with
Appended Signature
If hashes are equal,
signature is authentic
Digital Signatures
42
Requirements for Hash Function
h
(H1)
condenses message
M
of arbitrary length into
a fixed length ‘digest’
h(M)
(H2)
is one

way
(H3)
is collision free

it is computationally
infeasible to construct messages
M, M
'
with
h(M) = h(M
'
)
H3 implies a restriction on the size of
h(M)
.
Digital Signatures
43
DSA
•
Proposed by NIST in 1991
•
Explicitly requires the use of a hash
function
–
SHA

1
•
Very different set of functional
capabilities than RSA
Digital Signatures
44
DSA Set Up
•
System parameters
–
select a 160

bit prime
q
–
choose a 1024

bit prime
p
so that
q

p

1
–
choose g
Z
p
*
and compute
a
=
g
(p

1)/q
mod
p
–
if
a=1
repeat with different
g
•
User keys
–
select random secret key
x
(1
q

ㄩ1
–
compute public key
y = a
x
mod p
Digital Signatures
45
Signing with DSA
•
To sign message
m
–
hash message
m
to give
h(m)
(
1
栨h)
q

ㄩ
–
generate random secret
k
⠱
k
q

ㄩ
–
compute
r = (a
k
mod p) mod q
–
compute
k

1
mod q
–
compute
s =
k

1
{h(m) + ar}
mod q
–
signature on
m
is
(r,s)
Digital Signatures
46
DSA Signature Verification
•
To verify
(r,s)
–
check that
1
r
q

ㄠ
and
1
s
q

1
–
compute
w = s

1
mod q
–
compute
u
1
= wh(m) mod q
–
compute
u
2
= rw
mod q
–
accept signature if
–
(a
u
1
y
u
2
mod p) mod q = r
Digital Signatures
47
Security of DSA
•
Depends on
–
taking discrete logarithms in
GF(p)
(GNFS)
–
the logarithm problem in the cyclic subgroup
of order
q
•
algorithms for this take time proportional to
q
1/2
•
we choose
q
2
160
and
p
2
1024
–
other concerns follow the case of El Gamal
signatures
Digital Signatures
48
Performance of DSA
•
Using the subgroup of order
q
gives
good improvements over El Gamal
signatures
–
for signature
–
one (partial) exponentiation mod
p,
all other
operations less significant
–
also there are opportunities for pre

computation
–
for verification
–
two (partial) exponentiations mod
p,
all other
operations less significant
Digital Signatures
49
DSA and RSA
•
set a unit of time to be that required for one
1024

bit multiplication
•
use
e=2
16
+1
and CRT for RSA
•
pre

computation with DSA not included
•
also a difference in the sizes of the
signatures
RSA
DSA
Sign
384
240
Verify
17
480
Digital Signatures
50
Signing and Verifying
•
Which is more important

signature
or verification performance?
–
depends on the application!
•
certificates:
sign once but verify
very often
•
secure E

mail:
perhaps sign and verify
once
•
document storage:
sign once but maybe
never verify
Digital Signatures
51
Digital Signatures for Short Messages
Padding /
Redundancy
Text
Padding /
Redundancy
Text
Signature
Signature
RSA
Verify
RSA
Private
Key
Public
Key
a) Construction
b) Deconstruction
SEND
Digital Signatures
52
Types of Digital Signature
1. Arbitrated Signatures
Mediation by third party, the arbitrator
signing
verifying
resolving disputes
2. True Signatures
Direct communication between sender and receiver
Third party involved only in case of dispute
Digital Signatures
53
Arbitrated Signatures
Require trusted arbitrator
•
Arbitrator is involved in
–
Signing process
–
Settlement of all disputes
–
No one else can settle disputes
–
Potential bottleneck
Digital Signatures
54
Example of Arbitrated Signature
Scheme
(
1
)
Requirement:
A wants to send B message
B wants assurance of contents,
that A was originator and that A
cannot deny either fact.
Assumption:
A and B agree to trust an
arbitrator (ARB) and to
accept
ARB’s decision as binding.
Digital Signatures
55
Example of Arbitrated Signature
Scheme
(
2
)
Cryptographic Assumption
1.
Will use symmetric Algorithm eg DES
2.
Will use MACs
3.
A has established a DES key KA
shared with ARB
4.
B has established a DES key KB
shared with ARB
Digital Signatures
56
Example of Arbitrated Signature
Scheme
(
3
)
A wants to send ‘signed’ message M to B
Simplified protocol
Note:
B has no way of checking MAC
KA
is correct.
May be necessary to include identities in messages.
1)
A
ARB : M
1
=M  MAC
KA
2)
ARB uses KA to check MAC
KA
3)
ARB
B : M
2
= M
1
 MAC
KB
4)
B uses KB to check MAC
KB
Digital Signatures
57
True Signature
True Signature Requirement
•
Only one person can sign but anyone
can verify the signature
Public Key Requirement
•
Anyone can encrypt a message but
only one person can decrypt the
cryptogram.
Digital Signatures
58
True Signature
It is ‘natural’ to try to adopt public
key systems to produce signature
schemes by using the secret key in
the signing process
Digital Signatures
59
Digital Signatures
Common Terminology identifies the
terms Digital Signature and True
Signature
Digital Signatures
60
The Decision Process
•
Do I need Cryptography?
•
Do I need Public Key Cryptography?
•
Do I need PKI?
•
How do I establish a PKI?
Digital Signatures
61
Often Heard
•
PKI has never really taken off
•
PKI is dead
•
I’ve got a PKI, what do I do with it?
•
Secure e

commerce needs PKI
Digital Signatures
62
Diffie Hellman Key Establishment
Protocol
General Idea
:
Use Public System
A and B exchange public keys: P
A
and
P
B
There is a publicly known function f which has 2
numbers as input and one number as output.
A computes f (S
A
, P
B
) where S
A
is A’s
private
key
B computes f (S
B
, P
A
) where S
B
is B’s
private
key
f is chosen so that f (S
A
, P
B
) = f (S
B
, P
A
)
So A and B now share a (secret) number
Digital Signatures
63
Diffie Hellman Key Establishment Protocol
For the mathematicians:
Agree: Prime p primitive element a
A
:
chooses random r
A
and sends
B
:
chooses random r
B
and sends
Key:
Clearly any interceptor who can find discrete
logarithms can break the scheme
In this case
Note: Comparison with El Gamal
(modp)
a
B
r
(modp)
a
A
r
(modp)
a
s
B
A
r
r
B
A
B
A
r
r
A
r
B
r
y
a
)
r
,
f(a
)
r
,
f(a
.
x
y)
f(x,
Digital Signatures
64
D

H Man in the Middle Attack
A
B
Fraudster
F
A
P
F
P
F
P
B
P
The Fraudster has agreed keys with both
A
and
B
A
and
B
believe they have agreed a common key
Digital Signatures
65
D

H Man

in

the

Middle Attack
A
B
Fraudster
F
a
p
r
A
(mod
)
a
(
p)
r
F
mod
a
(
p)
r
F
mod
a
(
p)
r
B
mod
The Fraudster has agreed keys with both
A
and
B
A
and
B
believe they have agreed a common key
For the mathematicians
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