Elliptic curve Cryptography and Diffie- Hellman Key exchange

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 1 Ver. I (Jan. - Feb. 2017), PP 56-61
www.iosrjournals.org
DOI: 10.9790/5728-1301015661 www.iosrjournals.org 56 | Page
Elliptic curve Cryptography and Diffie- Hellman Key exchange
Dr.S.Vasundhara
Asst. Prof of Mathematics G.Narayanamma Institute of Technology & science(For Women)
Shaikpet, Hyderabad. Telengana
Abstract: In this paper an introduction of Elliptic curve cryptography explained Then the Diffie- Hellman
algorithm was explained with clear examples.
Keywords: Cryptography Elliptic curve cryptography, Diffie-Hellman Key exchange.
I. Introduction
The history of cryptography is long and interesting. It has a very considerable turning point when two
researchers from Stanford, Whitfield Diffie and Martin Hellman, published the paper ―New Directions in
Cryptography‖ in 1976. They preface the new idea of public key cryptography in the paper.
Public-key cryptography and [4]symmetric-key cryptography are two main categories of cryptography.
The Well-known public-key cryptography algorithms are RSA (Rivest, et al. 1978), El-Gamal and Elliptic
Curve Cryptography. Presently, there are only three problems of public key cryptosystems that are considered to
be both secure and effective (Certicom, 2001). Table 1.1 shows these mathematical problems and the
cryptosystems that rely on such problems.
Mathematical problem Detail Cryptosystem
1 Integer Factorization problem
(IFP)
Given an integer n find its prime
factorization
RSA
2 Discrete Logarithm
problem(DLS)
Given integer g and h find x‘
such that =gx
mod n
Diffie-
Hellman(DH)
3 Elliptic curve discrete
logarithmic problem(ECDLP)
Given points P and Q on the
curve find ‗x‘ such that Q=xP
Diffie-
Hellman(DH)
Providing an equivalent level of security with smaller key size is an advantage of ECC compared to
RSA. It is very efficient to[1] implement ECC.ECC obtains lower power consumption, and faster computation.
It also gains small memory and bandwidth because of its key size length (Dormale, Bulens and Quisquater
2004), (Huang 2007). Such attributes are mainly fascinating in security applications in which calculative power
and integrated circuit space are limited. Wireless devices and smart cards present a good example for the
constrained devices with limited resources. Cryptography companies such as Certicom Corporation have already
implemented ECC in their products for some commercial purposes which are RFID and Zigbee. This company
has an agreement with NSA on a set of cryptographic algorithms called suite B. This suite uses Elliptic curves
and works over the prime field.
A modular arithmetic performs a main role in public key cryptographic systems (Dormale, et al. 2004).
Some of these PKC are the Diffie-Hellman keys exchange algorithm (Diffie and Hellman 1976), the
decipherment operation in the RSA algorithm (Quisquater and Couvreur 1982), the US Government Digital
Signature Standard (FIPS 2000), and also elliptic curve cryptography (Koblitz 1987).
Diffie–Hellman:
Diffie–Hellman key exchange (D–H)[11] is a specific method of exchanging cryptographic keys. It is
one of the earliest practical examples of key exchange implemented within the field of cryptography. The
Diffie–Hellman key exchange method allows two parties that have no prior knowledge of each other to jointly
establish a shared secret key over an insecure communications channel. This key can then be used to encrypt
subsequent communications using a symmetric key cipher.
The scheme was first published by Whitfield Diffie and Martin Hellman in 1976, although it had been
separately invented a few years earlier within GCHQ, the British signals intelligence agency, by Malcolm J.
Williamson but was kept classified. In 2002, Hellman suggested the algorithm be called Diffie–Hellman–Merkle
key exchange in recognition of Ralph Merkle's contribution to the invention of public-key
cryptography (Hellman, 2002).
Although Diffie–Hellman key agreement itself is an anonymous (non-authenticated) key-agreement
protocol, it provides the basis for a variety of authenticated protocols, and is used to provide perfect forward

secrecy in Transport Layer Security's ephemeral modes (referred to as EDH or DHE depending on the cipher
suite).
The method was followed shortly afterwards by RSA, an implementation of public key cryptography
using asymmetric algorithms.
In 2002, Martin Hellman wrote:
The system...has since become known as Diffie–Hellman key exchange. While that system was first
described in a paper by Diffie and me, it is a public key distribution system, a concept developed by Merkle, and
hence should be called 'Diffie–Hellman–Merkle key exchange' if names are to be associated with it. I hope this
small pulpit might help in that endeavor to recognize Merkle's equal contribution to the invention of public key
cryptography. [1]U.S. Patent 4,200,770, now expired, describes the algorithm and credits Hellman, Diffie, and
Merkle as inventors.
II. Description
Diffie–Hellman establishes a shared secret[7] that can be used for secret communications by
exchanging data over a public network. The following diagram illustrates the general idea of the key exchange
by using colors instead of a very large number. The key part of the process is that Alice and Bob exchange
their secret colors in a mix only. Finally this generates an identical key that is mathematically difficult
(impossible for modern supercomputers to do in a reasonable amount of time) to reverse for another party that
might have been listening in on them. Alice and Bob now use this common secret to encrypt and decrypt their
sent and received data. Note that the yellow paint is already agreed by Alice and Bob.
Here is an explanation which includes the encryption's mathematics:
The simplest, and original, implementation of the protocol uses the multiplicative group of integers modulo p,
where p is prime and g is primitive root mod p. Here is an example of the protocol, with non-secret values in
blue, and secret values in boldface red:

Table: 1.1
1.Alice and Bob agree to use a prime number p=23 and base g=5.
1. Alice chooses a secret integer a=6, then sends Bob A = ga
mod p
 A = 56
mod 23
 A = 15,625 mod 23
 A = 8
2. Bob chooses a secret integer b=15, then sends Alice B = gb
mod p
 B = 515
mod 23
 B = 30,517,578,125 mod 23
 B = 19
4, Alice computes s = B a
mod p
 s = 196
mod 23
 s = 47,045,881 mod 23
 s = 2
5. Bob computes s = A b
mod p
 s = 815
mod 23
 s = 35,184,372,088,832 mod 23
 s = 2
3. Alice and Bob now share a secret: s = 2. This is because 6*15 is the same as 15*6. So somebody who had
known both these private integers might also have calculated s as follows:
 s = 56*15
mod 23
 s = 515*6
mod 23
 s = 590
mod 23
 s = 807,793,566,946,316,088,741,610,050,849,573,099,185,363,389,551,639,556,884,765,625 mod 23
 s = 2
Both Alice and Bob have arrived at the same value, because (ga
)b
and (gb
)a
are equal mod p. Note that
only a, b and gab
= gba
mod p are kept secret. All the other values – p, g, ga
mod p, andgb
mod p – are sent in the
clear. Once Alice and Bob compute the shared secret they can use it as an encryption key, known only to them,
for sending messages across the same open communications channel. Of course, much larger values of a, b,
and p would be needed to make this example secure, since it is easy to try all the possible values of gab
mod 23.
There are only 23 possible integers as the result of mod 23. If p were a prime of at least 300 digits,
and a and b were at least 100 digits long, then even the best algorithms known today could not find a given
only g, p, gb
mod p and ga
mod p, even using all of mankind's computing power. The problem is known as
the discrete logarithm problem. Note that g need not be large at all, and in practice is usually either 2, 3 or 5.
Here's a more general description of the protocol:
Alice and Bob agree on a finite cyclic group G and a generating element g in G. (This is usually done
long before the rest of the protocol; g is assumed to be known by all attackers.) We will
Write the group G multiplicatively.
Alice picks a random natural number a and sends ga
to Bob.
Bob picks a random natural number b and sends gb
to Alice.
Alice computes (gb
)a
.
Bob computes (ga
)b
.
Both Alice and Bob are now in possession of the group element gab
, which can serve as the shared secret key.
The values of (gb
)a
and (ga
)b
are the same because groups are power associative. (See also exponentiation.)
In order to decrypt a message m, sent as mgab
, Bob (or Alice) must first compute (gab
)-1
, as follows:
Bob knows |G|, b, and ga
. A result from group theory establishes that from the construction of G, x|G|
= 1 for
all x in G.
Bob then calculates (ga
)|G|-b
= ga(|G|-b)
= ga|G|-ab
= ga|G|
g-ab
= (g|G|
)a
g-ab
=1a
g-ab
=g-ab
=(gab
)-1
.
Alice Bob
Secret Public Calculates Sends Calculates Public Secret
a p, g p,g b
a p, g, A ga
mod p = A A p, g b
a p, g, A B gb
mod p = B p, g, A, B b
a, s p, g, A, B Ba
mod p = s Ab
mod p = s p, g, A, B b, s

When Alice sends Bob the encrypted message, mgab
, Bob applies (gab
)-1
and recovers mgab
(gab
)-1
= m(1) = m.
Here is a chart to help simplify who knows what. (Eve is an eavesdropper—she watches what is sent between
Alice and Bob, but she does not alter the contents of their communications.)
Let s = shared secret key. s = 2
Let g = public base. g = 5
Let p = public (prime) number. p = 23
Let a = Alice's private key. a = 6
Let A = Alice's public key. A = ga
mod p = 8
Let b = Bob's private key. b = 15
Let B = Bob's public key. B = gb
mod p = 19
Table:1.2
Alice
Knows
doesn't
know
p = 23 b = ?
base g = 5
a = 6
A = 56
mod 23 = 8
B = 5b
mod 23 = 19
s = 196
mod 23 = 2
s = 8b
mod 23 = 2
s = 196
mod 23 = 8b
mod
23
s = 2
Bob
Knows
doesn't
know
p = 23 a = ?
base g = 5
b = 15
B = 515
mod 23 = 19
A = 5a
mod 23 = 8
s = 815
mod 23 = 2
s = 19a
mod 23 = 2
s = 815
mod 23 = 19a
mod 23
s = 2
Eve
knows
doesn't
know
p = 23 a = ?
base g = 5 b = ?
s = ?
A = 5a
mod 23 = 8
B = 5b
mod 23 = 19
s = 19a
mod 23
s = 8b
mod 23
s = 19a
mod 23 = 8b
mod 23
Note: It should be difficult for Alice to solve for Bob's private key or for Bob to solve for Alice's private key. If
it is not difficult for Alice to solve for Bob's private key (or vice versa), Eve may simply substitute her own
private / public key pair, plug Bob's public key into her private key, produce a fake shared secret key, and solve
for Bob's private key (and use that to solve for the shared secret key. Eve may attempt to choose a public /
private key pair that will make it easy
for her to solve for Bob's private key). A demonstration of Diffie-Hellman (using numbers too small for
practical use) is given here.
operation with more than two parties
Diffie-Hellman key agreement is not limited to negotiating a key shared by only two participants. Any number
of users can take part in an agreement by performing iterations of the agreement protocol and exchanging
intermediate data (which does not itself need to be kept secret). For example, Alice, Bob, and Carol could
participate in a Diffie-Hellman agreement as follows, with all operations taken to be modulo :
The parties agree on the algorithm parameters and .
The parties generate their private keys, named , , and .
Alice computes and sends it to Bob.
Bob computes and sends it to Carol.
Carol computes and uses it as her secret.
Bob computes and sends it to Carol.
Carol computes and sends it to Alice.
Alice computes and uses it as her secret.
Carol computes and sends it to Alice.
Alice computes and sends it to Bob.

Bob computes and uses it as his secret.
An eavesdropper has been able to see , , , , , and , but cannot use any
Combination of these to reproduce .
To extend this mechanism to larger groups, two basic principles must be followed:
Starting with an ―empty‖ key consisting only of , the secret is made by raising the current value to every
participant‘s private exponent once, in any order (the first such exponentiation yields the participant‘s own
public key).
Any intermediate value (having up to exponents applied, where is the number of
participants in the group) may be revealed publicly, but the final value (having had all exponents applied)
constitutes the shared secret and hence must never be revealed publicly. Thus, each user must obtain their copy
of the secret by applying their own private key last (otherwise there would be no way for the last contributor to
communicate the final key to its recipient, as that last contributor would have turned the key into the very secret
the group wished to protect).
These principles leave open various options for choosing in which order participants contribute to keys.
The simplest and most obvious solution is to arrange the participants in a circle and have keys rotate
around the circle, until eventually every key has been contributed to by all participants (ending with its
owner) and each participant has contributed to keys (ending with their own). However, this requires that
every participant perform modular exponentiations.
By choosing a more optimal order, and relying on the fact that keys can be duplicated, it is possible to reduce
the number of modular exponentiations performed by each participant to using a divide-and-
conquer-style approach, given here for eight participants:
1. Participants A, B, C, and D each perform one exponentiation, yielding ; this value is sent to E, F, G,
and H. In return, participants A, B, C, and D receive .
2. Participants A and B each perform one exponentiation, yielding , which they send to C and D,
while C and D do the same, yielding , which they send to A and B.
3. Participant A performs an exponentiation, yielding , which it sends to B; similarly, B
sends to A. C and D do similarly.
4. Participant A performs one final exponentiation, yielding the secret , while B
does the same to get ; again, C and D do similarly.
5. Participants E through H simultaneously perform the same operations using as their starting point.
Upon completing this algorithm, all participants will possess the secret , but each
participant will have performed only four modular exponentiations, rather than the eight implied by a simple
circular arrangement.
Security: The protocol is considered secure against eavesdroppers if G and g are chosen properly. The
eavesdropper ("Eve") would have to solve the Diffie–Hellman problem to obtain gab
. This is currently
considered difficult. An efficient algorithm to solve the discrete logarithm problem would make it easy to
compute a or b and solve the Diffie–Hellman problem, making this and many other public key cryptosystems
insecure.
The order of G should be prime or have a large prime factor to prevent use of the Pohlig–Hellman
algorithm to obtain a or b. For this reason, a Sophie Germain prime q is sometimes used to calculate p=2q+1,
called a safe prime, since the order of G is then only divisible by 2 and q. g is then sometimes chosen to generate
the order q subgroup of G, rather than G, so If Alice and Bob use random number generators whose outputs are
not completely random and can be predicted to some extent, then Eve's task is much easier.
The secret integers a and b are discarded at the end of the session. Therefore, Diffie–Hellman key
exchange by itself trivially achieves perfect forward secrecy because no long-term private keying material exists
to be disclosed.
In the original description, the Diffie–Hellman exchange by itself does not provide authentication of
the communicating parties and is thus vulnerable to a man-in-the-middle attack. A person in the middle may
establish two distinct Diffie–Hellman key exchanges, one with Alice and the other with Bob, effectively

masquerading as Alice to Bob, and vice versa, allowing the attacker to decrypt (and read or store) then re-
encrypt the messages passed between them. A method to authenticate the communicating parties to each other is
generally needed to prevent this type of attack. Variants of Diffie-Hellman, such as STS, may be used instead to
avoid these types of attacks.
Other uses
Password-authenticated key agreement
When Alice and Bob share a password, they may use a password-authenticated key agreement (PAKE)
form of Diffie–Hellman to prevent man-in-the-middle attacks. One simple scheme is to make the generator g the
password. A feature of these schemes is that an attacker can only test one specific password on each iteration
with the other party, and so the system provides good security with relatively weak passwords. This approach is
described in
ITU-T Recommendation X.1035, which is used by the G.hn home networking standard.
Public Key:
It is also possible to use Diffie–Hellman as part of a public key infrastructure. Alice's public key is
simply . To send her a message Bob chooses a random b, and then sends
Alice (un-encrypted) together with the message encrypted with symmetric key .
Only Alice can decrypt the message because only she has a. A preshared public key also prevents man-in-the-
middle attacks.
In practice, Diffie–Hellman is not used in this way, with RSA being the dominant public key algorithm.
This is largely for historical and commercial reasons, namely that RSA created a Certificate Authority that
became Verisign. Diffie–Hellman cannot be used to sign certificates, although the ElGamal and DSA signature
algorithms are related to it. However, it is related to MQV, STS and the IKE component of the IP sec protocol
suite for securing Internet Protocol communications.
References
[1]. Cole, Eric, Jason Fossen, Stephen Northcutt, Hal Pomeranz. SANS Security Essentials with CISSP CBK, Version 2.1.USA: SANS
Press, 2003.
[2]. J.Edge,an introduction to elliptic curve ,cryptography, http://Iwn.net/Articles/174127/.2006.
[3]. N.Koblitz,A course in Number theory and cryptography,2nd
ed.,brookes/Cole,1997.
[4]. .J.H.Silverman,The Arithmetic of Elliptic curves, Springer –Verlag,1986.
[5]. RSA‖ Wikipedia.wikipedia,n.d.web.09 feb 2011.Stalings,William. Cryptography and network security.fourth,pearson,2009.print.
[6]. Alfredj Menezes, paul c,vanoorschot and scott A.vanstone,guide to Elliptic curve Cryptography ,1996.
[7]. Diffie, W., and M. E. Hellman. ―New directions in cryptography.‖ IEEE Transactions on Information Theory,, 1976: 644- 654.
[8]. Stalling, W. Network and Internet work Scurity. IEEE Press, 1995.
[9]. Tata, E. ―Elliptic Curve Cryptography, An Implementation Guide.‖ In Anoop MS. India: Anoop, MS, 2007
[10]. Rivest, R. L., A. Shamir, and L. Adleman. ―A Method for Obtaining Digital Signatures and Public-Key Cryptosystems.‖
Communications of the ACM, 1978: 120-126.
[11]. Diffie, W., and M. E. Hellman. ―New directions in cryptography.‖ IEEE Transactions on Information Theory,, 1976: 644- 654.

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