AN EFFICIENT GROUP AUTHENTICATION FOR GROUP COMMUNICATIONS

International Journal of Network Security & Its Applications (IJNSA), Vol.5, No.3, May 2013
DOI : 10.5121/ijnsa.2013.5302 09
AN EFFICIENT GROUP AUTHENTICATION FOR
GROUP COMMUNICATIONS
Lein Harn1
and Changlu Lin2
1
Department of Computer Science Electrical Engineering, University of Missouri-Kansas
City, MO 64110, USA
harnl@umkc.edu
2
Key Laboratory of Network Security and Cryptology, Fujian Normal University, Fujian,
35007, P. R. China
cllin@fjnu.edu.cn
ABSTRACT
Group communication implies a many-to-many communication and it goes beyond both one-to-one
communication (i.e., unicast) and one-to-many communication (i.e., multicast). Unlike most user
authentication protocols that authenticate a single user each time, we propose a new type of authentication,
called group authentication, that authenticates all users in a group at once. The group authentication
protocol is specially designed to support group communications. There is a group manager who is
responsible to manage the group communication. During registration, each user of a group obtains an
unique token from the group manager. Users present their tokens to determine whether they all belong to
the same group or not. The group authentication protocol allows users to reuse their tokens without
compromising the security of tokens. In addition, the group authentication can protect the identity of each
user.
KEYWORDS
User authentication; Group communication; Secret sharing; Ad hoc network; Strong t -consistency
1. INTRODUCTION
User authentication is one of the most important security services in computer and
communication application. Knowledge based authentication (e.g., password) [16,9] and key
based authentication (e.g., public/private key) [7,12] are the two most popular approaches.
Knowledge based authentication has some security flaws. Most users like to use simple and short
passwords. However, Internet hackers can easily crack simple passwords. Public-key based
authentication needs a certificate authority (CA) to provide the authenticity of public keys. In
addition, public-key computations involve large integers. Computational time is one of the main
concerns for public-key based authentication.
All user authentication protocols [10,6] are one-to-one type of authentication where the prover
interacts with the verifier to prove the identity of the prover. For example, the RSA digital
signature [13] is used to authenticate the signer of the signature. In this approach, the verifier
sends a random challenge to the prover. Then, the prover digitally signs the random challenge and
returns the digital signature of the challenge to the verifier. After successfully verifying the digital
signature, the verifier is convinced that the prover is the one with the identity of the public key
used to verify the digital signature. In wireless communications, when a mobile subscriber wants
to establish a connection with the base station, the subscriber and the base station interact to

10
establish mutual authentication. Mutual authentication can prevent an illegitimate subscriber from
using the service and prevent the fake base station from harming the subscriber.
Network applications are no longer just one-to-one communication; but involve multiple users
( 2) . Group communication [14,2] implies a many-to-many communication and it goes beyond
both one-to-one communication (i.e., unicast) and one-to-many communication (i.e., multicast).
In this paper, we propose a new type of authentication, called group authentication, which
authenticates all users in a group at once. The group authentication protocol is specially designed
to support group communications. The group authentication is defined to involve multiple users
and users want to convince each other that they all belong to the same group without revealing
their identities. In the group authentication, each user acts as both the prover and the verifier.
Group authentication is extremely important in an ad hoc network because this network is
temporarily established by multiple users and these users want to use this network to exchange
secret information.
Devising protocols to provide group authentication in ad hoc networks is extremely challenging
due to highly dynamic and unpredictable topological changes. As a result, there are two popular
models to provide group authentication services in an ad hoc network. The first model involves a
centralized authentication server (AS) [11,3] and the second model has no AS [5,4]. In the first
model, AS manages the access rights of the network. For example, Bhakti et al. [3] proposed to
adopt Extensible Authentication Protocol (EAP) in the IEEE 802.1x standard for wireless ad hoc
network. This approach requires to set up the AS and have mobile users to access to the AS
service. In fact, in some situations, the second model is the only way to provide group
authentication. For example, in an ad-hoc network communication, there has no AS service
available to mobile users. In the second model, each user needs to take in charge of authenticating
other users. In a straightforward approach, if there are n users in the group, each user can use the
one-to-one authentication protocol for 1n  times to authenticate other users. Computational time
is one of the major concerns in this approach.
In this paper, we introduce a special type of group authentication which provides an efficient way
to authenticate multiple users belonging to the same group without revealing identity of each user.
Our proposed protocol is no longer a one-to-one type of authentication. It is a many-to-many type
of authentication. Unlike most user authentication protocols that authenticate a single user each
time, our proposed protocol authenticates all users of a group at once. In our proposal, each user
needs to register with a group manager (GM) to become a group user. Like the trusted dealer in
Shamir's ( , )t n secret sharing scheme [15], the GM needs to select a secret polynomial and
compute token for each user. Based on these tokens, our protocol can establish group
authentication for all users at once. The group authentication protocol allows users to reuse their
tokens without compromising the security of tokens. Our proposed protocol supports existing
wireless communication network including wireless ad hoc network.
The rest of this paper is organized as follows. In next section, we include some preliminaries.
In Section 3, we introduce the model of our proposed group authentication. In Section 4, we
present basic one-time group authentication protocol; in Section 5, we present group
authentication protocol without revealing tokens. We conclude in Section 6.

11
2. PRELIMINARIES
2.1. Review of Shamir's secret sharing scheme [15]
In Shamir's ( , )t n secret sharing scheme based on the polynomial, there are n shareholders and a
mutually trusted dealer. The scheme consists of two algorithms:
a) Share generation algorithm: the dealer first picks a random polynomial of degree 1t  ,
 1
1 1 0( ) modt
i tf x a x a x a p
    , such that the secret s satisfies 0(0)f a s 
and all coefficients, a0,a1,…..at-1 ZP, p is a prime with p s . The dealer computes shares,
( )if x , for 1,2, ,i n  , and distributes each share ( )if x to shareholder iU secretly.
b) Secret reconstruction algorithm: it takes any t or more than t shares, for example, j
shares (i.e., t j n  ), 1 1 2 2( , ( )),( , ( )), ,( , ( ))j jx f x x f x x f x , as inputs, and outputs the
secret s using Lagrange interpolating formula as
 
1 1,
( ) mod .
jj
r
i
i r r i i r
x
s f x p
x x  



 
We note that the above algorithms satisfy the basic requirements of the secret sharing scheme,
that are, (1) with the knowledge of any t or more than t shares, shareholders can reconstruct the
secret s ; and (2) with the knowledge of any 1t  or fewer than 1t  shares, shareholders cannot
obtain the secret s . Shamir's secret sharing scheme is unconditionally secure since the scheme
satisfies these two requirements without making any computational assumption. For more
information on this scheme, please refer to the original paper [15].
2.2. Harn and Lin's definition on strong t -consistency [8]
Benaloh [1] presented a notion of t -consistency to determine whether a set of shares is generated
from a polynomial of degree 1t  at most. Recently, Harn and Lin [8] proposed a new definition
of strong t -consistency which is the extension of Benaloh's definition.
Definition 1 (Strong t -consistency [8]). A set of n shares (i.e., t n ) is said to be strong t -
consistent if (a) any subset of t or more than t shares can reconstruct the secret, and (b) any
subset of fewer than t shares cannot reconstruct the secret.
It is obvious that if shares in Shamir's secret sharing scheme are generated by a polynomial with
degree 1t  exactly, then shares satisfy the security requirements of a ( , )t n secret sharing
scheme and these shares are also strong t -consistent.
Checking strong t -consistency of n shares can be executed very efficiently by using Lagrange
interpolating formula. In fact, to check whether n shares are strong t -consistent or not, it only
needs to check whether the interpolation of n shares yields a polynomial with degree 1t 
exactly. If this condition is satisfied, we can conclude that all shares are strong t -consistent.
However, if there are some illegitimate shares, the degree of the interpolating polynomial of these
n shares is more than 1t  with very high probability. In other words, these n shares are most
likely to be not strong t -consistent. The property of strong t -consistency will be used in Section
5 of our protocol to check strong t -consistency of n shares without revealing tokens.

12
3. MODEL
3.1. Entities
a) Group Manager (GM): A group manager is responsible to register users to form a group.
The responsibility of GM is to issue a secret token to each user during registration. Later,
authentication is based on the secret tokens. Since tokens are used in authentication,
identities of users are protected. In order to prevent malicious users to reveal their tokens to
attackers, each token is a unique integer. The secret tokens are shares of the polynomial
generated by the GM.
b) Group Users: Join a group and become a group user, each user needs to register with the
GM. After being successfully registered, each user receives a secret token from the GM.
Each user with a unique token can prevent malicious users to give their tokens to
impersonators.
c) Attackers: We consider two types of attackers, the inside attackers and the outside attackers.
The inside attackers are users who are legitimate users and own legitimate tokens from the
GM. We consider that the insider attackers may collude to forge tokens for non-users. The
outside attackers are impersonators who do not own any tokens and try to impersonate users
to fail the authentication protocol. We also assume that the GM does not collude with any
user. If the GM colludes with any user by revealing the secret of the GM to the user, the
colluded user can do harm to the group. In addition, we assume all users act honestly in the
authentication. If any use acts dishonestly by revealing a invalid value, the authentication is
failed.
3.2. Authentication outcomes
There are only two possible outcomes of a group authentication; that are, either “yes” or “no”. If
the outcome is “yes”, it means that all users belong to the same group; otherwise, there are
impersonators.
4. BASIC ONE-TIME GROUP AUTHENTICATION PROTOCOL
In the following discussion, we assume that there are n users, 1 2, , , nM M M , registered at the
GM to form a group.
4.1. System set up
During registration, GM constructs a random ( 1)t  -th (i.e., t n ) degree polynomial ( )f x
with (0)f s , and computes secret tokens of users as ( )i iy f x , for 1,2, ,i n  , where ix
is the public information associated with user iM . GM sends each token iy to user iM secretly.
GM makes ( )H s publicly known, where H is a one-way function.
Remark 1. The threshold t is an important security parameter that affects the security of group
authentication protocols. Using a ( , )t n secret sharing scheme to issue tokens in the registration
can prevent up to 1t  inside attackers, who are legitimate users, colluded together to forge
tokens.

13
4.2. Basic one-time group authentication protocol
From now on, we assume that there are j users with their tokens 1 2( ), ( ), , ( )jf x f x f x where
t j n  , who want to execute the group authentication protocol.
The basic idea of this protocol is that each user releases the token obtained from the GM during
registration. If all released tokens are valid, the interpolation of the released tokens can
reconstruct the secret s . The published one-way hash of the secret is used to compare with the
one-way hash of the reconstructed secret.
Theorem 1. Protocol 1 can detect any number of illegitimate users.
Proof. If there is illegitimate user who does not own a valid token on the polynomial ( )f x , the
reconstructed secret will be different from the secret s . Thus, Protocol 1 can detect any number
of illegitimate users. □
Protocol 1: One-time group authentication protocol
 Step 1. Each user iM reveals his token ( )if x , to all other users simultaneously.
 Step 2. After knowing all tokens, ( )if x , for 1,2, ,i j  , following Lagrange interpolating
formula, each user computes  
1 1,
( ) mod .
jj
r
i
i r r i i r
x
s f x p
x x  

 

  If ( ) ( )H s H s  , all
users have been authenticated successfully; otherwise, there are illegitimate users.
Remark 2. This is a one-time authentication protocol since the secret and tokens are revealed to
all users in this protocol. The authentication is no longer a one-to-one authentication and it is a
many-to-many authentication. The proposed protocol is very efficient to authenticate multiple
users belonging to the same group without revealing identity of each user.
5. GROUP AUTHENTICATION PROTOCOL WITHOUT REVEALING TOKENS
In Protocol 1, since tokens are revealed to all users, each token can only be used for one-time
authentication. In addition, the secret s is also exposed to users in Protocol 1. In the following
discussion, we propose a way to protect tokens. In addition, the secret does not need to be
recovered in each authentication. Our authentication is based on the property of strong t -
consistency in Section 2.2.
5.1. Group authentication protocol without revealing tokens
In the following protocol, it can be achieved authentication without revealing tokens and the
secret. The basic idea of our approach uses the property of strong t -consistency. Let each user
select a random polynomial with ( 1)t  -th degree and generate shares for other users. Then, each
user releases the additive sum of his own token obtained from the GM during the registration and
sum of shares of polynomials generated by users. Due to the property of secret

14
Protocol 2: Group authentication protocol without revealing tokens
 Step 1. Each user iM selects a random polynomial, ( )if x , with ( 1)t  -th degree. For the
polynomial ( )if x , user iM computes shares as ( )i rf x , for 1,2, , ,r j r i   , for other
users. User iM sends each share, ( )i rf x to user rM secretly.
 Step 2. After receiving ( )r if x for 1,2, ,r j  , each user uses his token ( )if x to compute
 ''
1
( ) ( ) mod
j
i i r i
r
y f x f x p

   . Each user releases his value iy  .
 Step 3. After knowing ''
iy , for 1,2, ,i j  , each user checks whether they are strong t -
consistent. If they are not strong t -consistent, there are illegitimate users; else, all users have
been successfully authenticated belonging to the same group.
sharing homomorphism in Section 2.2, the released sums are shares of the secret polynomial
( )f x of tokens and sum of polynomials generated by users. If all users act honestly and own
valid tokens, the released sums should be strong t -consistent; otherwise, the released sums are
not strong t -consistent. Since users do not need to reconstruct the secret in the protocol and the
tokens have not been revealed directly, the dealer does not need to publish the one-way of the
secret s during system set up and the tokens can be reused.
Theorem 2. Protocol 2 can detect any number of illegitimate users.
Proof. Due to the property of secret sharing homomorphism, each released value, ''
iy in Step 2 is
the share of additive sum of polynomials,  
1
( ) ( ) mod
j
r
r
f x f x p

  , with ( 1)t  -th degree.
Thus, in Step 3, all released values, ''
iy , for 1,2, ,i j  , are strong t -consistent. If there is any
illegitimate user who does not own a valid token, ( )if x , the released values, ''
iy , for
1,2, ,i j  , are not strong t -consistent with very high probability. □
Remark 3. In Step 2, the token ( )if x cannot be computed from the revealed value
 ''
1
( ) ( ) mod
j
i i r i
r
y f x f x p

   . Therefore, the tokens are protected unconditionally and can
be reused for multiple authentications.
5.2. Computational complexity
The most time-consuming operation for each user is to check the strong t -consistency of released
values ''
iy for 1,2, ,i j  , in Step 3 of Protocol 2. Following our discussion presented in
Section 2.2, checking strong t -consistency needs to compute the interpolating polynomial of
values ''
iy . The polynomial interpolation becomes the main computational task in our proposed
protocol. However, the modulus p in our polynomial interpolation is much smaller than the

15
modulus in most public-key cryptosystems, such as RSA cryptosystem [13]. In addition, not like
conventional user authentication protocol that authenticates one user at a time, this proposed
authentication protocol authenticates all users at once. Thus, the proposed authentication protocol
is very efficient in comparing with all existing authentication protocols.
6. CONCLUSIONS
We propose a special type of group authentication which is specially designed for group
communications such as the ad hoc wireless communication network. The proposed group
authentication protocol is no longer a one-to-one type of user authentication and it is a many-to-
many type of authentication that authenticates multiple users at once. We first propose an basic
one-time group authentication protocol and then propose a general group authentication protocol
without revealing tokens. Our proposed group authentication is very efficient since the
computation is based on the computation of linear polynomial.
ACKNOWLEDGEMENTS
This research is supported by the National Natural Science Foundations of China under Grant No.
61103247 and the Natural Science Foundation of Fujian Province under Grant No. 2011J05147.
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16
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Authors
Lein Harn received the B.S. degree in electrical engineering from the National Taiwan
University in 1977, the M.S. degree in electrical engineering from the State University of
New York-Stony Brook in 1980, and the Ph.D. degree in electrical engineering from the
University of Minnesota in 1984. In 1984, he joined the Department of Electrical and
Computer Engineering, University of Missouri- Columbia as an assistant professor, and in 1986, he moved
to Computer Science and Telecommunication Program (CSTP), University of Missouri, Kansas City
(UMKC). While at UMKC, he went on development leave to work in Racal Data Group, Florida for a year.
His research interests include cryptography, network security, and wireless communication security. He has
published a number of papers on digital signature design and applications and wireless and network
security. He has written two books on security. He is currently investigating new ways of
using secret sharing in various applications.
Changlu Lin received the BS degree and MS degree in mathematics from the Fujian
Normal University, P.R. China, in 2002 and in 2005, respectively, and received the Ph.D
degree in information security from the state key laboratory of information security,
Graduate University of Chinese Academy of Sciences, P.R. China, in 2010. He works
currently for the School of Mathematics and Computer Science, and the Key Laboratory of Network
Security and Cryptology, Fujian Normal University. He is interested in cryptography and network security,
and has conducted research in diverse areas, including secret sharing, public key cryptography and their
applications.

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