Patent Publication Number: US-7715556-B2

Title: Key establishment method and system using commutative linear function

Description:
CROSS-REFERENCE TO RELATED PATENT APPLICATIONS 
     This application claims the benefit of Korean Patent Application No. 10-2005-0113850, filed on Nov. 26, 2005, in the Korean Intellectual Property Office, the disclosure of which is incorporated herein in its entirety by reference. 
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a method and system for securely establishing a key for a security service such as a cryptographic service, and more particularly, to a key establishment method and system using commutative linear functions. 
     2. Description of the Related Art 
     In the field of information technologies, various types of cryptographic mechanisms are used to protect data to be transmitted or individual privacy. The safety and reliability of a cryptographic mechanism depend on the safety and management of a cryptographic algorithm to be used, and the management and protection of keys to be used. 
     Insecure key management leads to a fatal threat to the safety of a cryptographic mechanism regardless of the type of a cryptographic algorithm, and therefore, key management is the most basic factor of the cryptographic mechanism. In key management, it is most important to securely distribute keys to the other party concerned in cryptographic communications without disclosing the keys to a third party. 
     It is the most simple key establishment method in which all of nodes share one key. However, this method is disadvantageous in that when one of the nodes is damaged, the shared key is exposed, thus allowing a person who has no authority for overall network traffic to perform a decoding operation. 
     Also, there is another key establishment method in which each node stores a pairwise key to be shared with each of the other nodes. If the number of all of nodes belonging to a network is n, each node stores n−1 pairwise keys for the other nodes. However, this method is disadvantageous in that the greater n is, the greater the number of pairwise keys to be stored, and further, it is difficult to add a node to a network. 
     Conventionally, a public key-based key establishment method is very often used in a general network. This method requires each node to store only its public key and secret key, thus solving problems related to key storing. However, this method requires a lot of amount of computation. Therefore, this method is difficult to be applied to an Ad-hoc environment that undergoes a limitation to resources, and in particular, a sensor network environment. 
     Recently, many researches have been conducted into key establishment in the sensor network environment that is very limited to resources available. In particular, most of key establishment methods are designed based on the Blom (EUROCRYPT 84) method and the Blundo (CRYPTO 92) method. Such key establishment methods are performed in a hop-by-hop fashion without using an end-to-end method. 
     Meanwhile, the key establishment methods in the hop-by-hop fashion in the sensor network are confronting many problems more and more. In particular, the Blom method and the Blundo method require finite field multiplication to be performed several tens times, thereby increasing the load on a sensor node that is limited to resources. 
     SUMMARY OF THE INVENTION 
     The present invention provides a key establishment method and system for guaranteeing end-to-end security that allows each node to establish a key while reducing memory consumption and computational complexity. 
     The present invention also provides a computer readable medium in which a program for executing the key establishment method in a computer is stored. 
     According to an aspect of the present invention, there is provided a key establishment method comprising a server defining a set of linear functions which use elements of a first finite field as coefficients and satisfy a commutative rule, selecting a first linear function from the set, and selecting a predetermined element from a second finite field; the server selecting a second linear function corresponding to each node from the set; generating combination function based on the first linear function and the second linear function; generating a value of the second linear function using the selected element as a factor; and transmitting the combination function and the value of the second linear function to a corresponding node; and each node exchanging the value of the second linear function received from the server with the other nodes, computing a value of the combination function using the exchanged value as a factor, and establishing the computed value as a shared key among the nodes. 
     According to another aspect of the present invention, there is provided a key establishment system comprising a key generating unit defining a set of linear functions which use elements of a first finite field as coefficients and satisfy a commutative rule, selecting a first linear function from the set, and selecting a predetermined element from a second finite field; and a key allocating unit selecting a second linear function corresponding to each node from the set, generating a combination function based on the first and second linear functions, generating a value of the second linear function using the selected element as a factor, and transmitting the combination function and the value of the second linear function to a corresponding node. 
     Accordingly, each node is capable of establishing a key while reducing computational complexity and memory consumption. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The above and other aspects and advantages of the present invention will become more apparent by describing in detail exemplary embodiments thereof with reference to the attached drawings in which: 
         FIG. 1  is a flowchart of a key establishment method according to an embodiment of the present invention; 
         FIG. 2  is a diagram illustrating a key establishment method performed from a server to node according to an embodiment of the present invention; 
         FIG. 3  is a method of establishing a key to be shared between nodes according to an embodiment of the present invention; and 
         FIG. 4  is a block diagram of a key establishment system according to an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Hereinafter, a key establishment method and system using commutative linear functions, according to embodiments of the present invention, will be described in greater detail with reference to the accompanying drawings. 
     First, a set F to be used in the present invention is as follows: 
                     F   =     {         ∑     i   =   0       m   -   1       ⁢       α   i     ⁢     x     2   i           |       α   i     ∈     G   ⁢           ⁢     F   ⁡     (   2   )             }       ,           (   1   )               
wherein elements
 
             f   =         ∑     i   =   0       m   -   1       ⁢       α   i     ⁢     x     2   i           ∈   F           
and s,tεGF(2 m ) satisfy an equation ƒ(s+t)=ƒ(s)+ƒ(t). That is, an arbitrary element of the set F is comprised of a linear function GF(2 m )→GF(2 m ). Also, two functions ƒ,gεF satisfy ƒ∘g(x)=g∘ƒ(x), i.e., ƒ(g(x))=g(ƒ(x)).
 
       FIG. 1  is a flowchart of a key establishment method according to an embodiment of the present invention. Referring to  FIG. 1 , a server  200  of  FIG. 2  randomly selects elements ƒεF and αεGF(2 m ), and maintains them as secret information (S 100 ). The secret information must be protected not to be exposed to an attacker or nodes that are not concerned. 
     The server  200  selects h i εF as a secret value of each node i, and allocates ƒ∘h i (x) mod(x 2     m   −x) and h i (α) to each of first through n th  nodes i  210 ,  220 , . . . ,  230 , using the secret information ƒεF and αεGF(2 m ) which are selected in operation S 100  (S 110 ).  FIG. 2  illustrates an example of a key assignment method. 
     After receiving ƒ∘h i (x)mod(x 2     m   −x) and h i (α), each node i discloses h i (α), and maintains ƒ∘h i (x)mod(x 2     m   −x) as a secret value so that an attacker or the other nodes cannot recognize it. 
     Given predetermined values from the server  200 , each node i establishes a shared key among the other nodes (S 120 ). A method of establishing a key to be shared between nodes will now be described in greater detail with reference to  FIG. 3 . 
     The first node  210  stores ƒ∘h 1 (x)mod(x 2     m   −x) and h 1 (α), which are given from the server  200  in operation S 110 , in its memory, and the second node  220  stores ƒ∘h 2 (x)mod(x 2     m   −x) and h 2 (α) in its memory. 
     The first node  210  transmits h 1 (α) to the second node  220 , and the second node  220  transmits h 2 (α) to the first node  210 . The first node  210  computes ƒ∘h 1 (h 2 (α)) using received h 2 (α). ƒ∘h 1 (h 2  (α))=ƒ∘h 1 ∘h 2 (α)=ƒ∘h 2 (h 1 (α)) is drawn from the commutative characteristics of the above linear functions. Thus, the first and second nodes  210  and  220  set ƒ∘h 1 ∘h 2 (α) as a shared key between them. 
     A key establishment method according to the present invention has many advantages compared to the prior art. 
     Since h i (α) of a node i is an element of GF(2 m ) and ƒ∘h i (x)mod(x 2     m   −x) is a polynomial having coefficients of m GF(2), each node i stores 2m bits. 
     Next, the computational complexity of a key establishment method according to the present invention will now be described. 
     As described above, the first node  210  computes ƒ∘h 1 (h 2 (α)) to obtain a shared key to be shared with the second node  220 . 
     Since may be ƒ∘h 1 (x)mod(x 2     m   −x)εF, h 2 (α)εGF(2 m ), ƒ∘h 1 (h 2 (α)) may be computed by computing 
                 p   ⁡     (   γ   )       =         ∑     i   =   0       m   -   1       ⁢       α   i     ⁢     γ     2   i       ⁢           ⁢   when   ⁢           ⁢   p       =         ∑     i   =   0       m   -   1       ⁢       α   i     ⁢     x     2   i           ∈   F         ,     γ   ∈       GF   ⁡     (     2   m     )       .             
Here, γ,γ 2 , . . . , γ 2     m-1   , which is a square of γ, needs to be computed.
 
     A normal basis is introduced to effectively express a square operation in GF(2 m ) which is a finite field. When expressing an element in GF(2 m ) using the normal basis, a square of the element may be expressed by simply using a shift operation. 
     Based on the characteristics of the normal basis and that a coefficient α i  of P is 0 or 1, P(γ) can easily be computed by performing the shift operation m−1 times and performing an addition operation on GF(2 m ) having m elements. In this case, computational complexity is very lower than in a conventional key establishment method, and the present invention can be embodied as hardware that is much smaller than a conventional scheme. 
     Also, in the case of the conventional key establishment method, when direct key sharing is not allowed in the sensor network, a key is transmitted in the hop-by-hop fashion, and thus, the key is disclosed to an intermediate node. However, in the key establishment method according to the present invention, only a public value h i (α) is transmitted to a destination node, and therefore, a shared key is not disclosed to an intermediate node or an attacker even if the public value h i (α) is exposed. That is, the key establishment method according to the present invention provides end-to-end security. 
     In addition, according to the present invention, it is easy to add or delete a node. When a new node j is added, a server randomly generates h j εF, and allocates h j (α) and ƒ∘h j (x)mod(x 2     m   −x) to the node j. In this case, the other nodes are not influenced by deletion of a node. 
       FIG. 4  is a block diagram of a key establishment system according to an embodiment of the present invention. Referring to  FIG. 4 , the key establishment system includes a server  400  comprised of a key generating unit  410  and a key allocating unit  420 , and more than one node given a key from the server  400 , i.e., first through n th  nodes  430 ,  440 , . . . ,  450 . 
     The key generating unit  410  defines a set 
               F   =     {         ∑     i   =   0       m   -   1       ⁢       α   i     ⁢     x     2   i           |       α   i     ∈     G   ⁢           ⁢     F   ⁡     (   2   )             }       ,         
that is a set of linear functions that use elements of a first finite field GF(2) as coefficients and satisfy commutative rules, selects a first linear function ƒεF from the set F, and selects a predetermined element αεGF(2 m ) from a second finite field.
 
     The key allocating unit  420  selects a second linear function h i εF indicating each node from the set F of the linear functions, defined by the key generating unit  410 ; generates a combination function ƒ∘h i (x)mod(x 2     m   −x) based on the selected second linear function h i εF and the first linear function ƒεF selected by the key generating unit  410 ; generates the values h i (α) of the second linear functions, each using as a factor a predetermined element selected by the key generating unit  410 , and transmits the combination function ƒ∘h i (x)mod(x 2     m   −x) and the value h i (α) to each of the nodes  430 ,  440 , . . . ,  450 . 
     A key setting unit  432  of the first node  430 , a key setting unit  442  of the second node  440 , and a key setting unit  452  of the n th  node  450  respectively receive the value h i (α) of the second linear function, exchange the received value h i (α) with those of the other nodes, compute ƒ∘h i (h j (α))=ƒ∘h i ∘h j (α)=ƒ∘h j (h i (α)) using the exchanged value h i (α) as a factor of the combination function ƒ∘h i (x)mod(x 2     m   −x) and set ƒ∘h i (h j (α))=ƒ∘h i ∘h j (α)=ƒ∘h j (h i (α)) as a shared key between them. 
     The present invention can be embodied as computer readable code in a computer readable medium. Here, the computer readable medium may be any recording apparatus capable of storing data that is read by a computer system, e.g., a read-only memory (ROM), a random access memory (RAM), a compact disc (CD)-ROM, a magnetic tape, a floppy disk, an optical data storage device, and so on. Also, the computer readable medium may be a carrier wave that transmits data via the Internet, for example. The computer readable medium can be distributed among computer systems that are interconnected through a network, and the present invention may be stored and implemented as a computer readable code in the distributed system. 
     As compared to the prior art, a key establishment method according to the present invention has the following advantages. 
     1. It is possible to establish a shared key between nodes with low memory consumption. 
     2. It is possible to establish a shared key between nodes with a very small amount of computation. In particular, the key establishment method according to the present invention is accomplished by performing only a shift operation and an addition operation on GF(2 m ). Therefore, this method can be realized as hardware that is far smaller than in the prior art. 
     3. It is possible to effectively add a node to or delete a node from a network, and further support a network with a large number of elements. 
     4. It is possible to provide end-to-end security that a conventional sensor network does not support. 
     As described above, a key establishment method according to the present invention can be used for key establishment for cryptographic mechanisms, and security services, such as a certification service, in various environments. In particular, the present invention provides a key establishment method optimized for an Ad-hoc network environment that is limited to resources. 
     While this invention has been particularly shown and described with reference to exemplary embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.