Provided are a key establishment method and system using commutative
linear functions. In the method, a server defines a set of linear
functions that use elements of a first finite field as coefficients and
satisfy a commutative rule, selects a first linear function from the set,
and selects a predetermined element from a second finite field. Next, the
server selects a second linear function corresponding to each of nodes
from the set, generates a predetermined combination function based on the
first and second linear functions, generates a value of the second linear
function using the selected element as a factor, and transmits the
combination function and the value of the second linear function to a
corresponding node. Each node receives the value of the second linear
function from a server, exchanges the received values with the other
nodes, computes a value using the exchanged value as a factor of the
combination function, and establishes the computed value as a shared key
between the nodes. Therefore, each node can perform key establishment
with a small amount of computation and low memory consumption, while
guaranteeing end-to-end security.