In a network with capacity
h for multicast, information
Xh=(
X1,
X2, …,
Xh) can be transmitted from a source node to sink nodes without error by a linear network code. Furthermore, secret information
Sr=(
S1,
S2, …,
Sr) can be transmitted securely against wiretappers by
k-secure network coding for
k≤
h-
r. In this case, no information of the secret leaks out even if an adversary wiretaps
k edges, i. e. channels. However, if an adversary wiretaps
k+1 edges, some
Si may leak out explicitly. In this paper, we propose strongly
k-secure network coding based on strongly secure ramp secret sharing schemes. In this coding, no information leaks out for every (
Si1,
Si2, …,
Sir-j) even if an adversary wiretaps
k+
j channels. We also give an algorithm to construct a strongly
k-secure network code directly and a transform to convert a nonsecure network code to a strongly
k-secure network code. Furthermore, some sufficient conditions of alphabet size to realize the strongly
k-secure network coding are derived for the case of
k<
h-
r.
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