A New Security Measure in Secret Sharing Schemes and Secure Network Coding

Linear secret sharing schemes protect secret information from leakage and destruction by encoding secret information into multiple shares, where the secret information can be reconstructed by collecting a certain subsets of shares. Perfect Security,
$\alpha $
-strong Security, and Individual Security (IS) have been given as security measures of linear secret sharing schemes. Consider the threshold for each security measure, which is defined as the maximum allowable size of the set of leaked shares. Kurihara et al. have revealed that thresholds for Perfect Security and
$\alpha $
-strong Security are characterized in terms of a relative code parameter Relative Generalized Hamming Weight (RGHW). However, the threshold for IS is not yet characterized. In this paper, we focus on individual elements of secret information and give the threshold for IS (Individual Security Threshold IST) as a new security measure, where each element of secret information cannot be uniquely determined from subsets of shares with size less than or equal to the IST. We show that the IST can be characterized in terms of RGHW as well as Perfect Security and
$\alpha $
-strong Security. We also give a precoding method for secret information that can guarantee IST above a certain value in any existing linear secret sharing schemes. Moreover, we extend the notion of the IST to universal secure network coding (USNC) and present the Universal IST. We also show that the Universal IST can be represented by the code parameter Relative Generalized Rank Weight (RGRW) similarly to the IST of the linear secret sharing schemes.