Consider the following: A whole system depends on the security of one private key and if this key is, for any reason, compromised, the whole system will be compromised.
What's a suitable name for such an attack?
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The whole system depends on the security of one private key (..that gets..) compromised.
Give a suitable name for such an attack.
I would call this class of attacks: Master key compromise.
Examples of such attacks (as asked in comment) include
It is not an attack model while it is a security feature for a scheme. The adversary might be able to compromise these lack of secrecy by different attacks to obtain the master key and find other session keys. Take a look at this question. Definitions of secrecy