The phrase "not breathe a word" is commonly used to describe the act of keeping something a secret. However, there are several other synonym phrases that can be used to convey the same message. For example, "keep mum" denotes the idea of not sharing information and is similar to "hush-hush," which suggests that something is kept covert or confidential. "Keep under wraps" is another phrase that represents the action of keeping something concealed, while "take to the grave" means keeping a secret until death. All of these phrases convey the idea of not sharing information or keeping something a secret.