If you were fired for cheating, even though it is a personal matter to you, there is no law against airing your dirty linen in public. You cannot sue for defamation because truth is an absolute defense. You cannot sue for invasion of privacy because you have no privacy on this issue. The whole workplace knows. If a client asks what happened, the employer is allowed to, but is not required to, disclose the truth in this case.
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Although the law of defamation may be slightly different in each state, generally truth is a defense. That means that if the statement made by your previous employer is true, then you would lose your case.
Defamation cases are tough and it is hard to say from the facts in your posting. I am linking you to a guide on defamation law below to help you decide whether or not to pursue this.
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This answer is offered for informational purposes only. It is not offered as, and does not constitute, legal advice. Laws vary widely from state to state. You should rely only on the advice given to you during a personal consultation by a local attorney who is thoroughly familiar with state laws and the area of practice in which your concern lies.