An electronic signature does not have to be cryptographically based at all to be legally valid in united-states law. There are no US laws that I can find specifying this level of detail on what makes an electronic signature valid. A particular contract might specify what key servers are requited.
The Law of E-Signatures in the United States and Canada (an article published by the law firm of Baker And Mckenzie) says:
The E-Sign Act is a federal law that applies to interstate commerce, namely transactions across states of the United States and with foreign nations. ... UETA provides a framework for states to enact state law concerning the enforceability of e-signatures and the validity of electronic records. Forty-seven states and the District of Columbia, Puerto Rico and the US Virgin Islands have adopted some form of UETA.
In general terms, both the UETA and the E-Sign Act define an "electronic signature" as "an electronic sound, symbol, or process attached to or logically associated with a record and executed or adopted by a person with the intent to sign the record" and an "electronic record" as a record that is "created, generated, sent, communicated, received, or stored by electronic means". UETA and the E-Sign Act provide that: (a) a record or signature may not be denied legal effect or enforceability solely because it is in electronic form; (b) a contract may not be denied legal effect or enforceability solely because an electronic record was used in its formation; (c) if a law requires a record to be in writing, an electronic record satisfies the law; and (d) if a law requires a signature, an electronic signature satisfies the law.
15 U.S. Code § 7001 (part of the ESIGN act) provides that:
(a) In generalNotwithstanding any statute, regulation, or other rule of law (other than this subchapter and subchapter II), with respect to any transaction in or affecting interstate or foreign commerce—
(a) (1) a signature, contract, or other record relating to such transaction may not be denied legal effect, validity, or enforceability solely because it is in electronic form; and
(a) (2) a contract relating to such transaction may not be denied legal effect, validity, or enforceability solely because an electronic signature or electronic record was used in its formation.