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If I understand your question correctly, you are essentially asking if points in Edwards and Montgomery curves can be represented in Weierstrass coordinates. This is true; in fact, any elliptic curve over a prime field can be represented in Weierstrass form $\mathcal{E}_{w}^{a, b} : y^2 = x^3 + ax + b$, and by extension its points can too. The question, ...


Your idea seems secure, but it is more complicated than it needs to be. If you only need to know if the "secure information" in the document matches some public information (stored on an insecure computer), it is enough to calculate a preimage resistant hash of the document (e.g. SHA-256). The document needs only to be unguessable, like a 256-bit random ...

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