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Let (f, g, F, G) be an NTRU digital signature private key and let
be the associated public key. Suppose that (s, t) is the signature on the document D = (D1, D2), so in particular, the vector (s, t) is in the NTRU lattice LNTRU h.
(a) Prove that for every vector w ∈ ZN, the vector
is in the NTRU lattice .
(b) Let f −1 be the inverse of f in the ring R[x]/(xN −1) (cf. Table 7.6). Prove that the vector
is a signature on a document of the form D = (0, D2 + D3) for some D3 that depends on D1.
(c) Conclude that anyone who can sign documents of the form (0, D ) is also able to sign documents of the form (D1, D2). Hence in the NTRU digital signature scheme we might as well assume that the document being signed is of the form (0, D2). This has several benefits, including speeding the computation of v1 and v2.