# Is Lamport-Diffie EUF-CMA secure in the standard model?

Is the Lamport-Diffie signature secure in the standard model? I ask this question because I am reading the Postquantum-Cryptography book and the reductions the authors use one-way functions and not hash functions. According to my understanding, ROM is used to replace hash function for random oracles, but in Lamport-Diffie only pre-image resistance functions are sufficient.

However, you can easily construct an adversary which performs 2 interactions with the signature scheme in the EUF-CMA model and achieves an advantage of 1 from that. Such an adversary would e.g. query $$0^n$$ and $$1^n$$ as messages, revealing all private keys.
Also however, we can still construct a security statement, namely $$\mathbf{Adv}^{\textsf{EUF-CMA}}_{\text{LD-OTS}_F}(\mathcal A;q,n)\leq\begin{cases}\mathbf{Adv}^{\textsf{OWF}}_{F}(\mathcal A';n)&q\leq 1\\1&\text{else}\\\end{cases}$$ where $$n$$ is the security parameter, $$q$$ is the number of queries, $$\mathcal A'$$ is an existentially qualified adversary against the underlying OWF $$F$$ and $$\mathcal A$$ is the polynomially-bounded universally qualified adversary. The order of quantifiers is for all $$\mathcal A$$ exists $$\mathcal A'$$.