Exact algorithms (e.g. in coding theory, cryptography) using the field of rational numbers

Question

I noticed that most algorithms in coding theory or cryptography are based on the integers and some arithmetic results (e.g. RSA) or on the finite fields (e.g. Elliptic curve cryptography or BCH codes).

The field $\mathbb{Q}$ is of infinite cardinality, which I suppose makes it less interesting for most applications (e.g. elliptic curves may have an infinite number of points, we cannot easily control the memory required for exact representation, etc.).

Despite this, I was wondering if there were algorithms using rational numbers and their properties as a field (or some extension of $\mathbb{Q}$) with real usage?

Answer 1

Not a direct answer to “algorithms over $\mathbb{Q}$”, but there are constructions of error-correcting codes over fields of characteristic zero, which is the closest I know of. I’ll throw some references below — but you are right that in general, if it’s not a discrete set, it’s difficult to utilize for information transmission applications (coding, cryptography, etc.).

• D. Augot, P. Loidreau, and G. Robert, “Rank metric and Gabidulin codes in characteristic zero,” in 2013 IEEE International Symposium on Information Theory, pp. 509–513, IEEE, 2013.
• S. Muelich, S. Puchinger, D. M ̈odinger, and M. Bossert, “An alternative decoding method for Gabidulin codes in characteristic zero,” in 2016 IEEE International Symposium on Information Theory (ISIT), pp. 2549–2553, IEEE, 2016.
• S. Muelich, S. Puchinger, and M. Bossert, “Low-rank matrix recovery using Gabidulin codes in characteristic zero,” Electronic Notes in Discrete Mathematics, vol. 57, pp. 161–166, 2017.
• C. Sippel, C. Ott, S. Puchinger, and M. Bossert, “Reed-Solomon codes over fields of characteristic zero.” in 2019 IEEE International Symposium on Information Theory (ISIT), pp. 1537–1541, IEEE, 2019.