Paper 2017/548

Fully Homomorphic Encryption from the Finite Field Isomorphism Problem

Yarkın Doröz, Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman, Berk Sunar, William Whyte, and Zhenfei Zhang


If $q$ is a prime and $n$ is a positive integer then any two finite fields of order $q^n$ are isomorphic. Elements of these fields can be thought of as polynomials with coefficients chosen modulo $q$, and a notion of length can be associated to these polynomials. A non-trivial isomorphism between the fields, in general, does not preserve this length, and a short element in one field will usually have an image in the other field with coefficients appearing to be randomly and uniformly distributed modulo $q$. This key feature allows us to create a new family of cryptographic constructions based on the difficulty of recovering a secret isomorphism between two finite fields. In this paper we describe a fully homomorphic encryption scheme based on this new hard problem.

Available format(s)
Public-key cryptography
Publication info
Preprint. MINOR revision.
Finite field isomorphismfully homomorphic encryptionlattice-based cyrptopgraphy
Contact author(s)
ydoroz @ wpi edu
2017-06-08: received
Short URL
Creative Commons Attribution


      author = {Yarkın Doröz and Jeffrey Hoffstein and Jill Pipher and Joseph H.  Silverman and Berk Sunar and William Whyte and Zhenfei Zhang},
      title = {Fully Homomorphic Encryption from the Finite Field Isomorphism Problem},
      howpublished = {Cryptology ePrint Archive, Paper 2017/548},
      year = {2017},
      note = {\url{}},
      url = {}
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