In mathematical cryptography, a Kleinian integer is a complex number of the form ${\displaystyle m+n{\frac {1+{\sqrt {-7}}}{2}}}$, with m and n rational integers. They are named after Felix Klein.

The Kleinian integers form a ring called the Kleinian ring, which is the ring of integers in the imaginary quadratic field ${\displaystyle \mathbb {Q} ({\sqrt {-7}})}$. This ring is a unique factorization domain.

• Eisenstein integer
• Gaussian integer

• Conway, John Horton; Smith, Derek A. (2003), On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, A. K. Peters, Ltd., ISBN 978-1-56881-134-5. (Review).
• Dimitrov, V. S.; Järvinen, K. U.; Jacobson, M. J.; Chan, W. F.; Huang, Z. (2006), "FPGA Implementation of Point Multiplication on Koblitz Curves Using Kleinian Integers", Cryptographic Hardware and Embedded Systems - CHES 2006, Lecture Notes in Computer Science, vol. 4249, pp. 445–459, doi:10.1007/11894063_35, ISBN 978-3-540-46559-1

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