SCIS 2022
2022 Symposium on
Cryptography and Information Security
Osaka, Japan & Online, Jan. 18 – 21, 2022
The Institute of Electronics,
Information and Communication Engineers

cCopyright ⃝2022
The Institute of Electronics,
Information and Communication Engineers

Mathematical Structure of Finsler Encryption and Signature
Tetsuya NAGANO ∗Hiroaki ANADA †Keywords: Finsler encryption, Public-key encryption, Digital signature, Diﬀerential geometry

Abstract

is solved, then we can obtain the plaintext v.
According to the above decryption algorithm, weFinsler encryption was defined by Nagano and Anada
have two mappings. One of them is a projection prat [3]. The basic structure of this encryption is supfrom R9 to R2 , and the other is a linear mapping ppkτported by the diﬀerential geometry, especially, Finsler
from R2 to R2 . If the cipher is made by satisfying ageometry(cf.[1],[2]). Thus, all items of Finsler encrypcertain condition, then ppkτ is regular. Therefore thetion are initially introduced by a real and smooth manimapping ppkτ−1 ◦ pr is constructed and gives the plainfold and have continuous real forms. In general, encryptext v from the cipher. However, ppkτ−1 ◦ pr depends
tion system is constructed in a discrete mathematicalon the pair (plaintext, its ciphertext).
system(cf.[5]). By quantization, however, we succeed inFinally, we introduce a digital signature system basedchanging their forms into integer and rational forms.
on this Finsler encryption. It is very easy. For the keys
In this paper, firstly, we study the mathematicalof Finsler encryption (P K, SK), we should put only thestructure of the encryption and decryption algorithm ofencryption key P K as the the signature(secret) key sk,Finsler encryption([4]). A plaintext is a 2-dimensional
namely, sk := P K, and the decryption key SK as thevector v. The space of plaintext is the first quadrant
verification(public) key vk, namely, vk := SK, respecof the tangent plane R2 . The algorithm of encoding is,tively. In addition, we state the encryption strength is
first, transforming by the linear parallel displacement(cf.[2])similarly to one of Multivariate Public Key Cryptsyswith parameter βi for a vector v, next, calculating thetems(MPKC).
energy of a transformed vector and dividing it. Lastly,
we transforms a obtaining vector from dividing by usReferences
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trix of linear parallel displacement with τ , we prepare
from Diﬀerential Geometry with Example”, Inthree the energy forms of vector obtained by the linear
novative Security Solutions for Information Techparallel displacement with parameter βi for a vector v.
nology and Communications 2021,Springer Nature,
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Finsler Encryption”, 2021-CSEC-95, No.6 of IPSJ
formal solution is input the energy equation(one of seSIG Technical Report.
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Cryptography, Second Edition, CRC Press, Florida∗ University of Nagasaki, 1-1-1, Manabino, Nagayocho, Nagasaki(2014).
prefecture, Japan (hnagano@sun.ac.jp)† ∗ (anada@sun.ac.jp)1