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Public key cryptosystems are cryptosystems in which the encryption key is public, whereas the decryption key is private. This book is dedicated to the public key cryptosystems including digital signature algorithms, which have been developed based on mathematical fundamentals and the security of the public key cryptosystems is also based on the hardness of some mathematical problems. In Section 1.2, we first provide some basic properties of modular arithmetic since this operation is part of many public key cryptosystems. We also give the so-called fast powering algorithm related to modular arithmetic to be used for the solution of a fundamental mathematical problem in many public key cryptosystems. The “Chinese remainder theorem” is related to the solution of a set of linear congruences, and it is used in some public key cryptosystems to speed up the decryption process. Although the concepts “quadratic residue” and “quadratic nonresidue” are applicable only in some public key cryptosystems, it is still given in Section 1.2 to provide a mathematical background. Since many public key cryptosystems work in groups, fields, or rings, Section 1.3 is devoted to the basic properties of groups, fields, and rings including some other topics such as “group homomorphism”, “polynomial ring”, “primitive root of a finite field”, and “ring of convolution polynomials”. Divisibility, greatest common divisors, and prime numbers are also critical in many public key cryptosystems; thus, Section 1.4 has been devoted to these topics including sub-topics “Fermat’s little theorem”, “Euclidean algorithm”, and “extended Euclidean algorithm”. The “Euclidean algorithm” and “extended Euclidean algorithm” can also be extended to polynomials, which are provided in Section 1.5. The main reason for providing Section 1.5 is that some special public key cryptosystems use polynomials as part of encryption and decryption steps. The security of some public key cryptosystems is based on the hardness of factoring a large integer number into two prime numbers; thus, Section 1.6 includes basic selected topics related to “integer factorization related selected mathematical fundamentals”. “Elliptic curve cryptography” is a distinctive area in cryptography and elliptic curve analogs of various public key cryptosystems and digital signature algorithms can be developed. Section 1.7 has been devoted to some selected basics of elliptic curves including some subtopics such as “elliptic curves as an Abelian group”, “point addition in elliptic curves”, and “bilinear pairings, Weil pairings, distortion maps, and modified Weil pairings on elliptic curves”. “Lattice-based cryptography” is also another distinctive area; thus, the basics of lattices will be given in Section 1.8 in comparison to vector spaces. Moreover, the lattice-related properties including fundamental domain, Hadamard’s inequality, Hadamard ratio, and fundamental lattice problems will also be provided in this section. Finally, some selected basics of probability will be introduced in Section 1.9 since randomness is part of many public key cryptosystems to increase security. The topics provided in this chapter of the book are applicable in different parts of Chapters 3 to 6.