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**Handbook of applied cryptography.**
*(English)*
Zbl 0868.94001

CRC Press Series on Discrete Mathematics and its Applications. Boca Raton, FL: CRC Press. xxviii, 780 p. (1997).

This is an encyclopedic volume on the current status of the theory and practice of modern cryptography. The nature of cryptography and cryptographic protocols are fully presented, giving both a thorough treatment of the underlying theory and the practicalities of implementation. Thus both lower level topics such as random number generation and efficient implementation of number theoretic algorithms as well as more theoretical topics as zero-knowledge proofs, are given extensive treatment. The material has what the authors refer to as a functional organization where material of interest to an end user, such as authentication, is addressed in a single chapter. This is as opposed to what might be termed an academic organization where systems and protocols depending on the same concept, such as zero-knowledge, might be given in a single chapter.

To briefly cover the contents of the volume, the introductory chapters one to three give a broad overview of the field of modern cryptography and the mathematical background required for its study, including the number-theoretic background for factorization, a discussion of the discrete logarithm problem, the subset sum problem and the factorization of polynomials over finite fields. The fourth chapter considers primality testing and prime generation as it relates to choosing parameters for public key systems. The next three chapters deal with the generation of pseudorandom sequences, stream ciphers and block ciphers. The public key encryption systems of RSA, ElGamal, McEliece and knapsack, among others, are considered in chapter 8. All of the important hash functions and their use in data integrity and message authentication systems are discussed, followed by chapters on identification and entity authentication and digital signatures. Chapters 12 and 13 address the problems of key establishment and key management. Techniques for the efficient implementation of multi-precision integer arithmetic, including modular arithmetic, greatest common divisor algorithms, the Chinese remainder theorem, and exponentiation are given in Chapter 14. The final chapter gives a comprehensive review of the important patents and standards. The single appendix lists the titles of papers in the Proceedings of all the Asiacrypt/Auscrypt, Crypto, Eurocrypt and Fast Software Encryption Conferences as well as the Table of Contents of the Journal of Cryptology. An extensive bibliography of some 1276 references is given. It is of interest that there is no treatment of elliptic curve cryptosystems since two of the authors are major contributors to that field.

The material is exceptionally well organized and thoroughly treated. As Professor Rivest notes in his Foreword, ”I am happy to ... inform the reader that he/she is looking at a landmark in the development of the field.” The volume is a major contribution to the field of cryptography that will be serve as the standard reference for both theoretical researchers and practitioners alike for the foreseeable future.

To briefly cover the contents of the volume, the introductory chapters one to three give a broad overview of the field of modern cryptography and the mathematical background required for its study, including the number-theoretic background for factorization, a discussion of the discrete logarithm problem, the subset sum problem and the factorization of polynomials over finite fields. The fourth chapter considers primality testing and prime generation as it relates to choosing parameters for public key systems. The next three chapters deal with the generation of pseudorandom sequences, stream ciphers and block ciphers. The public key encryption systems of RSA, ElGamal, McEliece and knapsack, among others, are considered in chapter 8. All of the important hash functions and their use in data integrity and message authentication systems are discussed, followed by chapters on identification and entity authentication and digital signatures. Chapters 12 and 13 address the problems of key establishment and key management. Techniques for the efficient implementation of multi-precision integer arithmetic, including modular arithmetic, greatest common divisor algorithms, the Chinese remainder theorem, and exponentiation are given in Chapter 14. The final chapter gives a comprehensive review of the important patents and standards. The single appendix lists the titles of papers in the Proceedings of all the Asiacrypt/Auscrypt, Crypto, Eurocrypt and Fast Software Encryption Conferences as well as the Table of Contents of the Journal of Cryptology. An extensive bibliography of some 1276 references is given. It is of interest that there is no treatment of elliptic curve cryptosystems since two of the authors are major contributors to that field.

The material is exceptionally well organized and thoroughly treated. As Professor Rivest notes in his Foreword, ”I am happy to ... inform the reader that he/she is looking at a landmark in the development of the field.” The volume is a major contribution to the field of cryptography that will be serve as the standard reference for both theoretical researchers and practitioners alike for the foreseeable future.

Reviewer: I.F.Blake (Palo Alto)

### MSC:

94-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to information and communication theory |

94A60 | Cryptography |

### Keywords:

cryptography; primality testing; public key systems; pseudorandom sequences; stream ciphers; block ciphers; hash functions; authentication; digital signatures; key management; multi-precision integer arithmetic; identification
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\textit{A. J. Menezes} et al., Handbook of applied cryptography. Boca Raton, FL: CRC Press (1997; Zbl 0868.94001)

### Digital Library of Mathematical Functions:

§27.16 Cryptography ‣ Applications ‣ Chapter 27 Functions of Number Theory### Online Encyclopedia of Integer Sequences:

Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime.Degrees of primitive irreducible trinomials: n such that 2^n - 1 is a Mersenne prime and x^n + x^k + 1 is a primitive irreducible polynomial over GF(2) for some k with 0 < k < n.

Smallest odd number for which Miller-Rabin primality test on bases <= n-th prime does not reveal compositeness.

Values of k corresponding to A073571.

Irreducible trinomials: numbers n such that x^n + x^k + 1 is an irreducible polynomial (mod 2) for some k with 0 < k < n.

Primitive irreducible trinomials: x^n + x^k + 1 is a primitive irreducible polynomial (mod 2) for some k with 0 < k < n.

Smallest k such that trinomial x^A001153(n) + x^k + 1 over GF(2) is primitive.

Values of k corresponding to A073726.

Number of prime divisors (counted with multiplicity) of n such that the primitive irreducible trinomial x^n + x^k + 1 is a primitive irreducible polynomial (mod 2) for some k with 0 < k < n (A073726).

Totients of the Blum integers.

Irregular triangle read by rows: row n lists values of k in range 1 <= k <= n/2 such x^n + x^k + 1 is irreducible (mod 2), or -1 if no such k exists.

Irregular triangle read by rows: row n lists values of k in range 1 <= k <= n-1 such x^n + x^k + 1 is irreducible (mod 2), or -1 if no such k exists.