SIAM REVIEWVol. 30, No. 3, September 1988

(C) 1988 Society for Industrial and Applied Mathematics011

BOOK REVIEWS

EDITED BY GEORGE HANDELMAN

Nonlinear Diffusive Waves. By P. L. Sach-dev. Cambridge University Press, Cam-bridge, UK, 1987. viii + 246 pp. $49.50.ISBN 0-521-26593-2.

This monograph focuses on scalarequations of Burgers’ type and their gener-alizations. The title was chosen to distinguisha third type of nonlinear wave that is diffu-sive--a class epitomized by Burgers’ equa-tions u, + uux au-.. Chapter 2 begins witha heuristic derivation of Burgers’ equation.Special solutions to initial value problemsare found by utilizing the Forsyth-Hopf-Cole transformation, specifically the travel-ing shock wave, the single hump, and theN-wave. Boundary value problems insemi-infinite domains are also discussed.

Chapter 3 is devoted to generalizedBurgers’ equations. These are derived byusing singular perturbation and multiplescaling techniques as applied to theNavier-Stokes equations, for example. Non-planar generalized equations are obtainedand their N-waves determined by singularperturbation methods. Various generaliza-tions of the Forsyth-Hopf-Cole transfor-mation are found to be useful in reducingthe Navier-Stokes momentum equations(Cole), and wider classes are obtained by Chuand the author (Quart. Appl. Math 23,(1965)).

In Chapter 4 the author turns to theutilization of subgroups (scaling and trans-lation) of the symmetry group for nonlineardiffusion equations. The main reason forintroducing this chapter is to compare thesolutions with and without convection andto pave the way for stability analysis. Fisher’sequation, various nonlinear diffusion equa-tions, and a model from plasma physics areexamined. Finally, knowledge unattainableby analytic means is sought by numericalsimulation in Chapter 5. The numerical toolsare pseudospectral methods and implicit fi-

Publishers are invited to send books for re-view to Professor George Handelman, DepartmentofMathematical Sciences, Rensselaer PolytechnicInstitute, Troy, New York 12180-3590.

nite difference procedures. A reasonable bib-liography and indices round out the volume.

The author has done a creditable job incollecting and distilling information appro-priate to his narrow mission. The book isrecommended for this collection to engineersand scientists with problems of Burgers’ typeand certain limited nonlinear diffusionequations.

W. F. AMESGeorgia Institute of Technology

The Simplex Method: A Probabilistic Analy-sis. By Karl Heinz Borgwardt. Springer-Verlag, Berlin, 1987. xii + 268 pp. $3800,paper. ISBN 0-387-17096-0 (US). Algo-rithms and Combinatorics, Study andResearch Texts, Vol. 1.

The author has been studying the ex-pected number of steps used by the simplexalgorithm on randomly generated problemsin a series of papers since 1977. This booksynthesizes the preceding work and attemptsto make it easier to understand. The authorhas given numerous diagrams to convey geo-metric insight (this is especially helpful whenmultiple integrals related to volumes of partsof n-spheres are involved), a numericalexample to illustrate the version of thesimplex algorithm under study, and anappendix dealing with bounds on gammaand beta functions. The work is still ratherdemanding, and involves quite a bit of"hard analysis."

The author discusses the relationshipof his model to those used by others [1 ]-[3],[5], especially in a thoughtful section entitled"What is the ’Real World Model’?" How-ever, this will probably only be understoodby those already familiar with the papersdiscussed. A more accessible survey is [4].

There is also a brief but technical dis-cussion of the Karmarkar and ellipsoid al-gorithms. think the current consensus isthat the Karmarkar algorithm is significantlybetter than the simplex algorithm on somevery large practical problems, but nobody

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BOOK REVIEWS 51

claims the simplex algorithm is no longeruseful.

Two excerpts illustrate the author’sfriendly but occasionally awkward style: "Ihave used the ’we’-form in the book in orderto include the reader into the considerationsand to let him participate. And, of course,my English is not perfect. Please do notmind!"

"It was not my aim to demonstrate theconsiderations very briefly and concisely, be-cause have the seldom chance to motivateand to make ideas plausible. Simultaneously,want to demonstrate the proofs explicitly,

leaving little effort to the reader."

REFERENCES

I. ADLER, R. KARP, AND A. SHAMIR, A simplexmethod solving an m x d linear program inO(min(m2, d2)) expected number of pivotsteps, Math. Oper. Res., to appear.

[2] C. BLAIR, Random linearprograms with manyvariables andfew constraints, Math. Program-ming, 34 (1986), pp. 62-71.

[3] M. HAIMOVICH, The simplex algorithm is verygood! Tech. Report, Columbia Univ., NewYork, April 1983.

[4] A. SHAMIR, The efficiency of the simplexmethod: a survey, Management Sci., 33(1987), pp. 301-334.

[5] S. SMALL, On the average speed ofthe simplexmethod, Math. Programming, 27 (1983), pp.241-262.

CHARLES BLAIRUniversity ofIllinois

Primality and Cryptography. By EvangelosKranakis. B. G. Teubner, Stuttgart; JohnWiley, Chichester, 1986. xvi + 235 pp.$41.95. ISBN 0-471-90934-3 (Wiley). A vol-ume in the Wiley-Teubner Series in Com-puter Science.

The author lectured at Yale in 1984and, soon thereafter, produced this smallbook. It is an attempt to get an in-depthtreatment of the number theoretic and prob-abilistic background of public key cryptog-raphy and related topics into print quickly.It is deeper and more specialized than thestandard cryptographic texts [DES0],[KOS1], [BE82], [ME82]. Its terminology ismore mathematical. "Indices" are men-tioned more often than "discrete loga-rithms," for example. Section describes the

basic computational number-theoretic kitand some of its cryptographic uses. Section2 treats primality tests up through Adelman-Pomerance-Rumeley. Section 3 is a shortexcursion into probabilistic notions relevantto number-theoretic pseudorandom numbergenerators. Section 4 describes many suchgenerators. Section 5 extends this descrip-tion by treating various public key crypto-systems, for example, RSA, Rabin-Williams and Merkle-Hellman (with anoutline of Shamir’s attack) and describes theGoldwasser-Micali quadratic-residue-basedsecurity arguments. Section 6 contains sev-eral complexity-theoretic results, especiallythe XOR Lemma and its applications.

The book gives evidence ofhow quicklyit was written. In the remarks on thresholdschemes (pp. 8-10, 37), for example, nomention of Shannon perfect security ap-pears. Most of the relevant literature is ig-nored. And the single example given, theAsmuth-Bloom Chinese Remainder Theo-rem Scheme, widely known since 1980[AS83], [DE80, p. 183], is credited to anothersource in 1983.

It is, however, misleading to find faultwith parts of a work which, taken as a whole,is a valuable and timely addition to the openliterature of cryptography. The book will bemost useful to the mathematically sophisti-cated individual who has learned the gist ofseveral recent cryptography protocols, cryp-tosystems, topics in key management, andrelated notions, and who wants a mathemat-ically oriented sourcebook to either explainthe underlying mathematical theory or elsepoint to the part of the literature containingthe explanation. It is not a text or a history.Neither is it light reading. It is for those whowant proofs or algorithms, rather than justan overview.

REFERENCES

[AS83] C. ASMUTH AND J. BLOOM, A modularapproach to key safeguarding, IEEE Trans.Inform. Theory, IT-29 (1983), pp. 208-210.

[BE82] H. BEKER AND F. PIPER, Cipher Systems:The Protection of Communications, Wiley-Interscience, New York, 1982.

[DE80] D. E. R. DENNING, Cryptography andData Security, Addison-Wesley, Reading,MA, 1980.

[KO81 A. G. KONHEIM, Cryptography:A Primer,3rd printing, Wiley-Interscience, New York,1982.

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