July 24, 2009
Elementary Cryptanalysis: A Mathematical Approach, second edition
Abraham Sinkov; revised and updated by Todd Feil
228 pp., hardcover, 2009. Series: Anneli Lax New Mathematical Library, Vol. 22
Many people who are acquainted with cryptology, either through cloak-and-dagger tales or through newspaper cryptograms, may not know that aspects of this endeavor can be treated systematically—and by means of elementary mathematical concepts and methods.
In Elementary Cryptanalysis, first published in 1966, Abraham Sinkov (1907-1998) explains some of the fundamental techniques at the heart of the cryptanalytic endeavor from which much more sophisticated techniques have evolved, especially since the advent of computers. Topics relevant in these discussions include modular arithmetic, number theory, linear algebra of two dimensions with matrices, combinatorics, and statistics.
This second edition, which has been revised and updated by Todd Fiel, includes a discussion of public-key cryptosystems and the RSA method and of perfectly secure systems and one-time pads.
Although Sinkov wrote the original book more than 40 years ago, he “presents the best discussion available on how to break columnar ciphers of unequal length,” Fiel points out in his preface. “These and other topics have been usurped by the more modern methods . . . but they still present excellent examples of how a little mathematics can give you a great deal of insight into the solution of a problem,” he notes. “They also are a source of darn good puzzles.”
2.1 Mixed alphabets (p. 32). Most of the cryptographic problems encountered in magazines and newspapers use randomly generated alphabets. A disadvantage of such alphabets, from the point of view of communications, is the difficulty of committing the cipher sequence to memory. If the sequence has to be written out, there is the danger that it may be lost or stolen. This suggests the idea of generating a cipher sequence in some systematic way which will make it easy to construct the sequence and yet create a sufficient amount of mixing to gain most of the advantages of randomness. Many ingenious schemes have been devised for deriving such sequences; readers can no doubt think up some of their own. Two of the most commonly used methods will be briefly described as illustrations. The first method produces what is called a keyword mixed sequence. The cryptographer is required to remember just one important word—the keyword.
Preface to the First Edition. Preface to the Second Edition. Part I. Monoalphabetic Ciphers Using Additive Alphabets: 1. The Caesar Cipher 2. Modular Arithmetic 3. Additive Alphabets 4. Solution of Additive Alphabets 5. Frequency Considerations 6. Multiplications 7. Solution of Multiplicative Alphabets 8. Affine Ciphers. Part II. General Monoalphabetic Substitution: 1. Mixed Alphabets 2. Solution of Mixed Alphabet Ciphers 3. Solution of Five-Letter Groupings 4. Monoalphabets with Symbols. Part III. Polyalphabetic Substitution: 1. Polyalphabetic Ciphers 2. Recognition of Polyalphabetic Ciphers 3. Determination of Number of Alphabets 4. Solutions of Additive Subalphabets 5. Mixed Plain Sequences 6. Matching Alphabets 7. Reduction to a Monoalphabet 8. Mixed Cipher Sequences 9. General Comments. Part IV. Polygraphic Systems: 1. Linear Transformations 2. Multiplication of Matrices—Inverses 3. Involutory Transformations 4. Recognition of Digraphic Ciphers 5. Solution of a Linear Transformation 6. How to Make the Hill System More Secure. Part V. Transposition: 1. Columnar Transposition 2. Completely Filled Rectangles 3. Incompletely Filled Rectangles 4. Probable Word Method 5. General Case 6. Identical Length Messages. Part VI. RSA Encryption: 1. Public-key Encryption 2. The RSA Method 3. Creating the RSA Keys 4. Why RSA Works—Fermat's Little Theorem 5. Computational Considerations 6. Maple and Mathematica for RSA 7. Breaking RSA and Signatures. Part VII. Perfect Security—One-time Pads: 1. One-time Pads 2. Pseudo-Random Number Generators A. Tables B. ASCII Codes C. Binary Numbers. Solutions to Exercises. Further Readings. Index.
About the Authors:
Abraham Sinkov graduated from the City College of New York and received his master’s in mathematics from Columbia University and his doctorate from George Washington University. In 1930, he joined the U.S. Army’s new Signal Intelligence Service (SIS). He was instrumental in breaking the Japanese diplomatic codes in the mid 1930s. After World War II, he continued working for the SIS, which later became the Armed Forces Security Agency, then the National Security Agency (NSA). He retired from the NSA in 1963 and subsequently taught mathematics at Arizona State University.
Todd Feil (Denison University), who has written several books, graduated from Milliken University and received his doctorate in mathematics from Bowling Green State University.
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