Each chapter is punctuated with "Exercises for the Reader;" complete solutions for these are included in an appendix. Carefully crafted exercise sets are also provided at the end of each chapter, and detailed solutions to most odd-numbered exercises can be found in a designated appendix.
The computer implementation section at the end of every chapter guides students through the process of writing their own programs. A supporting website provides an extensive set of sample programs as well as downloadable platform-independent applet pages for some core programs and algorithms.
As the reliance on cryptography by business, government, and industry continues and new technologies for transferring data become available, cryptography plays a permanent, important role in day-to-day operations.
This self-contained sophomore-level text traces the evolution of the field, from its origins through present-day cryptosystems, including public key cryptography and elliptic curve cryptography. It will be very suitable for undergraduate students. There is adequate material in the book for teaching one or two courses on cryptography.
The author has provided many mathematically oriented as well as computer-based exercises. I strongly recommend this book as an introductory book on cryptography for undergraduates. As someone who has taught cryptography courses in the past, I was particularly impressed with the scaled-down versions of DES and AES that the author describes Stanoyevitch's writing style is clear and engaging, and the book has many examples illustrating the mathematical concepts throughout.
One of the many smart decisions that the author made was to also include many computer implementations and exercises at the end of each chapter. It is clear that Stanoyevitch designed this book to be used by students and that he has taught this type of student many times before.
The book feels carefully structured in a way that builds nicely By presenting the necessary mathematics as needed, An Introduction to Cryptography superbly fills that void. Although it is intended for the undergraduate student needing an introduction to the subject of cryptography, it contains enough optional, advanced material to challenge even the most informed reader, and provides the basis for a second course on the subject.
Beginning with an overview of the history of cryptography, the material covers the basics of computer arithmetic and explores complexity issues. The author then presents three comprehensive chapters on symmetric-key cryptosystems, public-key cryptosystems, and primality testing.
There is an optional chapter on four factoring methods: Pollard's p-1 method, the continued fraction algorithm, the quadratic sieve, and the number field sieve.
Another optional chapter contains detailed development of elliptic curve cryptosystems, zero-knowledge, and quantum cryptography. He illustrates all methods with worked examples and includes a full, but uncluttered description of the numerous cryptographic applications. He includes a number of illustrative and motivating examples, as well as optional topics that go beyond the basics presented in the core data.
With an extensive index and a list of symbols for easy reference, An Introduction to Cryptography is the essential fundamental text on cryptography. Cryptography has become essential as bank transactions, credit card infor-mation, contracts, and sensitive medical information are sent through inse-cure channels.
This book is concerned with the mathematical, especially algebraic, aspects of cryptography. It grew out of many courses presented by the authors over the past twenty years at various universities and covers a wide range of topics in mathematical cryptography.
It is primarily geared towards graduate students and advanced undergraduates in mathematics and computer science, but may also be of interest to researchers in the area. This is a substantially revised and updated introduction to arithmetic topics, both ancient and modern, that have been at the centre of interest in applications of number theory, particularly in cryptography.
As such, no background in algebra or number theory is assumed, and the book begins with a discussion of the basic number theory that is needed. The approach taken is algorithmic, emphasising estimates of the efficiency of the techniques that arise from the theory, and one special feature is the inclusion of recent applications of the theory of elliptic curves. Extensive exercises and careful answers are an integral part all of the chapters.
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Enigma Machine Simulators and Software — The definitive online catalogue of Enigma software and everything else including purchasing parts. The RSA Cryptosystem 51 Solutions 56 Index 60 1. Important Sets Before we start with the main topics, we need to review some notation: Definition 1. Note that the order that we write the elements of the set does not matter, all that matters is the content, i.
Loosely speaking, [classical] number theory, the subject of this course, is the study of integers. But, for purists, the study of those sets are only relevant if it yields results or ideas of importance to the integers. It is then not very surprising that the rational numbers often show up in number theory, and it is also often considered to have its own interest in the field.
Although it would seem that the set of real numbers is not very relevant to the study of integers, it does show up often. Even more, the set of complex numbers [for those who have heard of them], which is usually denoted by C, is quite important to number theory. Long Division We will deal mostly with integers in this course, as it is the main object of study of number theory. We will need to know long division [also called division algorithm] of integers. Example 2. We also call the quotient and 2 is the remainder.
If you forgot, review and practice! Here is another example. We can still perform the long division. Note that, as specified in equation 2. On the other hand, the quotient might be negative. If we multiply what we get [i. The new quotient is the negative of the old minus one, and the new remainder is the difference between the divisor [which is 15 in the above example] and the old remainder. This is, again, how it works in general.
The new quotient is the negative of the old one, and the new remainder equal to the old one. This is how it works in general. The new quotient is the old plus one, and the new remainder is the difference between the old divisor [which is 15 in the above example] and the old remainder.
One can also look at long division with a geometric point of view. For simplicity, I will use smaller and easier numbers. Say I want to divide 19 by 5. We mark all multiples of 5 [the divisor] in the real line [marked with circles in the picture below]. So, the quotient is 3 and the remainder is 4. It also works for negatives. Hence, the quotient is 4 and the remainder is 1, i. Finally, you can also use a simple calculator to perform long division.
Discard the decimals [if you have any], and that is your quotient, say q. Here is an example: say you want find the quotient q and remainder r of when divided by We finish this section with some more terminology: Definition 2. It might be useful to say a few words on what a theorem is. A theorem is a statement [or proposition] whose validity can be deduced from its assumptions by logical steps.
So, it is something that you can deduce [the key word here] from other facts. Note that we must always have a Proposition or Theorem, and never a Corollary, following a Lemma. In the same way, a Corollary always comes after a Proposition or a Theorem, but never after a Lemma. So, here is a simple but useful theorem: Theorem 3. Let a, b, and d be integers, and suppose that d a.
But, as you will see, we will derive many results from it, and so it has great importance to this text. The language in math has to be very precise, and sometimes, in everyday conversation, we are not as careful. Here is an example. Do it again! So, eating your meat is a sufficient condition for you to eat your pudding [i. So, eating your meat is a necessary condition to eat pudding.
Also, a mathematician is a skeptic by nature. Although in high-school, and very often in college too, one accepts the validity of theorems on faith, we should try to see why they actually hold. A mathematician always want to see a proof, that is, a precise argument which leaves no doubt about the veracity of the theorem.
We shall soon see a proof of the theorem above. Example 3. So, these numbers satisfy the conditions asked by the theorem, namely, these numbers are all integers, and 5 10 [corresponding to d a].
Note that the above is not a proof! But, if you want to think of only positive d, you can drop it. It takes even more time to get used to writing some yourself, but it is a very important part of mathematics. Also, many law schools like to admit math majors precisely because of their skills with proofs, which are basically ironclad arguments, which are crucial to [good] lawyers.
Finally, we have to be careful when applying a theorem. So, what is wrong here? The problem is that for us to indeed be able to use the theorem, the conditions asked by the theorem must be satisfied. So, we cannot apply the theorem to this case! Hence, before applying a theorem, make sure to check that all conditions are satisfied. Let a, b, and c be integers. Try to justify your answer. Can you tell if the forecast of yesterday said if it would rain or not?
Can we be sure whether or not it will rain tomorrow? Simple Divisibility Criteria Here are a few divisibility criteria with which you might be familiar: 1 A number is divisible by 2 [i. Example: is even since it ends in 4, and is odd, since it ends in 1. Note that if the number is too large, we can repeat the process until we get a number small enough to be easily decided if it is divisible by 3. In fact, a slick application of Theorem 3.
Example: is divisible by 5, since the last digit is 5, while is not, since the last digit is neither 0 nor 5. The key factor here is that 2 and 3 are relatively prime, as defined in Definition 6.
Example: is not divisible by 6, since it is not divisible by 2 [since it ends with 3, an odd number]. Just as with 3, you can also repeat the process for large numbers.
Example: is divisible by 10, but is not. The truth is that they follow from our Theorem 3. But, I will fight my own instincts here and, instead of giving a formal proof, I will just give you a couple of examples that contain the heart of the proofs. These should be enough to convince you and give you an idea of how the actual proof goes. The case of 3 and 9 are also similar. The case, of 9 is very similar.
The names already tell us what they mean: the GCD of two integers a and b is the largest integer that divides a and b [at the same time], and the LCM is the smallest positive integer that is a multiple of a and of b [at the same time].
We shall denote them gcd a, b and lcm a, b respectively. Here are a few examples: a b gcd a, b lcm a, b 5 7 1 35 6 12 6 12 18 27 9 54 53 1 12 12 6 So, how does one compute the GCD and LCM?
We will see soon a simple way using factorization of prime numbers, which is fine for small numbers, but not efficient enough for large numbers. One better way, at least to compute the GCD, is to use a succession of long divisions. It was still used in schools in Europe as a text book in the recent past.
The American publisher Dover still has it on print in the United States. The Elements collects most of the mathematical knowledge of its time [around BC].
Note then that Euclid [the author] collected all the work, he did not, necessarily, do it all himself. So, let me show you how the Euclidean Algorithm works by showing it in action. Example 5. Say we want to compute the GCD of and And we repeat: take the new dividend [i. This means that the algorithm is over and the GCD of and is the last non-zero remainder, i. The last non-zero remainder is the GCD. But be careful to always take the dividend of the division not the quotient!
One question that you may ask is if we indeed always get to the point where we get an exact division. Another question, which is a bit harder to answer, is why this procedure indeed gives us the GCD.
What is really behind this is Theorem 3. Let d denote the GCD of a and b. Then, of course, d a and d b. We have established that d divides b and r1 , and by Theorem 3.
And, in the same way, d must divide r3 , r4 , etc. Proceeding this way, we get finally that rn divides a and b.
So, rn is a common divisor of a and b. How does one compute it efficiently? As with the GCD, factorization into primes works well for small numbers, or numbers that can be factored easily.
This is also a very efficient way of doing it. This is easy to see, as the divisors of a number and its negative are the same. The Extended Euclidean Algorithm The computations performed in the Euclidean Algorithm can be used to prove the following result: Theorem 6.
So, be careful! As mentioned before, the proof comes from the Euclidean Algorithm, and in fact gives the way to actually compute x and y [as in the statement]. Example 6. The actual proof will be omitted, but hopefully one will be able to see that the idea works in general just from this particular example. The EEA gives us many nice results. We will now show a few of its applications. Corollary 6. Let a and b be integers. Proof of Corollary 6. Then, it clearly divides ax and by also. Then, by Theorem 3.
Since the only positive divisor of 1 is 1 itself, we have that the only possibility for d is 1. Hence, the GCD of a and b is 1 [as it is the only common divisor]. If n divides a and b, then n divides d.
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