Abstract
As Julius Caesar surveys the unfolding battle from his hilltop outpost, an exhausted and disheveled courier bursts into his presence and hands him a sheet of parchment containing gibberish:
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Notes
- 1.
The plaintext is the original message in readable form and the ciphertext is the encrypted message.
- 2.
A cipher wheel with mixed up alphabets and with encryption performed using different offsets for different parts of the message is featured in a fifteenth century monograph by Leon Batista Alberti [63].
- 3.
The word cryptic, meaning hidden or occult, appears in 1638, while crypto- as a prefix for concealed or secret makes its appearance in 1760. The term cryptogram appears much later, first occurring in 1880.
- 4.
In cryptography, it is traditional for Bob and Alice to exchange confidential messages and for their adversary Eve, the eavesdropper, to intercept and attempt to read their messages. This makes the field of cryptography much more personal than other areas of mathematics and computer science, whose denizens are often X and Y !
- 5.
Do you see how we got 1013 years? There are 60 ⋅ 60 ⋅ 24 ⋅ 365 s in a year, and 26! divided by 106 ⋅ 60 ⋅ 60 ⋅ 24 ⋅ 365 is approximately 1013. 107.
- 6.
The assertion that a large number of possible keys, in and of itself, makes a cryptosystem secure, has appeared many times in history and has equally often been shown to be fallacious.
- 7.
A Study in Scarlet (Chap. 2), Sir Arthur Conan Doyle.
- 8.
The first known recorded description of the fast powering algorithm appeared in India before 200 BC, while the first reference outside India dates to around 950 AD. See [66, page 441] for a brief discussion and further references.
- 9.
Note that log2(A) means the usual logarithm to the base 2, not the so-called discrete logarithm that will be discussed in Chap. 2.
- 10.
Finite fields are also sometimes called Galois fields, after Évariste Galois, who studied them in the nineteenth century. Yet another notation for \(\mathbb{F}_{p}\) is \(\mathrm{GF}(p)\), in honor of Galois. And yet one more notation for \(\mathbb{F}_{p}\) that you may run across is \(\mathbb{Z}_{p}\), although in number theory the notation \(\mathbb{Z}_{p}\) is more commonly reserved for the ring of \(p\)-adic integers.
- 11.
You may wonder why Theorem 1.24 is called a “little” theorem. The reason is to distinguish it from Fermat’s “big” theorem, which is the famous assertion that \(x^{n} + y^{n} = z^{n}\) has no solutions in positive integers x, y, z if n ≥ 3. It is unlikely that Fermat himself could prove this big theorem, but in 1996, more than three centuries after Fermat’s era, Andrew Wiles finally found a proof.
- 12.
The prime factorization of m is \(m = 15485207 = 3853 \cdot 4019\).
- 13.
We earlier defined the order of p in a to be the exponent of p when a is factored into primes. Thus unfortunately, the word “order” has two different meanings. You will need to judge which one is meant from the context.
- 14.
In classical terminology, a code is a system in which each word of the plaintext is replaced with a code word. This requires sender and receiver to share a large dictionary in which plaintext words are paired with their ciphertext equivalents. Ciphers operate on the individual letters of the plaintext, either by substitution, transposition, or some combination. This distinction between the words “code” and “cipher” seems to have been largely abandoned in today’s literature.
- 15.
A bit is a 0 or a 1. The word “bit” is an abbreviation for binary digit.
- 16.
ASCII is an acronym for American Standard Code for Information Interchange.
- 17.
There are in fact many primes in the interval 2159 < p < 2160. The prime number theorem implies that almost 1 % of the numbers in this interval are prime. Of course, there is also the question of identifying a number as prime or composite. There are efficient tests that do this, even for very large numbers. See Sect. 3.4.
- 18.
The history is actually somewhat more complicated than this; see our brief discussion in Sect. 2.1 and the references listed there for further reading.
- 19.
If the number of letters in the message is not an even multiple of 25, then extra random letters are appended to the end of the message.
References
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Hoffstein, J., Pipher, J., Silverman, J.H. (2014). An Introduction to Cryptography. In: An Introduction to Mathematical Cryptography. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1711-2_1
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