Book: Computational Number Theory and Modern Cryptography
Author: Song Y. Yan
ISBN: 978-1-118-18858-3 (Wiley, 2013)
Chapter 6 is the warm-up. Yan says cryptography starts from secret-key cryptography, then he spends most of the pages teaching you the vocabulary you need for RSA. I did not mind. The number theory so far had no Alice. Now there is Alice, Bob, and Eve.
Book: Computational Number Theory and Modern Cryptography
Author: Song Y. Yan
ISBN: 978-1-118-18858-3 (Wiley, 2013)
Chapter 5 is the other hard problem. Integer factorization got the last chapter. Discrete logs get this one. Yan puts IFP, DLP, and ECDLP on the same shelf. Three infeasible jobs. Three reasons a public key can exist.
Book: Computational Number Theory and Modern Cryptography
Author: Song Y. Yan
ISBN: 978-1-118-18858-3 (Wiley, 2013)
The last three sections of Chapter 4 share one equation. If you can find x and y with x^2 congruent to y^2 mod n, and x is not plus or minus y mod n, then gcd(x-y, n) splits n. Sometimes gcd(x+y, n) does it. That is Fermat’s difference of squares, industrialized.
Book: Computational Number Theory and Modern Cryptography
Author: Song Y. Yan
ISBN: 978-1-118-18858-3 (Wiley, 2013)
Chapter 4 is the problem crypto actually rests on. Yan writes IFP as: input a composite n, output one nontrivial factor f. Not the full factorization. Just a split. Recurse with a primality test if you want the prime picture.
Book: Computational Number Theory and Modern Cryptography
Author: Song Y. Yan
ISBN: 978-1-118-18858-3 (Wiley, 2013)
Last post was the fast maybe. This stretch of Yan is the receipt, then the theorem that closed the complexity question. Section 3.3 is elliptic curve primality. Section 3.4 is AKS. One of them is what you run when you need a proof. The other is why textbooks can now say PTP is in P.
Book: Computational Number Theory and Modern Cryptography
Author: Song Y. Yan
ISBN: 978-1-118-18858-3 (Wiley, 2013)
I hit Chapter 3 of Yan and the tone shifts. Part 1 was definitions and theorems. Now he wants algorithms. He lists four problems that sit under modern crypto: primality testing, integer factorization, discrete logs, and elliptic curve discrete logs. This chapter is the first one. And it is the one that actually got solved.
This stretch of Computational Number Theory and Modern Cryptography by Song Y. Yan, ISBN 978-1-118-18858-3, is the plot twist chapter. Section 2.5 is still integers. Section 2.6 draws a cubic and says the same group game now lives on a curve.
Still in Computational Number Theory and Modern Cryptography by Song Y. Yan, ISBN 978-1-118-18858-3. After the linear congruence chapter, he asks a question that looks cute and then ruins your week.
I am reading Computational Number Theory and Modern Cryptography by Song Y. Yan, ISBN 978-1-118-18858-3, the 2013 Wiley edition. Section 2.4 is the chapter that finally feels like the book’s operating system. Not a side topic. The kernel. If you skip this and jump to RSA, you will fake your way through every later proof. I did that once in undergrad. It did not stick. Remainders look like grade school. Then they become the room where public key crypto keeps its furniture.
Section 2.3 of Computational Number Theory and Modern Cryptography by Song Y. Yan (ISBN 978-1-118-18858-3) looks like a catalog. Arithmetic functions. tau, sigma, phi, lambda, mu. If you skim, it feels like homework. If you are here for crypto, one of those letters is the whole plot.
Section 2.2 of Computational Number Theory and Modern Cryptography by Song Y. Yan (ISBN 978-1-118-18858-3) is the oldest material in the book, and it is still the part your laptop uses. Divisibility, primes, gcd, Euclid. Yan says people have studied this for at least 3000 years. The Greeks already cared about even and odd, perfect numbers, amicable numbers, and primes. Some of those questions are still open. That is wild.
I used to skip this chapter. Groups, rings, fields. The unsexy stuff. Then RSA showed up and I was lost. Same with elliptic curve crypto. Computational Number Theory and Modern Cryptography by Song Y. Yan (ISBN 978-1-118-18858-3) starts Chapter 2 with this exact trap.
Sections 1.3 and 1.4 of Computational Number Theory and Modern Cryptography are the moment the textbook stops warming up. Song Y. Yan, ISBN 978-1-118-18858-3, finally names the problems the rest of the book will live inside. I read this as the whole map on one table. Computational number theory, the hard problems, then modern crypto after the 1970s.
Chapter 1 of Computational Number Theory and Modern Cryptography is orientation week. Song Y. Yan, ISBN 978-1-118-18858-3, starts with integers. Not apps. Not HTTPS. Integers. I rolled my eyes, then I remembered this 2013 Wiley book is trying to show why those integers are the internet’s load-bearing walls.
I picked up Computational Number Theory and Modern Cryptography because I was tired of the one-line version of internet security. Song Y. Yan wrote it. The ISBN is 978-1-118-18858-3. Wiley and Higher Education Press published it in 2013. It is a graduate textbook. It is dry in places. It is formula-heavy. And the spine of the argument is still the security model your phone uses when it opens a lock icon.