part i preliminaries
1 introduction
1.1 what is number theory?
1.2 what is computation theory?
1.3 what is computational number theory?
1.4 what is modern cryptography?
1.5 bibliographic notes and further reading
references
2 fundamentals
2.1 basic algebraic structures
2.2 divisibility theory
2.3 arithmetic functions
2.4 congruence theory
2.5 primitive roots
2.6 elliptic curves
2.7 bibliographic notes and further reading
references
part ii computational number theory
3 primality testing
3.1 basic tests
3.2 miller-rabin test
3.3 elliptic curve tests
3.4 aks test
3.5 bibliographic notes and further reading
references
4 integer factorization
4.1 basic concepts
4.2 trial divisions factoring
4.3 p and p - 1 methods
4.4 elliptic curve method
4.5 continued fraction method
4.6 quadratic sieve
4.7 number field sieve
4.8 bibliographic notes and further reading
references
5 discrete logarithms
5.1 basic concepts
5.2 baby-step giant-step method
5.3 pohlig-hellman method
5.4 index calculus
5.5 elliptic curve discrete logarithms
5.6 bibliographic notes and further reading
references
part iii modern cryptography
6secret-key cryptography
6.1 cryptography and cryptanalysis
6.2 classic secret-key cryptography
6.3 modern secret-key cryptography
6.4 bibliographic notes and further reading
references
7 integer factorization based cryptography
7.1 rsa cryptography
7.2 cryptanalysis of rsa
7.3 rabin cryptography
7.4 residuosity based cryptography
7.5 zero-knowledge proof
7.6 bibliographic notes and further reading
references
8 discrete logarithm based cryptography
8.1 diffie-heuman-merkle key-exchange protocol
8.2 e1gamal cryptography
8.3 massey-omura cryptography
8.4 dlp-based digital signatures
8.5 bibliographic notes and further reading
references
9 elliptic curve discrete logarithm based cryptography
9.1 basic ideas
9.2 elliptic curve diffie-hellman-merkle key exchange scheme
9.3 elliptic curve massey-omura cryptography
9.4 elliptic curve eigamal cryptography
9.5 elliptic curve rsa cryptosystem
9.6 menezes-vanstone elliptic curve cryptography
9.7 elliptic curve dsa
9.8 bibliographic notes and further reading
references
part iv quantum resistant cryptography
10 quantum computational number theory
10.1 quantum algorithms for order finding
10.2 quantum algorithms for integer factorization
10.3 quantum algorithms for discrete logarithms
10.4 quantum algorithms for elliptic curve discrete
logarithms
10.5 bibliographic notes and further reading
references
11 quantum resistant cryptography
11.1 coding-based cryptography
11.2 lattice-based cryptography
11.3 quantum cryptography
11.4 dna biological cryptography
11.5 bibliographic notes and further reading
references
index
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