09-03-2026, 10:57 PM
An Introduction to Mathematical Cryptography
Authors: Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman
Publication date: 11 September 2014
Publisher: Springer New York
An Introduction to Mathematical Cryptography is a mathematically rigorous introduction to the foundations of modern public-key cryptography. Rather than treating cryptographic algorithms merely as programming techniques, Hoffstein, Pipher, and Silverman develop the mathematics that explains why the algorithms work and why they are difficult to break. The book assumes mainly basic linear algebra and introduces the necessary tools from number theory, algebra, probability, combinatorics, and information theory as they arise.
The first part develops classical public-key ideas, including Diffie–Hellman key exchange, discrete logarithms, integer factorization, RSA, and digital signatures. It also studies the mathematics underlying attacks on these systems, including primality testing, factorization algorithms and collision methods. The treatment therefore connects computational difficulty directly with cryptographic security rather than simply presenting encryption algorithms as black boxes.
The later chapters move into more advanced mathematics. A substantial section introduces elliptic curves and elliptic-curve cryptography, including pairing-based methods. Another major chapter develops lattices and lattice-based cryptography, including the NTRU cryptosystem. The second edition expands the treatment of digital signatures and adds material on lattice-based signatures, rejection sampling, digital cash and homomorphic encryption. This makes the book particularly interesting today because lattice cryptography forms an important mathematical foundation for post-quantum cryptography.
Main topics
Key takeaways
BOOK
Authors: Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman
Publication date: 11 September 2014
Publisher: Springer New York
An Introduction to Mathematical Cryptography is a mathematically rigorous introduction to the foundations of modern public-key cryptography. Rather than treating cryptographic algorithms merely as programming techniques, Hoffstein, Pipher, and Silverman develop the mathematics that explains why the algorithms work and why they are difficult to break. The book assumes mainly basic linear algebra and introduces the necessary tools from number theory, algebra, probability, combinatorics, and information theory as they arise.
The first part develops classical public-key ideas, including Diffie–Hellman key exchange, discrete logarithms, integer factorization, RSA, and digital signatures. It also studies the mathematics underlying attacks on these systems, including primality testing, factorization algorithms and collision methods. The treatment therefore connects computational difficulty directly with cryptographic security rather than simply presenting encryption algorithms as black boxes.
The later chapters move into more advanced mathematics. A substantial section introduces elliptic curves and elliptic-curve cryptography, including pairing-based methods. Another major chapter develops lattices and lattice-based cryptography, including the NTRU cryptosystem. The second edition expands the treatment of digital signatures and adds material on lattice-based signatures, rejection sampling, digital cash and homomorphic encryption. This makes the book particularly interesting today because lattice cryptography forms an important mathematical foundation for post-quantum cryptography.
Main topics
- Diffie–Hellman key exchange and discrete logarithms
- RSA and integer factorization
- Digital signatures: RSA, ElGamal and DSA
- Primality testing and factorization algorithms
- Probability and information theory
- Elliptic curves and elliptic-curve cryptography
- Pairing-based cryptography
- Lattices and lattice reduction
- NTRU and lattice-based cryptography
- Lattice-based signatures
- Digital cash
- Homomorphic encryption
Key takeaways
- Mathematics is the central focus: this is much closer to a number-theory/algebra textbook than to a practical cybersecurity manual.
- Excellent bridge between pure and applied mathematics: concepts such as modular arithmetic, finite groups, elliptic curves and lattices acquire direct cryptographic applications.
- Suitable for advanced undergraduate or beginning graduate students, particularly in mathematics or computer science; reviews highlighted its clarity, examples and well-chosen exercises.
- Especially valuable for mathematicians: the elliptic-curve and lattice chapters show particularly well how sophisticated mathematical structures become practical cryptographic systems.
BOOK
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