An Introduction to Mathematical Cryptography (2nd Edition) by Joseph H. Silverman, Jeffrey Hoffstein, Jill Pipher

By Joseph H. Silverman, Jeffrey Hoffstein, Jill Pipher

This self-contained advent to trendy cryptography emphasizes the math at the back of the speculation of public key cryptosystems and electronic signature schemes. The ebook specializes in those key subject matters whereas constructing the mathematical instruments wanted for the development and safeguard research of various cryptosystems. merely easy linear algebra is needed of the reader; ideas from algebra, quantity concept, and chance are brought and built as required. this article offers a fantastic creation for arithmetic and computing device technology scholars to the mathematical foundations of recent cryptography. The publication comprises an in depth bibliography and index; supplementary fabrics can be found online.

The booklet covers numerous themes which are thought of critical to mathematical cryptography. Key subject matters include:

* classical cryptographic structures, comparable to Diffie–Hellmann key trade, discrete logarithm-based cryptosystems, the RSA cryptosystem, and electronic signatures;

* primary mathematical instruments for cryptography, together with primality trying out, factorization algorithms, chance idea, details idea, and collision algorithms;

* an in-depth remedy of significant cryptographic recommendations, corresponding to elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem.

The moment variation of An advent to Mathematical Cryptography features a major revision of the fabric on electronic signatures, together with an past advent to RSA, Elgamal, and DSA signatures, and new fabric on lattice-based signatures and rejection sampling. Many sections were rewritten or accelerated for readability, in particular within the chapters on info conception, elliptic curves, and lattices, and the bankruptcy of extra subject matters has been elevated to incorporate sections on electronic funds and homomorphic encryption. a number of new routines were incorporated.

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Additional resources for An Introduction to Mathematical Cryptography (2nd Edition) (Undergraduate Texts in Mathematics)

Example text

This is trivial, if we use the lattice isomorphism α = 1 ω1 . In this section we write (as usual) for ω and for ω ω∈L\{0} . ω∈L\{0} We define several constants associated to a lattice in C. 3. Let L be a lattice in C. a) We define the (convergent) quantities g2 (L) := 60 ω 1 , ω4 g3 (L) := 140 ω 1 . ω6 b) The discriminant of the lattice is (L) = g23 (L) − 27g32 (L). c) The j -invariant of the lattice is j (L) = 123 g23 (L) . (L) d) If the lattice L is given as L = Z + τ Z with τ ∈ H, we write g2 (τ ) := g2 (L), g3 (τ ) := g3 (L), (τ ) := (L), j (τ ) := j (L).

Let φ : E1 → E2 be an isogeny of degree d. There exists a unique isogeny φˆ : E2 → E1 with φˆ φ = d (= multiplication with d on E1 ), φ φˆ = d (= multiplication with d on E2 ). The isogeny φˆ is called the dual isogeny to φ. One has ˆ = deg(φ). deg(φ) Proof. e). 33. Let φ : E1 → E2 be a non-constant isogeny. Then for every Q ∈ E2 , φ −1 (Q) = degs φ, in particular ker φ = degs φ. If φ is separable then ker φ = deg φ. 5 Isogenies and endomorphisms of elliptic curves 29 Proof. 10. Examples for endomorphisms are the multiplication maps.

25. The formulas for φ1 , 1 , and ˜ 1 are trivial. 20: ψn+1 (z)ψn−1 (z) ψn (z)2 p(z)ψn (z)2 − ψn−1 (z)ψn+1 (z) = . 2 Weierstraß ℘-function and so we obtain the asserted formula for n (z) = ψn (z)3 p˜ (nz) = 51 n: 1 (ψn−1 (z)2 ψn+2 (z) − ψn−2 (z)ψn+1 (z)2 2ψ2 (z) − ψ2 (z)ψn (z)(a1 φn (z) + a3 ψn (z)2 )). 21 and the recursion formula for ψ2n (z) and get ψ2n (z) ψn (z)4 1 ψn−1 (z)2 ψn+2 (z) − ψn−2 (z)ψn+1 (z)2 = ψ2 (z) ψn (z)3 ℘ (nz) = and we obtain the recursion formula for ˜ n . The rest of the theorem is trivial.

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