By Luther Martin
If you are trying to absolutely comprehend and successfully enforce identity-based encryption (IBE) expertise, this authoritative source is a brilliant selection. beforehand, information on IBE have merely been present in bulky, hard-to-follow magazine articles and convention court cases. this is often the 1st e-book to provide you entire, easy-to-understand suggestions at the topic. evaluating and contrasting IBE with conventional public-key applied sciences, the ebook essentially explains how and why IBE structures are safe. you discover a wealth of sensible thoughts, algorithms and various labored examples that enable you create a safe IBE process.
From easy mathematical options and homes, houses of elliptic curves, divisors and the Tate pairing, and cryptography and computational complexity, to comparable cryptographic algorithms, numerous IBE schemes, hierarchical IBE and grasp mystery making plans, and calculating pairings, this finished quantity serves as a transparent advisor to this more and more vital safeguard expertise. This hands-on quantity comprises exact pseudocode for identity-based encryption algorithms and assisting algorithms that aid prevent time and simplify your demanding tasks within the box.
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Additional resources for Introduction to Identity-Based Encryption (Information Security and Privacy Series)
Fundamentals of Number Theory, Mineola, NY: Dover Books, 1977. , Abstract Algebra, New York: Wiley, 2001. , G. Seroussi, and N. : Cambridge University Press, 1999. 3 Properties of Elliptic Curves Many IBE algorithms rely on the properties of certain functions on elliptic curves called pairings. This chapter describes the basic properties of elliptic curves that will be used in following chapters to understand properties of pairings and how to calculate them. Further detail on the connection between elliptic curves and elliptic functions can be found in ; additional detail on the algebraic structure of groups of points on elliptic curves can be found in ; further details on the efficient implementation of algorithms involving points on elliptic curves can be found in .
Note that an elliptic curve may be nonsingular over one field and singular over another. Note that this definition of the discriminant is always even so it is always zero in a field of characteristic 2. In this case, the Weierstrass normal form needs to be replaced with a different form for which the discriminant is not always zero. 1 (i) The elliptic curve y 2 = x 3 + x + 1 is nonsingular over the real numbers because it has discriminant ⌬ = −16(31) = −496. 5 Graph of the elliptic curve y 2 = x 3 − 3x + 3 for which ⌬ > 0.
P ka k Then the Jacobi symbol ͩͪ a n is defined to be ͩ ͪ ͩ ͪ ͩ ͪ ͩ ͪ ͩ ͪ a n k = i=1 a pi ai = a1 a p1 a p2 a2 ... 10. 7 Let a, b be integers and n ≥ 3 and m ≥ 3 be odd integers. Then Jacobi symbols have the following properties. 8 If p and q and distinct odd primes and n = pq, then a ∈ ޚn* is a quadratic residue modulo n if and only if a is a quadratic residue modulo p and a is a quadratic residue modulo p. 2 Computing Jacobi Symbols Suppose that n is an odd integer and we can write a = 2k b where b is an odd integer.