, and are all fields under usual addition and multiplication with 0 = 0 and 1 = 1.
1.
The two-element ring B in Example 5.11(3) is a field.
2.
For p being a prime number, is a field under addition and multiplication modulo p with
0 = 0 and 1 = 1.
3.
We shall see more examples of fields in a moment.
Note that under integer addition and multiplication is not a field because any non-zero element does not have a multiplicative inverse in (a violation of the Inverse Axiom). Also, for n being a composite, is not a field too since we have seen that contains zero-divisors (a violation of the Closure Axiom).
Sometimes there will be no need for us to care about the difference among a group, a ring or a field. In such a situation we shall use an algebraic structure to refer to either of these
structures.
The notions of finite group, subgroup, quotient group and the order of group can be extended straightforwardly to rings and fields.
Definition 5.14: An algebraic structure is said to be finite if it contains a finite number of elements. The number of elements is called the order of the structure.
A substructure of an algebraic structure A is a non-empty subset S of A which is itself an
algebraic structure under the operation(s) of A. If S A then S is called a proper substructure of A.
Let A be an algebraic structure and B A be a substructure of A. The quotient structure of A modulo B, denoted by A/B, is the set of all cosets aoB with aranging over A, with the operation
defined by (aoB) (boB) = (aob) oB, and with the identity elements being0oB and1o B.
From Definition 5.14, a ring (respectively, a field) not only can have a subring (respectively, a subfield), but also can have a subgroup (respectively, a subring and a subgroup). We shall see such examples in §5.4.
• Table of Contents
Modern Cryptography: Theory and Practice By Wenbo Mao Hewlett-Packard Company
Publisher: Prentice Hall PTR Pub Date: July 25, 2003
ISBN: 0-13-066943-1 Pages: 648
Many cryptographic schemes and protocols, especially those based on public-keycryptography, have basic or so-called "textbook crypto" versions, as these versionsare usually the subjects for many textbooks on cryptography. This book takes adifferent approach to introducing
cryptography: it pays much more attention tofit-for-application aspects of cryptography. It explains why "textbook crypto" isonly good in an ideal world where data are random and bad guys behave nicely.It reveals the general unfitness of "textbook crypto" for the real world by demonstratingnumerous attacks on such schemes, protocols and systems under variousreal- world application scenarios. This book chooses to introduce a set of practicalcryptographic schemes, protocols and systems, many of them standards or de factoones, studies them closely, explains their working principles, discusses their practicalusages, and examines their strong (i.e., fit-for-application) security properties, oftenwith security evidence formally established. The book also includes self-containedtheoretical background material that is the foundation for modern cryptography.
5.4 The Structure of Finite Fields
Finite fields find wide applications in cryptography and cryptographic protocols. The pioneer work of Diffie and Hellman in public-key cryptography, the Diffie-Hellman key exchange protocol [98] (§8.3), is originally proposed to work in finite fields of a particular form. Since the work of Diffie and Hellman, numerous finite-fields-based cryptosystems and protocols have been proposed: the ElGamal cryptosystems [102], the Schnorr identification protocol and signature scheme [257], the zero-knowledge undeniable signatures of Chaum, and the zero-knowledge proof protocols of Chaum and Pedersen [73], are well-known examples. Some new
cryptosystems, such as the Advanced Encryption Standard [219] (§7.7) and the XTR cryptosystems [175], work in finite fields of a more general form. Finite fields also underlie elliptic curves which in turn form the basis of a class of cryptosystems (e.g., [166]).
Let us now conduct a self-contained course in the structure of finite fields.
5.4.1 Finite Fields of Prime Numbers of Elements
Finite fields with the simplest structure are those of orders (i.e., the number of elements) as prime numbers. Yet, such fields have been the most widely used ones in cryptography.
Definition 5.15: Prime FieldA field that contains no proper subfield is called a prime field. For example, is a prime field whereas is not, since is a proper subfield of . But is an infinite field. In finite fields, we shall soon see that a prime field must contain a prime number of elements, that is, must have a prime order.
Definition 5.16: Homomorphism and IsomorphismLet A, B be two algebraic structures. A mapping f : A B is called a homomorphism of A into B if f preserves operations of A. That is, if
ois an operation of A and , an operation of B, then x, y A, we have f(x) oy) = f(x) f(y). If f is a one-to-one homomorphism of A onto B, then f is called an isomorphism and we say that A and B are isomorphic.
If f : A B is a homomorphism and e is an identity element in A (either additive or multiplicative), then
so that f(e) is the identity element in B. Also, for any a A
so that f(a–1) = f(a)–1 for all a A. Moreover, if the mapping is one-one onto (i.e., A and B are isomorphic), then A and B have the same number of elements. Two isomorphic algebraic structures will be viewed to have the same structure.
• Table of Contents
Modern Cryptography: Theory and Practice By Wenbo Mao Hewlett-Packard Company
Publisher: Prentice Hall PTR Pub Date: July 25, 2003
ISBN: 0-13-066943-1 Pages: 648
Many cryptographic schemes and protocols, especially those based on public-keycryptography, have basic or so-called "textbook crypto" versions, as these versionsare usually the subjects for many textbooks on cryptography. This book takes adifferent approach to introducing
cryptography: it pays much more attention tofit-for-application aspects of cryptography. It explains why "textbook crypto" isonly good in an ideal world where data are random and bad guys behave nicely.It reveals the general unfitness of "textbook crypto" for the real world by demonstratingnumerous attacks on such schemes, protocols and systems under variousreal- world application scenarios. This book chooses to introduce a set of practicalcryptographic schemes, protocols and systems, many of them standards or de factoones, studies them closely, explains their working principles, discusses their practicalusages, and examines their strong (i.e., fit-for-application) security properties, oftenwith security evidence formally established. The book also includes self-containedtheoretical background material that is the foundation for modern cryptography.