Principios y Orientaciones Curriculares para la Educación Parvularia

INPUT _{Integers a > b 0;} OUTPUT gcd(a, b). if b = 0 return( a ); 1. return( gcd(b, a mod b) ). 2.

In Alg 4.1, "a mod b" denotes the remainder of a divided by b. (In §4.3.2.5 we will formally define the modular operation and provide some useful facts on modular arithmetic.) The

• 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.

condition can be satisfied by replacing a, b with their absolute values, and by invoking gcd(|b|, |a|) in case |a| < |b|.

Now let us examine how Alg 4.1 works. For positive integers a b, we can always write

Equation 4.3.2

for some integer q 0 (the quotient of a divided by b) and 0 r < b (the remainder of a divided by b). Since by definition, gcd(a, b) divides both a and b, equation (4.3.2) shows that it must also divide r too. Consequently, gcd(a, b) equals gcd(b, r). Since the remainder r (of a divided by b) is denoted by a mod b, we have derived

gcd(a, b) = gcd(b, a mod b).

This is the fact we have used in Alg 4.1, namely, gcd(a, b) is defined by gcd(b, a mod b)

recursively. The series of recursive calls of gcd compute the following series of equations, each is in the form of (4.3.2) and is formed by a division between the two input values:

Equation 4.3.3

where rk = 0 (which causes the terminating condition in step 1 being met) and q1, q 2, …, qk, r 1, r2, …, rk–1 are non-zero integers. With rk = 0, the last equation in (4.3.3) means rk–1 divides rk–2, and in the last-but-one equation, it must also divide rk–3, …, eventually, as shown in the first equation in (4.3.3), rk–1 must divide both a and b. None of other remainders in other equations has this property (that's why they are called remainders, not a divisor; only rk–1 is a divisor in the last equation in (4.3.3)). Therefore, rk–1 is indeed the greatest common divisor of a and b, i.e., rk–1 = gcd(a, b).

For example, gcd(108, 42) will invoke the following sequence of recursive calls: gcd(108, 42) = gcd(42, 24) = gcd(24, 18) = gcd(18, 6) = gcd(6, 0) = 6.

• 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.

4.3.2.2 Extended Euclid Algorithm

Alg 4.1 has thrown away all the intermediate quotients. If we accumulate them during the computation of gcd(a, b), we can obtain something more than just gcd(a, b).

Let us see what we can obtain.

The first equation in (4.3.3) can be written as

Multiplying both sides of this equation with q2, we can obtain

Using this equation and the second equation in (4.3.3), we can derive

Equation 4.3.4

The same way of calculation can be carried out. In general, for i = 1, 2, …, k, we can derive

Equation 4.3.5

where l i, m i are some integers which are, as indicated in (4.3.4), certain form of accumulations

of the intermediate quotients. We have seen in §4.3.2.1 that following this way of calculation we will eventually reach rk = 0, and then we have

Equation 4.3.6

An algorithm that inputs a, b and outputs the integers li–1, mk–1 satisfying (4.3.6) is called

extended Euclid algorithm. Extended Euclid algorithm will have an extensive use in the rest of the book for computing division modulo integers. Let us now specify this algorithm, that is, find a general method for accumulating the intermediate quotients.

• 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.

Then for i = 1, 2, …, k – 1, the ith equation in (4.3.3) relates ri–1, ri and ri+1 by

Equation 4.3.7

Replacing ri–1 and ri in the right-hand side of (4.3.7) using equation (4.3.5), we derive

Equation 4.3.8

Comparing between (4.3.8) and (4.3.5), we obtain (for i = 0, 1, …, k – 1)

Equation 4.3.9

These two equations provide us with a general method for accumulating the intermediate quotients while computing greatest common divisor (see Alg 4.2).