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Ch. 1 Overview of cryptograph for key exchange, the security of which is based on the intractability of the discrete loga rithm problem. Although the authors had no practical realization of a public-key encryp tion scheme at the time, the idea was clear and it gencratcd extensive interest and activity in the cryptographic community. In 1978 Rivest, Shamir, and Adleman discovered the first practical public-key encryption and signature scheme, now referred to as rsa. The rsa scheme is based on another hard mathematical problem, the intractability of factoring large integers. This application of a hard mathematical problem to cryptography revitalized ef- forts to find more efficient methods to factor the 1980s saw major advances in this area but none which rendered the Rsa system insecure. Another class of powerful and practical public-key schemes was found by ElGamal in 1985. These are also based on the discrete logarithm problem One of the most significant contributions provided by public-key cryptography is the digital signature. In 1991 the first international standard for digital signatures(ISO/IEC 9796)was adopted It is based on the rsa public-key scheme. In 1994 the U.s. Govern ment adopted the Digital Signature Standard, a mechanism based on the ElGamal public key scheme The search for new public-key schemes, improvements to existing cryptographic mec hanisms, and proofs of security continues at a rapid pace. Various standards and infrastruc- tures involving cryptography are being put in place. Security products are being developed to address the security needs of an information intensive society The purpose of this book is to give an up-to-date treatise of the principles, techniques and algorithms of interest in cryptographic practice. Emphasis has been placed on those aspects which are most practical and applied. The reader will be made aware of the basic issues and pointed to specific related research in the literature where more indepth discus sions can be found due to the volume of material which is covered most results will be stated without proofs. This also serves the purpose of not obscuring the very applied nature of the subject. This book is intended for both implementers and researchers. It describes gorithms, systems, and their interactie Chapter I is a tutorial on the many and various aspects of cryptography. It does not attempt to convey all of the details and subtleties inherent to the subject. Its purpose is to introduce the basic issues and principles and to point the reader to appropriate chapters in the book for more comprehensive treatments. Specific techniques are avoided in this chapter 1.2 Information security and cryptography The concept of information will be taken to be an understood quantity. To introduce cryp tography, an understanding of issues related to information security in gencral is necessary Information security manifests itself in many ways according to the situation and require ment. Regardless of who is involved, to one degree or another, all parties to a transaction must have confidence that certain objectives associated with information security have been met. Some of these objectives are listed in Table 1.1 Over the centuries, an elaborate set of protocols and mechanisms has been created to deal with information security issues when the information is conveyed by physical doc uments.Often the objcctives of information sccurity cannot solely be achieved through mathematical algorithms and protocols alone, but require procedural techniques and abid ance of laws to achieve the desired result. For example, privacy of letters is provided by sealed envelopes delivered by an accepted mail service. The physical security of the en velope is, for practical necessity, limited and so laws are enacted which make it a criminal C1997 by CRC Press, Inc.-Scc accompanying notice at front of chapter 1.2 Information security and cryptography rivas keeping information secret from all but those who are autho- or confidentiality rized to see it data integrity ensuring information has not been altered by unauthorized or unknown means entity authentication corroboration of the identity of an entity(e. g, a person,a or identification computer terminal, a credit card, etc. sage corroborating the source of information; also known as data authentication origin authentication signature a means to bind information to an entity authorization conveyance to another entity of official sanction to do or be meth validation a means to provide timeliness of authorization to use or ma nipulate information or resources access control restricting access to resources to privileged entities certification endorsement of information by a trusted entit timestamping recording the time of creation or existence of information witnessing verifying the creation or existence of information by an entity other thar n the creator receipt acknow ledgement that information has been received confirmation acknowledgement that services have been provided ownership a means to provide an entity with the legal right to use or transfer a resource to others anonymity concealing the identity of an entity involved in some process non-repudiation preventing the denial of previous commitments or actions. revocation retraction of certification or authorization Table 1.1: Some information security objectives offense to open mail for which one is not authorized It is sometimes the case that security is achieved not through the information itself but through the physical document recording it. For example, paper currency requires special inks and material to prevent counterfeiting Conceptually, the way information is recorded has not changed dramatically over time Whereas information was typically stored and transmitted on paper, much of it now re- ides on magnetic media and is transmitted via telecommunications systems, some wire less. What has changed dramatically is the ability to copy and alter information. One can make thousands of identical copies of a piece of information stored electronically and each is indistinguishable from the original. With information on paper, this is much more diffi- cult. What is needed then for a society where information is mostly stored and transmitted electronic form is a means to ensure information security which is independent of the physical medium recording or conveying it and such that the objectives of information se curity rely solely on digital information itself. One of the fundamental tools uscd in information sccurity is the signature. It is a build ing block for many other services such as non-repudiation, data origin authentication, iden- tification, and witnessing, to mention a few. Having learned the basics in writing, an indi vidual is taught how to produce a handwritten signature for the purpose of identification At contract age the signature evolves to take on a very integral part of the person's identity This signature is intended to be unique to the individual and serve as a means to identify authorize, and validate With electronic information the concept of a signature needs to be Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstonc Ch. 1 Overview of cryptograph redressed; it cannot simply be something unique to the signer and independent of the in formation signed. Electronic replication of it is so simple that appending a signature to a document not signed by the originator of the signature is almost a triviality Analogues of the paper protocols"currently in use are required. Hopefully these new electronic based protocols are at least as good as those they replace. There is a unique op portunity for society to introduce new and more efficient ways of ensuring information se- curity. Much can be learned from the evolution of the paper based system, mimicking those aspects which have served us well and removing the inefficiencies Achieving information security in an electronic society requires a vast array of techni- cal and legal skills. There is, however, no guarantee that all of the information security ob jectives deemed necessary can be adequately met. The technical means is provided through cryptography 1.1 Definition Cryptography is the study of mathematical techniques related to aspects of in- formation security such as confidentiality, data integrity, entity authentication, and data ori- gin authentication Cryptography is not the only means of providing information security, but rather one set of techniques Cryptographic goals Of all the information security objectives listed in Table 1.1, the following four form a framework upon which the others will be derived: (1)privacy or confidentiality($1.5,1. 8) (2)data integrity (81.9); (3)authentication($1. 7); and (4)non-repudiation ($1.6) 1. Confidentiality is a service used to keep the content of information from all but those authorized to have it. Secrecy is a term synonymous with confidentiality and privacy There are numerous approaches to providing confidentiality, ranging from physical protection to mathematical algorithms which render data unintelligible 2. Data integrity is a service which addresses the unauthorized alteration of data. To assure data integrity, one must have the ability to detect data manipulation by unau thorized parties. Data manipulation includes such things as insertion, deletion, and substitution 3. Authentication is a service related to identification. This function applies to both enti- ties and information itself. Two parties entering into a communication should identify each other Information delivered over a channel should be authenticated as to origin date of origin, data content, time sent, etc. For these reasons this aspect of crypto- raphy is usually subdivided into two major classes: entity authentication and data origin authentication. Data origin authentication implicitly provides data integrity (for if a message is modified, the source has changed) 4. Non-repudiation is a service which prevents an entity from denying previous commit- ments or actions. When disputes arise due to an entity denying that certain actions situation is Fo may authorize the purchase of property by another entity and later deny such autho rization was granted a procedure involving a trusted third party is needed to resolve the dispute A fundamental goal of cryptography is to adequately address these four areas in both theory and practice. Cryptography is about the prevention and detection of cheating and other malicious activities This book describes a number of basic cryptographic tools (primitives)used to provide information security. Examples of primitives include encryption schemes($1.5 and $1.8), 1997 by CRC Press, Inc.-Scc accompanying notice at front of chapter 2 Information security and cryptography hash functions($1.9), and digital signature schemes($1.6). Figure 1. l provides a schematic listing of the primitives considered and how they relate. Many of these will be briefly intro- duced in this chapter, with detailed discussion left to later chapters. These primitives should Arbitrary length hash function Unkeyed One-way permutations Primitives Block ciphers Symmetric ciphers Stream Arbitrar th ciphers hash functions(MACs) Security Symmetric-key Primitives Primitives Signatures Pseudorandom sequences Identification primitives Publi Signatures Primitives Identification primiti Figure 1.1: A taxonomy of cryptographic primitives be evaluated with respect to various criteria such as 1. level/ security. This is usually difficult to quantify. Often it is given in terms of the number of operations required (using the best methods currently known)to defeat the intended objective. Typically the level of security is defined by an upper bound the amount of work necessary to defeat the objective. This is sometimes called work factor(see$1.13. 4) 2. functionality. Primitives will need to be combined to meet various information se- curity objectives. Which primitives are most effective for a given objective will be determined by the basic properties of the primitives 3. methods ofoperation Primitives, when applied in various ways and with various in puts, will typically exhibit different characteristics; thus, one primitive could provide Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstonc Ch. 1 Overview of cryptograph very different functionality depending on its mode of operation or usage 4. performance. This refers to the efficiency of a primitive in a particular mode of op eration. For example, an encryption algorithm may be rated by the number of bits per second which it can encrypt. 5. case of implementation. This refers to the difficulty of realizing the primitive in a practical instantiation. This might include the complexity of implementing the prim- itive in either a software or hardware environment The relative importance of various criteria is very much dependent on the application and resources available. for example in an environment where computing power is limited one may have to trade off a very high level of security for better performance of the system as a wholc Cryptography, over the ages, has been an art practised by many who have devised ad hoc techniques to meet some of the information security requirements The last twenty years have been a period of transition as the discipline moved from an art to a science. There re now several international scientific conferences devoted exclusively to cryptography and also an international scientific organization, the International Association for Crypto logic Research(IACR), aimed at fostering research in the area This book is about cryptography: the theory, the practice, and the standards 1.3 Background on functions While this book is not a treatise on abstract mathematics. a familiarity with basic mathe matical concepts will prove to be useful. One concept which is absolutely fundamental to cryptography is that of a function in the mathematical sense. A function is alternately re ferred to as a mapping or a transformation 1.3.1 Functions(1-1, one-way, trapdoor one-way) A set consists of distinct objects which are called elements of the set. For example, a set X might consist of the elements a, b, c, and this is denoted x=a, b, ck 1.2 Definition A function is defined by two sets X and Y and a rule f which assigns to each element in X precisely one element in Y. The set X is called the domain of the function and y the codomain. If a is an element of X (usually written e x) the image of a is the element in y which the rule f associates with a; the image y of c is denoted by y=f(a Standard notation for a function f from set X to set Y is f X-,Y. Ify E Y, then a preimage of y is an element E X for which f(e)=y. The set of all elements in Y which have at least one preimage is called the image of f, denoted Im(f) 1.3 Example (function) Consider the sets X=a,b,c,Y=1, 2, 3, 4, and the rule f from X to y defined as f(a)=2, f(b)=4, f(c=1. Figure 1.2 shows a schematic of the sets X, y and the function f. The preimage of the element 2 is a. The image of f is {1.2,4} Thinking of a function in terms of the schematic(sometimes called a functional dia gram)given in Figure 1. 2, each element in the domain X has precisely one arrowed line originating from it. Each element in the codomain y can have any number of arrowed lines incident to it (including zero lines) c1997 by CRC Press, Inc- See accompanying notice at front of chapter. 1.3 Background on functions X b Figure 1.2: A function f from a set X of three elements to a set Y of elements the: Often only the domain X and the rule f are given and the codomain is assumed to be image of f. This point is illustrated with two examples 1.4 Example (unction) Take X=1, 2.3,..., 10) and let be the rule that foreach E X, f(c)=rr, where rr is the remainder when x2 is divided by 11. Explicitly then f(1)=1f(2)=4f(3)=9f(4)=5f(5)=3 ∫(6)=3/(7)=5/(8)=9/(9)=4/(10)=1 The image of f is the set Y=(1, 3, 4, 5, 9) 1.5 Example (unction) Take X=1, 2. 3,., 1050] and let be the rule/ (a)=ra, where rr is the remainder when is divided by 100U + l for all E X. Here it is not feasible to write down f explicitly as in Example 1.4, but nonetheless the function is completely specified by the domain and the mathematical description of the rule f. 01-1 functions 1.6 Definition A function(or transformation) is 1- 1(one-to-one) if each element in the codomain y is the image of at most one element in the domain X 1.7 Definition A function(or transformation) is onto if each element in the codomain y is the image of at least one clement in the domain. equivalently, a function f: X-Y is (f)=Y 1. 8 Definition If a function f X>Y is 1-1 and Im(f)=Y, then f is called a bijection 1.9 Fact If f: X- Y is 1-1 then f: X,Im(f) is a bijection. In particular, if f: Y is 1-1, and X and y are finite sets of the same size, then f is a bijection In terms of the schematic representation, if f is a bijection, then each element in Y has exactly one arrowed line incident with it The functions described in examples 1.3 and 1.4 are not bijections. In Example 1. 3 the element 3 is not the image of any element in the domain. In Example 1. 4 each element in the codomain has two preimages 1.10 Definition If f is a bijection from X to y then it is a simple matter to define a bijection g from Y to X as follows: for eachye Y define g(y)=a where w e X and f (a)=y. This function g obtained from f is called the inverse function of f and is denoted by 9= f Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstonc Ch 1 Overview of Cryptography X 03Y X Figure 1.3: A bijection f and its inverse g=f 1.11 Example (inverse, function)LetX=a, b,c,d,e, and Y=(1, 2, 3, 4, 5), and consider the rule f given by the arrowed edges in Figure 1.3. f is a bijection and its inverse g is formed simply by reversing the arrows on the edges. The domain of g is Y and the codomain Note that if f is a bijection, then so is f. In cryptography bijections are used as the tool for encrypting messages and the inverse transformations are used to decrypt. This will be made clearer in 81.4 when some basic terminology is introduced. Notice that if the transformations were not bijections then it would not be possible to always decrypt to a unique message. ( One-way functions There are certain types of functions which play significant roles in cryptography. At th expense of rigor, an intuitive dcfinition of a one-way function is given 1.12 Definition A function f from a set X to a set Y is called a one-way function if f(a)is easyto compute for all x E X but for"essentially all elements y E Im(f)it is"com- putationally infeasible"to find any c E X such that f(a)=y 1.13 Note(clarification of terms in Definition 1. 12) (i) A rigorous definition of the terms"“easy”and“ computationally infeasible” IS neces sary but would detract from the simple idea that is being conveyed. For the purpose of this chapter, the intuitive meaning will suffice (ii) The phrase"for essentially all elements in Y refers to the fact that there are a few values yE Y for which it is easy to find an m E X such that y =f(r For example one may compute y=f(e) for a small number of a values and then for these, the inverse is known by table look-up. An alternate way to describe this property of a one-way function is the following: for a random y E Im(f) it is computationally infeasible to find any a E X such that f(e)=y The concept of a one-way function is illustrated through the following examples 1.14 Example(one-way function) Take X=(1, 2, 3,..., 16) and define f(r)=r2 for all C E X where Tr is the remainder when 3 is divided by 17. explicitly, 12345678910111213141 f( 9101 5111614874122 b Given a number between l and 16, it is relatively easy to find the image ofit under f. How ever, given a number such as 7, without having the table in front of you, it is harder to find 1997 by CRC Press, Inc.-Scc accompanying notice at front of chapter 1.3 Background on functions given that f()=7. Of course, if the number you are given is 3 then it is clear that=1 is what you need; but for most of the elements in the codomain it is not that easy. One must keep in mind that this is an example which uses very small numbers; the important point here is that there is a difference in the amount of work to compute f(a and the amount of work to find c given f(a). Even for very large numbers, f()can be computed efficiently using the repeated square-and-multiply algorithm(Algorithm 2. 143 whereas the process of findingc from f(a) is much harder 1.15 Example (one-way function) a prime number is a positive integer greater than 1 whose only positive integer divisors are I and itself. Select primes p= 48611, 9=53993, form m= pq= 2624653723, and let X=(1, 2, 3,.,n-1. Define a function f on X by f()=T'r for each E X, where Tm is the remainder when J is divided by T. For Instance,f(2489991)=1981394214 Since24899913=5881949859.m+1981394214 Computing f(a is a relatively simple thing to do, but to reverse the procedure is much more difficult; that is, given a remainder to find the value x which was originally cubed (raised to the third power). This procedure is referred to as the computation of a modular cube root with modulus n. If the factors of n are unknown and large, this is a difficult problem; how ever, if the factors p and g of n are known then there is an efficient algorithm for computing modular cube roots. (See $8.2.2(i) for details.) □ Example 1. 15 leads one to consider another type of function which will prove to be fundamental in later development (iii Trapdoor one-way functions 1.16 Definition A trapdoor one-way function is a one-way function f: X Y with the additional property that given some extra information(called the trapdoor information) it becomes feasible to find for any given y E Im(f), an E X such that f(e)=y Example 1. 15 illustrates the concept of a trapdoor one-way function. With the addi tional information of the factors of n= 2624653723(namely, p=48611 and q= 53993, each of which is five decimal digits long) it becomes much easier to invert the function The factors of 2624653723 are large enough that finding them by hand computation would be difficult. Of course, any reasonable computer program could find the factors relatively quickly. If, on the other hand, one selects p and q to be very large distinct prime numbers (each having about 100 decimal digits )then, by todays standards, it is a difficult problem even with the most powerful computers, to deduce p and q simply from n. This is the well known integer factorization problem(see $3.2) and a source of many trapdoor one-way unctions It remains to be rigorously established whether there actually are any(true)one-way functions That is to has yet definitively tions under reasonable(and rigorous)definitions of"easy"and"computationally infeasi- ble. Since the existence of one-way functions is still unknown the existence of trapdoor one-way functions is also unknown. However, there are a number of good candidates for one-way and trapdoor one-way functions. Many of these are discussed in this book, with emphasis given to those which are practical. g R. One-way and trapdoor one-way functions are the basis for public-key cryptography iScussed in $1. 8). The importance of these concepts will become clearer when their appli ation to cryptographic techniques is considered. It will be worthwhile to keep the abstract oncepts of this section in mind as concrete methods are presented Handbook of Applied Cryptography by A. Mcnczes, P. van Oorschot and S. Vanstonc


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