6 - 1 Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 1 SANS GIAC LevelOne Security Essentials Introduction to Encryption II Hello, the primary author of this course is Harish Bhatt with updates by Stephen Northcutt and Eric Cole. This is the second of two of the most important classes we have the privilege to teach as part of GIAC. In the first course, we went on a quick tour of some of the important issues and concepts in the field of cryptography. We saw that encryption is real, it is crucial, it is a foundation of so much that happens in the world around us today --and, most of it in a manner that is completely transparent to us. I guess you know that one of SANS’ mottos is to never teach anything in a class which the student can’t use at work the next day. One of our goals in this course is to help you be aware of how cryptography operates under the covers in some of the major cryptosystems which are used on a 24x7 basis in our world. Along the way, we’ll share some hard-earned pragmatic lessons we’ve learned, and hope that our experience will be of help to you. Enjoy! 6 - 2 Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 2 Why Do I Care About Crypto? U.S. Dept. of Commerce no longer supports DES . Distributed Denial of Service attack daemon found to be protected by “blowfish” --a DES-like block cipher . National Institute of Standards and Technology (NIST) is leading the development of AES --the replacement for DES . Mobile Code Communications in the presence of adversaries… Confidentiality ! !! ! Integrity ! !! ! Authentication ! !! ! Non-Repudiation Insecure Global Networks Privacy The Internet E-Business E-Commerce Smart Cards “Alice” “Bob” “Adversary” Digital Signatures Public Key Infrastructure (PKI) Digital Certificates Without cryptography there is no e-business, no viable e-commerce infrastructures, no military presence on the Internet and no privacy for the citizens of the world. There are numerous and continually increasing everyday instances in which we encounter cryptosystems at work and at play, often without even realizing it. The underlying cryptographic infrastructure actually works so well that we only take notice when it is absent, or implemented incorrectly! When you use a secure mobile telephone, all communications between you and the party on the other end are rapidly encrypted and decrypted on the fly, so that any eavesdropper will not be able to listen in on your conversation. Every once in a while we hear how the confidential communication of a public figure was intercepted and his or her privacy compromised. Yet another example of not using cryptographically enabled products. One of the more important emerging applications of cryptographically-enabled communications is at e-commerce-enabled web sites on the Internet and the World Wide Web. When supported with an enterprise-wide Public Key Infrastructure (PKI) a whole suite of new and innovative products and services is instantly enabled. Today, this is leading to new business opportunities, new capabilities being delivered to consumers, new functionality provided by organizations to their shareholders, fundamental changes in the way entire industries function, new legislation, tapping into global opportunities… 6 - 3 Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 3 • Concepts in Cryptography • Secret (Symmetric) Key Systems –Triple-DES –AES • Public (Asymmetric) Key Systems –RSA –ECC Course Objectives We begin this course by examining the conceptual underpinnings behind major cryptosystems that are in use today. In particular, we’ll look at Triple-DES which is a good alternative for the now obsolete DES algorithm, which is officially no longer considered to be secure. Next, we’ll stop by for a quick status update on the development activity that is currently underway throughout the global cryptographic community in connection with the new Advanced Encryption Standard (AES). Our next stop will be the RSA algorithm, which is a widely implemented public key cryptographic algorithm, and which came off-patent in September 2000. We’ll perform an exercise in which we’ll walk through a highly simplified version of the mathematical mechanism upon which the RSA algorithm is based. We’ll wrap up this course by considering the characteristics of emerging Elliptic Curve Cryptosystems (ECC), which are rapidly growing in popularity due to the proliferation of such devices as PDAs, mobile telephones, information appliances, ATMs, and smart cards. All right. Enough of the big picture. Let’s dive right into it… 6 - 4 Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 4 • What if… – we can find a mathematical “problem” that exhibits characteristics of one-way functions (with trapdoors)? – or, as mathematicians would prefer to say, a problem that is “impossible” to solve in polynomial time? Concepts in Cryptography 1 • Probability Theory • Information Theory • Complexity Theory • Number Theory • Abstract Algebra • Finite Fields • Hmm… – we could use it to build a new cryptosystem! Confidentiality Integrity of Data Authentication Non-Repudiation You’ll recognize the four important characteristics of cryptosystems that are at the top of this slide: Confidentiality, Integrity of Data, Authentication, and Non-Repudiation. We covered this material in Encryption I. OK. So we know that these are important characteristics that any good cryptosystem must have. But, how do we go about actually constructing such a cryptosystem? Where do we begin? Mathematics comes to our rescue. In general, there are many fields in mathematics that contain concepts that could prove to be useful as we seek to build a cryptosystem. Specifically, we find that the following branches of mathematics are particularly rich in ideas we could use: Probability Theory, Information Theory, Complexity Theory, Number Theory, Abstract Algebra, and Finite Fields. In Encryption I, we were introduced to one-way mathematical functions. We saw how such functions which have “trapdoors” have interesting properties that could prove to be useful in cryptography. We are using the term “trapdoor” to refer to a way to decrypt a message using a different key. So with public key cryptography, one would encrypt the message with a public key. The “trapdoor” would be the corresponding private key that would be used to decrypt or retrieve the message. If the one-way function deals with a “hard” mathematical problem – one that is impossible to solve in polynomial time – then it could be used to make things very difficult for any adversary who might be eavesdropping on our communications over an insecure public network like the global Internet. At the same time, the existence of a “trapdoor” could be used to provide an easy solution to the “intractable” problem for use by the sender and/or the recipient. Hmmm . 6 - 5 Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 5 Concepts in Cryptography 2 Tractable Problems “Easy” problems. Can be solved in polynomial time (i.e. “quickly”) for certain inputs Examples: • constant problems • linear problems • quadratic problems •cubic problems Intractable Problems “Hard” problems. Cannot be solved in polynomial time (i.e. “quickly”) Examples: • exponential or super-polynomial problems • factoring large integers into primes (RSA) • solving the discrete logarithm problem (El Gamal) • computing elliptic curves in a finite field (ECC) Computational Complexity deals with time and space requirements for the execution of algorithms. Problems can be classified as tractable or intractable. This is exactly the class of problems we are looking for! Following this train of thought, let’s see what hard or intractable problems are already well known in mathematics. These problems just might provide us with the building blocks upon which we could build our cryptosystem. Computational complexity is a branch of mathematics which studies time and space requirements for the execution of algorithms. It classifies problems as either tractable (easy to solve) or intractable (hard to solve). This is really neat, because its exactly what we’re looking for. It turns out that there are many well known intractable problems – the class of problems we’re interested in. These exponential or super-polynomial problems are “hard” problems which cannot be solved in polynomial time (i.e., quickly). Actually, it is more accurate to say that these problems are believed to be intractable by the worldwide mathematical community that is active in researching issues in the field of computation complexity. Three well known examples of intractable problems include: factoring large integers into their two prime factors (the basis for RSA); solving the discrete logarithm problem over finite fields (the basis for ElGamal); and computing elliptic curves over finite fields (the basis for Elliptic Curve Cryptosystems). Now, let’s examine each of these three important classes of intractable problems in greater detail, as each one of them forms the basis of important cryptosystems that are widely used all over the world today. 6 - 6 Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 6 Concepts in Cryptography 3 Example: RSA • based on difficulty of factoring a large integer into its prime factors • ~1000 times slower than DES • considered “secure” • de facto standard • patent expires in 2000 An Example of an Intractable Problem . Difficulty of factoring a large integer into its two prime factors • A “hard” problem • Years of intense public scrutiny suggest intractability • No mathematical proof so far Every middle school student knows how to factor integers. So, given an integer 15, they can immediately respond that the integer factors are 1x15 and 3x5. Easy enough! So why is this a hard problem? Why is it on our list of intractable problems? It turns out that the key here – no pun intended – is the word “large.” Factoring a large integer into its prime factors is decidedly non-trivial. In fact, there is no easy solution to the problem. This is the general consensus of the global community that actively researches such mathematical topics. It is important to note, however, that there is no unequivocal mathematical “proof” that this problem cannot be solved easily. It’s the years of public scrutiny of the problem that leads us to conclude that it is a hard problem which cannot be solved in polynomial time. For our purposes, this is good enough to build a cryptosystem upon. Actually .that’s already been done! The most widely used example is the RSA algorithm, which takes advantage of the intractability of the integer factorization problem to build the public key (asymmetric) cryptosystem which is widely used throughout the world. How about some of the other intractable problems we found from our brief survey of the field of mathematics? Can they also be used to construct cryptosystems? Great question! Glad you asked. 6 - 7 Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 7 Concepts in Cryptography 4 Examples • El Gamal encryption and signature schemes • Diffie-Hellman key agreement scheme • Schnorr signature scheme • NIST’s Digital Signature Algorithm (DSA) Another Intractable Problem . Difficulty of solving the discrete logarithm problem --for finite fields • A “hard” problem • Years of intense public scrutiny suggest intractability • No mathematical proof so far • The discrete logarithm problem is as difficult as the problem of factoring a large integer into its prime factors Another intractable problem that appears to have useful properties that we can use to build a cryptosystem upon is the difficulty of solving what is known as the discrete logarithm problem for finite fields. The mathematics behind this type of problem are complex and we will not attempt an explanation of the working mechanism in this brief course. It turns out that there is no easy solution to this problem either. Again, this is the general consensus of the global community that actively researches such mathematical topics. It is important to note, however, that there is no unequivocal mathematical “proof” that this problem cannot be solved easily. It’s the years of public scrutiny of the problem that leads us to conclude that it is a hard problem which cannot be solved in polynomial time. But, how does it compare with the previous intractable problem we looked at – the factorization of large integers into their two prime factors? There is evidence that the discrete logarithm problem is just as difficult. So, we should be able to use this problem in building a cryptosystem? Right? Absolutely! Again .that’s already been done! The following cryptosystems are all built upon the intractability of the discrete logarithm problem over finite fields: the ElGamal encryption and signature schemes, the Diffie-Hellman key agreement scheme, the Schnorr signature scheme, and the Digital Signature Algorithm (DSA) by the U.S. Department of Commerce’s National Institute of Standards and Technology (NIST). 6 - 8 Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 8 Concepts in Cryptography 5 Examples • Elliptic curve El Gamal encryption and signature schemes • Elliptic curve Diffie-Hellman key agreement scheme • Elliptic curve Schnorr signature scheme • Elliptic Curve Digital Signature Algorithm (ECDSA) Yet Another Intractable Problem . Difficulty of solving the discrete logarithm problem --as applied to elliptic curves • A “hard” problem • Years of intense public scrutiny suggest intractability • No mathematical proof so far • In general, elliptic curve cryptosystems (ECC) offer higher speed, lower power consumption, and tighter code Now, let’s take a quick look at yet another class of intractable problems. This one involves the difficulty of solving the discrete logarithm problem (we just discussed it in the previous slide) as applied to elliptic curves. So, how does this class of intractable problem compare with the previous intractable problem we’ve looked at – the factorization of large integers into their two prime factors, and solving the discrete logarithm problem over finite fields? Very well, thank you! And…it has a number of very attractive features to boot. Features that include high security levels even at low key lengths, high speed processing, and low power and storage requirements. These characteristics are very useful in crypto-enabling the many new devices that are rapidly appearing in the marketplace, e.g. mobile telephones, information appliances, smart cards, and even the venerable ATMs. Of course it has been broken a few times so they are still working on this one. 6 - 9 Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 9 Voila! We Can Now Build . Hash Digital Signature Original Document ---------- Ciphertext or plaintext Original Document ---------- Ciphertext or plaintext Digital Signature Hash Hash “Alice” first creates a Hash of the Original Document. Next, she encrypts the Hash with her Private Key to generate a Digital Signature. Finally, she transmits the Original Document and the Digital Signature to “Bob.” “Bob” first creates a Hash of the Original Document. Next, he decrypts the Digital Signature with Alice’s Public Key to regenerate the Hash that Alice originally created. Finally, he compares the two Hashes. A match indicates the Original Document was not tampered with. Bob compares the two hashes Hash Algorithm Same Hash Algorithm Alice encrypts with her Private Key Bob decrypts with Alice’s Public Key Authentication! Non-Repudiation! Integrity of Data! Confidentiality! Communications in the presence of adversaries… Confidentiality ! !! ! Integrity ! !! ! Authentication ! !! ! Non-Repudiation We started out by noting that communicating in the presence of adversaries meant constructing a cryptosystem that was capable of providing support for important requirements such as Confidentiality, Integrity of Data, Authentication, and Non-Repudiation. We briefly examined some of the well known intractable mathematical problems which could be used as building blocks upon which to construct our cryptosystem. But how do we make the connection between complex and abstract mathematical concepts, to crypto-enabled products we use routinely every day of our lives? While each type of cryptosystem addresses the specific details in its own unique way, the fundamental concepts behind the working crypto-mechanism that actually delivers the functionality that makes it possible to support Confidentiality, Integrity of Data, Authentication, and Non- Repudiation are fundamentally quite similar. This “big picture” slide puts it all together from the perspective of a message being sent by Alice over an insecure public network (like the global Internet) to Bob. Please study this slide carefully for a few moments, and trace the working mechanism that is at the foundation of many cryptosystems. See for yourself exactly how the users of the cryptosystem are able to tap into the Confidentiality, Integrity of Data, Authentication, and Non-Repudiation services that are supported by the cryptosystem. 6 - 10 Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 10 Exercise Mix-n-Match Game: Can you pair them up? 1. Authentication A. Used in generating a digital signature 2. Diffusion B. Ciphertext does not yield any information about the plaintext 3. Confidentiality C. Validate identity of a person or entity 4. Perfect Forward Secrecy D. Property of a cryptosystem that makes it technically impossible for a person or entity to fraudulently claim that it did not participate in a cryptographically-enabled transaction 5. Data Integrity E. Any relationship between the ciphertext and the plaintext is obscured 6. Hash Function F. Guarantee that messages have not been tampered with 7. Confusion G. Dissipate patterns and redundancies in the plaintext 8. Non-Repudiation H. Prevent unauthorized parties from eavesdropping All right, now. It’s time to get warmed up for the upcoming mathematical exercise on the mechanism of the RSA algorithm. Let’s play the Mix-n-Match Game! On the left hand side of this slide we have eight important concepts that are of significance in cryptography. On the right hand side of the slide, we have a description of these important concepts. The only problem is that they are not listed in the same order as the concepts on the left hand side. Your job is to mix-n-match the concepts on the left, to the descriptions on the right. If you have the ability to pause your audio, please pause and work on this exercise. If you do not have the ability to pause your audio,, just go on to the next slide and we will tell you the answers. [...]... operations often used to introduce confusion 8 Non-Repudiation D Property of a cryptosystem that makes it technically impossible for a person or entity to fraudulently claim that it did not participate in a cryptographically-enabled transaction Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 11 OK It’s time to see how we did on the Mix-n-Match Game… Most of the above cryptographic concepts... shown on the left, is applied three times, and two different crypto-variables are used Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 16 Earlier in today’s discussion and also in Encryption I, we noted that the Data Encryption Standard (DES) is no longer officially considered to be secure We also noted that the Advanced Encryption Standard (AES) is currently under development as we... throughout the world today, and AES is expected to be just as popular VULNERABILITIES Too early to tell… At the moment, the actual algorithm(s) that will be chosen as the AES has not yet been selected The NIST is spearheading the selection process Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 19 The Advanced Encryption Standard (AES) development process is a splendid opportunity to see first... Diffie-Hellman Problem (ECDHP) has been mathematically proven to be equivalent to the ECDLP y2 = x3 + ax + b (mod p) (an elliptic curve E over Zp where p>3 is prime) Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 28 In 1985, Neil Koblitz and Victor Miller independently proposed a new cryptosystem the Elliptic Curve Cryptosystem (ECC) whose security depends on the intractability... practical way to develop such algorithms is to perform the development process in an open manner, and under intense public scrutiny of the global cryptographic community Can you think of a recent example in which this was not followed? Countdown to AES ! • 1/2/1997, the quest for AES begins • 8/9/1999, five finalist algorithms announced • Announced winner Rijndeal Introduction to Encryption II – SANS... 12 DES • In 1992 is was proven that DES is not a group This means that multiple DES encryptions are not equivalent to a single encryption THIS IS A GOOD THING • If something is a group than – E(E(K,M)K2) = E(K3,M) • Since DES is not a group multiple encryptions will increase the security Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 13 As we know DES is no longer supported because... opportunity to see first hand what it takes to develop a cryptographic algorithm The development process is inherently complex, and the only realistic way to reduce the risk is to open up the development activity to all interested parties, and also to intense scrutiny by the global cryptographic community Visit the AES web site at NIST at http://www.nist.gov/aes/ to learn more about the effort, which is... no longer considered to be secure) So far, there have been no public reports claiming to have cracked Triple-DES Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 17 Triple-DES is a well known and widely implemented algorithm which has been intensely scrutinized by the global cryptographic community See the ANSI X9.52 standard for additional information on Triple-DES encryption Support... years, to the advanced mathematical ideas that serve as the foundation of many widely used cryptosystems in use today We also noted that each of the three classes of intractable problems we discussed had been successfully employed as building blocks for constructing cryptosystems There is a long, rich history behind modern cryptosystems This slide lists a few (by no means, all!) of the leading cryptographers... communicating in the presence of adversaries, and we want to make sure that the cryptosystem we are using supports our requirements for Confidentiality, Integrity of Data, Authentication, and Non-Repudiation Take about a minute to review and brush up on the above concepts All right Time to move on 6 - 11 Milestones in Cryptography Origins of Cryptography (traced as far back as 4000 years! RSA (Rivest, . 6 - 1 Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 1 SANS GIAC LevelOne Security Essentials Introduction to Encryption II Hello,. cryptosystems that are widely used all over the world today. 6 - 6 Introduction to Encryption II – SANS GIAC LevelOne ©2000, 2001 6 Concepts in Cryptography