A Quick Look at some Mathematics and Cryptography A Talk for CLIR at UConn
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1 A Quick Look at some Mathematics and Cryptography A Talk for CLIR at UConn Jeremy Teitelbaum September 5, 2014
2 Secret Key Ciphers A cipher is a method of communication in which the letters making up the message are scrambled (replaced by other letters, or rearranged) in order to conceal the text. A cipher algorithm is the rule that tells you how to do this. Usually a cipher algorithm consists of a general rule for swapping around the letters. Each particular message or set of messages is then scrambled (encrypted) with a particular key.
3 Simple Substitution A simple substitution cipher is the classic cipher in which the letters of the alphabet are rearranged. They key for such a cipher ooks like: Here is a sample secret message: abcdefghijklmnopqrstuvwxyz mlhinuafrgcswjyoxtezkdpqvb wepledgeanewtooldconnecticut pnosnianmjnpzyysihyjjnhzrhkz
4 Redundancy and Security Language has a lot of internal structure words and sentences aren t random. This internal structure is called redundancy. To get a sense of this, consider the following sentence. whldthstrthstbslfvdnt thtllmnrcrtdql thtthyrndwdbythrcrtr wthcrtnnlnblrghts. Can you figure out what the original message was?
5 Letter Frequencies The simplest example of statistical regularity is the unequal distribution of letters in English text. A substitution cipher rearranges the names associated with the bars but doesn t change the shape of the graph.
6 Digraphs The distribution of Two-Letter combinations is also uneven.
7 The Vigenère Cipher a b c d e f g h i j k l m n o p q r s t u v w x y z b c d e f g h i j k l m n o p q r s t u v w x y z a c d e f g h i j k l m n o p q r s t u v w x y z a b d e f g h i j k l m n o p q r s t u v w x y z a b c e f g h i j k l m n o p q r s t u v w x y z a b c d f g h i j k l m n o p q r s t u v w x y z a b c d e g h i j k l m n o p q r s t u v w x y z a b c d e f h i j k l m n o p q r s t u v w x y z a b c d e f g i j k l m n o p q r s t u v w x y z a b c d e f g h j k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i j l m n o p q r s t u v w x y z a b c d e f g h i j k m n o p q r s t u v w x y z a b c d e f g h i j k l n o p q r s t u v w x y z a b c d e f g h i j k l m o p q r s t u v w x y z a b c d e f g h i j k l m n p q r s t u v w x y z a b c d e f g h i j k l m n o q r s t u v w x y z a b c d e f g h i j k l m n o p r s t u v w x y z a b c d e f g h i j k l m n o p q s t u v w x y z a b c d e f g h i j k l m n o p q r t u v w x y z a b c d e f g h i j k l m n o p q r s u v w x y z a b c d e f g h i j k l m n o p q r s t v w x y z a b c d e f g h i j k l m n o p q r s t u w x y z a b c d e f g h i j k l m n o p q r s t u v x y z a b c d e f g h i j k l m n o p q r s t u v w y z a b c d e f g h i j k l m n o p q r s t u v w x z a b c d e f g h i j k l m n o p q r s t u v w x y
8 Vigenère Example liftupyourheadsoyegatesandbeyecastdownyeeverlastingdoors centrifugalforcemustbeappliedcentrifugalforcemustbeappliedc nmsmlxdiarsjouuskyytuispcojibggnlkltqtypjjvtpmmlbokdddca
9 Statistical Properties of Vigenère Cipher
10 The Enigma Machine
11 Modern Cryptography Communication over the internet is typically protected by a modern stream cipher such as the advanced encryption standard. 1. The encryption algorithm is published 2. Encrypting and decrypting a message requires a key (usually a large number ; for example, a string of 1024 zeros and ones.) 3. Any system is vulnerable to a brute force attack. That is: try every key! A system is secure if there s no way to break faster than that. 4. Even if the system is secure in this sense, it is vulnerable if one can somehow extract the key for example, by capturing an agent and making them reveal it, or by tricking someone into giving the key away. 5. There are other vulnerabilities (traffic analysis, timing attacks for example) unrelated to the mathematics of the algorithm.
12 A fundamental problem: Key Exchange Problem I am located in Palo Alto, and I wish to communicate over a secure channel with my stockbroker (secret agent; criminal conspirator) located in New York. We have never met. We both have access to a state of the art stream cipher. But how do we agree on a key? Solution We could meet in person, verify each other s identity, and share the key on a piece of paper. Otherwise?
13 Public Key Cryptography In public key cryptography (or trapdoor cryptography) the methods for putting a message INTO cipher and taking a message OUT of cipher are different, and one cannot be deduced from the other. Imagine that there is a system with two keys, K 1 and K 2. Knowing K 1, you can t figure out K 2, and vice versa. Message Secret Message K 1 K 2 Secret Message Message
14 Key Exchange with Public Key Cryptography 1. I PUBLISH the key K 1 so everyone can see it. (This is the public key. ) 2. I PROTECT the K 2 so only I know it. (This is the private key. ) 3. If you want to send me a message, youllook up my K 1 and use it to put your message into code. You send that coded message to me. Only I can read it, since only I have K 2.
15 Digital Certificates
16 RSA 1. Choose two very large prime numbers P and Q and let M = PQ. 2. Let T = (P 1)(Q 1). 3. Choose an integer A that has no factor in common with T. This is your public key. 4. Find an integer B so that AB 1 is a multiple of T. This is your private key. Example Here are some concrete numbers. Never mind how I found them for now: P = 101, Q = 103, M = 10403, T = 10200, A = 7, B = Notice that AB 1 = = = 6 T.
17 RSA (cont d) My compatriot wants to send me the secret number He looks up my public key, which is A = 7 together with the number M = He calculates the remainder when is divided by M. He finds the number 8119 and sends it to me. 2. I take the number 8119 and find the remainder when is divided by M. Amazingly, I get the secret number 1345!
18 Why does this work? Theorem (Euler) Let p and q be prime numbers and suppose a is an integer that is not evenly divisible by p or q. Then a (p 1)(q 1) 1 is a multiple of pq. Remember that we ve set things up so that AB = 1 + k(p 1)(Q 1) for some integer k. As a result, (x A ) B = x AB = x 1+k(P 1)(Q 1) = x(x (P 1)(Q 1) ) k = x(1 + jpq) and the remainder of this last number, when divided by PQ, is just x!
19 Why is this secure? To crack this algorithm, you need to be able to figure out the secret key B if all you know is the public key A and the modulus M. You know that M = PQ. If you knew P and Q then you could figure out T = (P 1)(Q 1). If you knew A and T you could figure out B.
20 Factoring is HARD IF P AND Q ARE LARGE, THE TIME IT TAKES TO FACTOR M AND FIND P AND Q IS PROHIBITIVE! Even using worldwide networks of computers working over a period of years, it s impossible to factor a general number of the form PQ into its prime factors if the number of digits of PQ is more than, say, 250. See Table of RSA Numbers
21 Thanks! The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic. C. F. Gauss, Disquisitions Arithmeticae, 1801 No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years. G. H. Hardy, A Mathematician s Apology, 1941 The era of electronic mail may soon be upon us; we must ensure that two important properties of the current paper mail system are preserved: (a) messages are private, and (b) messages can be signed. from the introduction to A Method for Obtaining Digital Signatures and Public Key Cryptosystems, by R. L. Rivest, A. Shamir, and L. Adelman, Communcations of the ACM, Vol. 21, Issue 2, Feb pp
22 Homework Crack this Simple Substitution Cipher qalj xj vxuez a.i.d. vlucol, ud juwkeuotdw, t qud ikba dwekbs qtwa blewutj hubwd tj wal ztdwetvkwtxj xh wal tjauvtwujwd xh dxkwa uiletbu, ujz tj wal clxoxctbuo elouwtxjd xh wal reldljw wx wal rudw tjauvtwujwd xh wauw bxjwtjljw. waldl hubwd dllilz wx il wx waexq dxil otcaw xj wal xetctj xh drlbtld wauw ipdwlep xh ipdwletld, ud tw aud vllj buoolz vp xjl xh xke celuwldw ratoxdxraled.
23 More Homework Vigenère Cipher Use the keyword blackboard and the Vigenère cipher to encrypt the message: It was many and many a year ago, In a kingdom by the sea, That a maiden there lived whom you may know By the name of Annabel Lee; And this maiden she lived with no other thought Than to love and be loved by me.
24 More homework Public/Secret Keys Suppose that P = 113 and Q = 127. If the associated public key is 5, what is the secret key? Use the public key to encrypt the secret number 12.
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