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Week 5 - Friday

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What did we talk about last time? Secure programs

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Alexander Barrow

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A vulnerability is a weakness in a security system A threat is a set of circumstances that can cause loss or harm Performing an attack is exploiting a vulnerability A control is a protection against an attack by reducing a vulnerability “A threat is blocked by control of a vulnerability.”

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Someone read something they weren’t supposed to Interception Something became unavailable or unusable Interruption Someone changed something they weren’t supposed to Modification Someone created fake things Fabrication

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As with traditional crime, an attacker must have these three things: Skills and tools to perform the attack Method Time and access to accomplish the attack Opportunity A reason to perform the attack Motive

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Confidentiality IntegrityAvailability

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You don’t want other people to be able to read your stuff Some of your stuff, anyway Cryptography, the art of encoding information so that it is only readable by those knowing a secret (key or password), is a principle tool used here Confidentiality is also called secrecy or privacy

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You don’t want people to mess up your stuff You want to know: That your important data cannot be easily changed That outside data you consider trustworthy cannot be easily changed either There are many different ways that data can be messed up, and every application has different priorities

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You want to be able to use your stuff Many attacks are based on denial of service, simply stopping a system from functioning correctly Availability can mean any of the following: The service is present in usable form There is enough capacity for authorized users The service is making reasonable progress The service completes in an acceptable period of time

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There are five common ways of dealing with attacks, many of which can be used together Prevent Remove the vulnerability from the system Deter Make the attack harder to execute Deflect Make another target more attractive (perhaps a decoy) Detect Discover that the attack happened, immediately or later Recover Recover from the effects of the attack

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Many different controls can be used to achieve the five methods of defense

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A system popularized by Ron Rivest uses Alice and Bob as the two parties communicating Carl or another “C” name can be used if three people are involved Trent is a trusted third party Eve is used for an evil user who often eavesdrops Mallory is used for a malicious user who is usually trying to modify messages

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Encryption takes a message and hides its meaning Decryption is the reverse process Encode and encipher can mean the same as encrypt Decode and decipher can mean the same decrypt A system for encrypting and decrypting messages is a cryptosystem Plaintext (often represented as P) is the original message Ciphertext (often represented as C) is the encrypted version E() and D() are used as functions to represent the encryption and decryption processes C = E(P) P = D(C)

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The algorithms for encryption often rely on a secret piece of information, called a key We can notate the use of a specific key in either of the two following ways: C = E K (P) C = E(K, P) In symmetric (or private key) encryption, the encryption key and the decryption key are the same In asymmetric (or public key) encryption, the encryption key and the decryption key are different

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Cryptography means “secret writing” A cryptographer is someone who specializes in using cryptography to make messages secret A cryptanalyst is someone who is trying to break the cryptography and discover the plaintext or the key A cryptanalyst could: Break a single message Find patterns in the encryption that allow future messages to be decrypted Discover information in the messages without fully decrypting them Discover the key Find weaknesses in the implementation of the encryption Find weaknesses in the encryption that may or may not be able to lead to breaks in the future

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There are two kinds of security for encryption schemes Unconditionally secure ▪ No matter how much time or energy an attacker has, it is impossible to determine the plaintext Computationally secure ▪ The cost of breaking the cipher exceeds the value of the encrypted information ▪ The time required to break the cipher exceeds the useful lifetime of the information We focus on computationally secure, because there is only one practical system that is unconditionally secure "I want them to remain secret for as long as men are capable of evil" -Avi from Cryptonomicon

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Cryptography is supposed to prevent people from reading certain messages Thus, we measure a cryptosystem based on its resistance to an adversary or attacker Kinds of attacks: Ciphertext only: Attacker only has access to an encrypted message, with a goal of decrypting it Known plaintext: Attacker has access to a plaintext and its matching ciphertext, with a goal of discovering the key Chosen plaintext: Attacker may ask to encrypt any plaintext, with a goal of discovering the key Others, less common

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Substitution ciphers cover a wide range of possible ciphers, including the shift cipher In a substitution cipher, each element of the plaintext is substituted for some corresponding element of the ciphertext Monoalphabetic substitution ciphers always use the same substitutions for a letter (or given sequence of letters) Polyalphabetic substitution ciphers use different substitutions throughout the encryption process

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We can map to a random permutation of letters For example: E(“MATH IS GREAT”) = “UIYP TQ ABZIY” 26! possible permutations Hard to check every one ABCDEFGHIJKLMNOPQRSTUVWXYZ INOVZHAPTRGEUFDWSBQYLKMJCX

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English language defeats us Some letters are used more frequently than others: ETAOINSHRDLU Longer texts will behave more and more consistently Make a histogram, break the cipher Digram analysis can help too

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The Vigenère cipher is a form of polyalphabetic substitution cipher In this cipher, we take a key word and repeat it, over and over, until it is as long as the message Then, we add the repetitions of keywords to our message mod 26

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Key: BENCH Plaintext: A LIMERICK PACKS LAUGHS ANATOMICAL BENCHBENCHBENCHBENCHBENCHBENCH ALIMERICKPACKSLAUGHSANATOMICAL BPVOLSMPMWBGXUSBYTJZBRNVVNMPCS

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The index of coincidence measures the differences in the frequencies in the ciphertext It is the probability that two randomly chosen letters from the ciphertext are the same IC = Period Large Expected IC

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If the IC indicates that a period of more than 1 is being used, look for repeated sequences Look at the gaps between long sequences Try to find the GCD of gaps between long sequences If you have a reasonable guess for the length of the key, break the ciphertext into groups based on the corresponding letter of the key If the IC is high (in the range of a single letter), then you have probably found the key length

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The rest is easy Try various shifts for each letter of the key so that high frequency letters (E, T, A) occur with high frequency and low frequency letters (Q, X, Z) occur with low frequency Guess and check

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A One-Time Pad is similar to the Vigenère cipher, except that the key is as long as the message What will this do to the index of coincidence? Any given ciphertext could be decrypted into any plaintext, provided that you have the right key

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Key: THISISTHESECRETPASSWORD Plaintext: SOMEBODY SHOUTED MCINTYRE SOMEBODYSHOUTEDMCINTYRE THISISTHESECRETPASSWORD LVUWJGWFWZSWKIWBCAFPMIH

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A One-Time Pad has the property of perfect secrecy or Shannon secrecy Perfect secrecy means that P(M) = P(M|C) Thus, learning the ciphertext tells you nothing about the plaintext

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You can only use it one time Otherwise, recovering the key is trivial Completely vulnerable to known plaintext attack The key is as long as the message If you have a way of sending a key that long securely, why not send the message the same way? Generating keys with appropriate levels of randomness presents a problem

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A block cipher is a symmetric key cipher that works on a block of data of a given size For compatibility with hardware, block sizes are often powers of two: 64 bits, 128 bits, 256 bits, etc. Block ciphers are a fundamental part of many modern cryptosystems To encrypt a message longer than a single block: First break the message into blocks Then, each block could be encrypted individually Or data from the first block can be used in the encryption of the second, and so on

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Data Encryption Standard DES is a typical block cipher It was chosen as the government's standard for encryption in 1976 (but has since been deprecated) DES works on blocks 64 bits in size DES uses a 56 bit key NSA helped design it… amidst some controversy

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DES has 16 rounds The book calls them cycles In each round, the input is broken into 2 halves, manipulated, and combined with part of the key Input Permutation Left 0 Right 0 f f Key 1 Left 1 Right Left 0 Right 0

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DES uses bitwise operations as well as lookup tables DES has 8 substitution boxes (S-boxes) which take 6 bits of data and give back 4

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The expansion permutation takes 32 input bits and expands them into 48 bits while permuting them 16 bits are repeated These 48 bits are XORed with the round key The resulting 48 bits are substituted through S-boxes which produces a 32 bit result The final 32 bits are permuted Expansion Permutation XOR with Key S-boxP-box

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The NSA tinkered with DES They shortened the key length from the original 128 bits of Lucifer to 56 They changed the S-boxes People were concerned that the NSA had introduced a trapdoor so that they could read messages Eventually, the NSA released information about the choice of S-boxes: No S-box is a linear or affine function of its input Changing 1 bit of the S-box input changes at least 2 bits of its output If a single bit is held constant, changing the others should not radically change the total number of 1s or 0s in the output

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DES is fast Easy to implement in software or hardware Encryption is the same as decryption Triple DES is still standard for many financial applications Resistant to differential and linear cryptanalysis (2 47 and 2 43 known pairs required, respectively)

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Short key size Brute force attack by EFF in 1998 in 56 hours then in 1999 in just over 22 hours Brute force attack by University of Bochum and Kiel in 9 days in 2006 (but, using a machine costing only $10,000) If you could check 1,000,000,000 keys per second (which is unlikely with a commodity PC), it would take an average of 417 days to recover a key

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Encrypt P 1 Decrypt C 1 Two pairs of plaintexts and ciphertexts are needed Encrypt P 1 with all possible keys and save them Decrypt C 1 with all possible keys If the result matches anything in the list, use the key to encrypt P 2 If that matches C 2, you win! On the left, I show all the decryptions, but only the encryptions need to be stored

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Although susceptible to a brute force attack, DES has no other major weaknesses Double DES can be defeated by an extension of the brute force attack What about triple DES? Let E K (X) and D K (X) be encryption and decryption using DES with key K Triple DES uses keys K1, K2, and K3 C = E K1 (D K2 (E K3 (M))) Setting K1 = K2 = K3 allows for compatibility with single DES systems Triple DES is still a standard for financial transactions with no known practical attacks

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Advanced Encryption Standard Block cipher designed to replace DES Block size of 128-bits Key sizes of 128, 192, and 256 bits Like DES, has a number of rounds (10, 12, or 14 depending on key size) Originally called Rijndael, after its Belgian inventors Competed with 14 other algorithms over a 5 year period before being selected by NIST

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AES keeps an internal state of 128 bits in a 4 x 4 table of bytes There are four operations on the state: Substitute bytes Shift rows Mix columns Add round key

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AES supports key sizes of 128, 192, and 256 bits Rijndael supports unlimited key size, in principle, as well as other block sizes 128 bit keys use 10 rounds, 192 use 12, and 256 use 14 Add round key First RoundNormal RoundLast Round

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Strengths Strong key size Fast in hardware and software Rich algebraic structure Well-studied, open standard Weaknesses Almost none A few theoretical attacks exist on reduced round numbers of AES No practical attacks other than side channel attacks

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Attacks that rely on timing, measuring cache, energy consumption, or other ways an implementation leaks data are called side channel attacks Several practical side channel attacks for AES do exist In 2005, Bernstein found a cache-timing attack that broke an OpenSSL implementation of AES using 200 million chosen plaintexts and a server that would give him precise timing data Later in 2005, Osvik et al. found an attack that recovered a key after 800 encryptions in only 65 milliseconds, with software running on the target machine In 2009, Saha et al. found an attack on hardware using differential fault analysis to recover a key with a complexity of 2 32 In 2010, Bangerter et al. found a cache-timing attack that required no knowledge of plaintexts or ciphertexts and could work in about 3 minutes after monitoring 100 encryptions

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DESAES Date Block size64 bits128 bits Key length56 bits128, 192, 256 bits Encryption primitivesSubstitution, permutation Substitution, shift, bit mixing Cryptographic primitivesConfusion, diffusion DesignOpen Design rationaleClosedOpen Selection processSecret Secret with public comment SourceIBM with NSA helpIndependent Belgians Security Broken if you’ve got the resources No practical attacks yet

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Sometimes, we need something other than a shared secret We want a public key that anyone can use to encrypt a message to Alice Alice has a private key that can decrypt such a message The public key can only encrypt messages, it cannot be used to decrypt messages

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RSA depends in large part on the difficulty of factoring large composite numbers (particularly those that are a product of only 2 primes) For those of you who aren't in Formal Methods, an integer p is prime if p > 1 p is not divisible by any positive integers other than 1 and itself

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Any integer greater than 1 can be factored into a unique series of prime factors: Example: 52 = 2 2 ∙ 13 Two integers a and b (greater than 1) are relatively prime or coprime if and only if a shares no prime factors with b

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The greatest common divisor or GCD of two numbers gives the largest factor they have in common Example: GCD( 12, 18 ) = GCD( 42, 56 ) = For small numbers, we can determine GCD by doing a complete factorization

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For large numbers, we can use Euclid's algorithm to determine the GCD of two numbers Algorithm GCD( a, b) 1. If b = 0 ▪ Return a 2. Else ▪ temp = a mod b ▪ a = b ▪ b = temp 3. Goto Step 1 Example: GCD( 1970, 1066)

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If p is prime and a is a positive integer not divisible by p, then: a p –1 1 (mod p)

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Euler’s totient function (n) (n) = the number of positive integers less than n and relatively prime to n (including 1) If p is prime, then (p) = p – 1 If we have two primes p and q (which are different), then: (pq) = (p)∙ (q) = (p – 1)(q – 1)

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Euler’s Theorem: For every a and n that are relatively prime, a (n) 1 (mod n) This generalizes Fermat’s Theorem because (p) = p – 1 if p is prime Proof is messier

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Named for Rivest, Shamir, and Adleman Take a plaintext M converted to an integer Create an ciphertext C as follows: C = M e mod n Decrypt C back into M as follows: M = C d mod n = (M e ) d mod n = M ed mod n

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TermDetailsSource MMessage to be encryptedSender CEncrypted messageComputed by sender nModulus, n = pqKnown by everyone pPrime numberKnown by receiver qPrime numberKnown by receiver eEncryption exponentKnown by everyone dDecryption exponentComputed by receiver (n)(n) Totient of nKnown by receiver

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To encrypt: C = M e mod n e is often 3, but is always publically known To decrypt: M = C d mod n = M ed mod n We get d by finding the multiplicative inverse of e mod (n) So, ed 1 (mod (n))

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We know that ed 1 (mod (n)) This means that ed = k (n) + 1 for some nonnegative integer k M ed = M k (n) + 1 M∙(M (n) ) k (mod n) By Euler’s Theorem M (n) 1 (mod n) So, M∙(M (n) ) k M (mod n)

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You can’t compute the multiplicative inverse of e mod (n) unless you know what (n) is If you know p and q, finding (n) is easy Finding (n) is equivalent to finding p and q by factoring n No one knows an efficient way to factor a large composite number

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Once you have great cryptographic primitives, managing keys is still a problem How do you distribute new keys? When you have a new user When old keys have been cracked or need to be replaced How do you store keys? As with the One Time Pad, if you could easily send secret keys confidentially, why not send messages the same way?

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We will refer to several schemes for sending data Let X and Y be parties and Z be a message { Z } k means message Z encrypted with key k Thus, our standard notation will be: X Y: { Z } k Which means, X sends message Z, encrypted with key k, to Y X and Y will be participants like Alice and Bob and k will be a clearly labeled key A || B means concatenate message A with B

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Typical to key exchanges is the idea of interchange keys and session keys An interchange key is a key associated with a particular user over a (long) period of time A session key is a key used for a particular set of communication events Why have both kinds of keys?

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If Bob and Alice have no prior arrangements, classical cryptosystems require a trusted third party Trent Trent and Alice share a secret key k Alice and Trent and Bob share a secret key k Bob Here is the protocol: 1. Alice Trent: {request session key to Bob} k Alice 2. Trent Alice: { k session } k Alice || { k session } k Bob 3. Alice Bob: { k session } k Bob

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Purpose Exchange a session key between two parties Weaknesses A trusted third party is required Protocols are complicated Some protocols have hard to spot security risks Practice looking for the holes in the protocols They always have a "man in the middle" aspect Always assume that Eve can completely control all communication

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Suddenly, the sun comes out! Public key exchanges should be really easy The basic outline is: 1. Alice Bob: { k session } e Bob e Bob is Bob's public key Only Bob can read it, everything's perfect! Problems can still happen if parties cannot get each other’s public keys reliably

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A cryptographic (or one-way) hash function (called a cryptographic checksum in the book) takes a variable sized message M and produces a fixed-size hash code H(M) Not the same as hash functions from data structures The hash code produced is also called a digest It can be used to provide authentication of both the integrity and the sender of a message It allows us to store some information about a message that an attacker cannot use to recover the message

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When two messages hash to the same value, this is called a collision Because of the pigeonhole principle, collisions are unavoidable The key feature we want from our hash functions is that collisions are difficult to predict

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Given a digest, should be hard to find a message that would produce it One-way property Preimage Resistance Given a message m, it should be hard to find a different message that has the same digest Second Preimage Resistance Should be hard to find any two messages that hash to the same digest (collision) Collision Resistance

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A small change in input should correspond to a large change in output Avalanching Hash function should work on a block of data of any size Applicability Output should be a fixed length Uniformity It should be fast to compute a digest in software and hardware No longer than retrieval from secondary storage Speed

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Instead of storing the actual passwords, Windows and Unix machines store the hash of the passwords When someone logs on, the operating system hashes the password and compares it to the stored version No one gets to see your original password!

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What’s the probability that Ahmad has the same password (or a password that hashes to the same value) as Bai Li? Very small! What’s the probability that anyone has the same password (or a password that hashes to the same value) as anyone else? Not nearly as small!

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If you are the administrator of a large system, you might notice that two people have the same password hash With people's password habits, the odds are very high that their passwords are the same To add to the semantic security of such schemes random salt (often 8 bits or so) is added to the end of a password When checking a password against the hash, the system tries all 2 8 possible values for the salt

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Message Digest Algorithm 5 Very popular hashing algorithm Designed by Ron Rivest (of RSA fame) Digest size: 128 bits Security Completely broken Reasonable size attacks (2 32 ) exist to create two messages with the same hash value MD5 hashes are still commonly used to check to see if a download finished without error

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Secure Hash Algorithm Created by NIST SHA-0 was published in 1993, but it was replaced in 1995 by SHA-1 The difference between the two is only a single bitwise rotation, but the NSA said it was important Digest size: 160 bits Security Mostly broken Attacks running in time exist SHA-2 is a successor family of hash functions 224, 256, 384, 512 bit digests Better security, but not as widely used Designed by the NSA SHA-3 is out now

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If we care about a group of k items which can have a value between 1 and n, the probability that two are the same is: Because this form is a little unwieldy, we have an approximation that is easier to punch into a calculator:

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If we want to find the number of items needed before there is greater than a 1/2 probability of collision we get: For large k, k(k-1) ≈ k 2, giving:

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If a hash value is made up of k bits 2 k can be big So, we need to check one hash against 2 k - 1 other hashes to have a 50% probability of matching But, by the birthday paradox We need a much smaller number to get a collision!

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Exam 1!

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Review Chapters 1 and 2 and all notes Keep working on Project 1 Finish Assignment 2 Due tonight by 11:59 Exam 1 In class on Monday

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