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Public Key Encryption Dan Fleck CS 469: Security Engineering 1 Coming up: Public Key Encryption These slides are modified with permission from Bill Young (Univ of Texas) Public Key Encryption • In 1976, Whitﬁeld Diﬃe and Martin Hellman proposed public key encryption (asymmetric encryption) in which diﬀerent keys are used for encryption and decryption. • In 1997, it was disclosed that asymmetric key algorithms had been developed in the early 1970’s by the British Government’s Communication Headquarters (GCHQ). They referred to the technique as non-secret encryption. 2 Coming up: Public Key Encryption Public Key Encryption Use a publicly disclosed key to encrypt and a secret key to decrypt. The requisite relationship is: P = {{P}K pub }K priv We’ll denote the public key for principal A by Ka and the private key will be denoted Ka−1 3 Coming up: Public Key Systems Public Key Systems Also, for some public key systems, RSA in particular, encryption and decryption commute and either key can be used in either function. That is: {{P}K }K -1 = P = {{P}K -1 }K This is crucial in some uses of RSA. But is not true for most public key cryptosystems. 4 Coming up: Public Key Systems Public Key Systems The basis of any public key system is the identiﬁcation of a oneway function: easily computed, but diﬃcult to invert without additional information. Example: It is easy to multiply two large primes p1 and p2. However, it is very diﬃcult to factor p1p2 to recover p1 and p2. But, given p1p2 and either of p1 or p2, it is straightforward to recover the other, simply by dividing. 5 Coming up: Eﬃciency of Encryption Eﬃciency of Encryption Public key systems largely solve the key distribution problem. Why? A public key encryption may take 10,000 times as long to perform as a symmetric encryption; the computation depends on more complex operations, not on simple bit-wise operations. Symmetric encryption remains the work horse of commercial cryptography, with asymmetric encryption playing some important special functions. 6 Coming up: Lessons Lessons • Devising an asymmetric encryption algorithm depends on identifying a one-way function, easy to compute but hard to invert. • Public key systems largely solve the key distribution problem. • Asymmetric algorithms are generally much less eﬃcient than symmetric algorithms. 7 Coming up: RSA Algorithm RSA Algorithm The Rivest-Shamir-Adelman (RSA) algorithm relies on the diﬃculty of factoring large numbers. Two keys, e and d, are used for encryption and decryption. The algorithm is such that: {{P}d }e = P = {{P}e }d A plaintext block P is encrypted as (Pe mod n). d is chosen so that: e d (P ) mod n = P An interceptor would have to factor Pe to recover the plaintext. The legitimate receiver knows d and merely computes, (P e )d mod n = P which is much easier. Coming up: Other Public Key Algorithms 8 Other Public Key Algorithms A public key system can be based on any one-way function. A rich source is the set of NP-complete problems. These are infeasible to solve, but a solution can be checked in polynomial time. Merkle and Hellman proposed a public key system based on the knapsack problem: given a set of integers and a target sum, ﬁnd a subset of the integers that sum to the target. The algorithm is theoretically very secure, but has practical weaknesses. 9 Coming up: Elliptic Curve Cryptography Elliptic Curve Cryptography • Elliptical Curve Cryptography (ECC) is a newer public key encryption system (1985) using elliptic curves instead of prime numbers • Some say 164 bit key is equivalent to other systems 1024 bit key (another say 256 bit = 3076 RSA bits) • 112-bit key ECC broken using 200 Playstation 3s running continuously for 3.5 months. • G=start point per curve • k=random private key, public key=kG Source:http://en.wikipedia.org/wiki/Elliptic_curve_cryptography#Cryptographic_schemes Coming up: Authentication with Public Keys 10 Authentication with Public Keys Assume Ka is A’s public key. Suppose B sends the following message to A: {M}Ka . What assurances does A have? 1. No-one intercepting the message could read it. Why? 2. He can’t be sure it actually came from B. Why not? Thus, encryption with the public key is a privacy transformation, but not an authenticity transformation. 11 10 Coming up: Authentication with Public Keys Authentication with Public Keys Using RSA, B send {M}Kb-1 to A. If A can decrypt it using Kb, what assurance is gained? 1. A is sure it originated with B. Why? 2. But someone intercepting the message might read it. Why? Thus, encryption with private key is an authenticity transformation, not a privacy transformation. Note this only works in RSA, because: {{P}d }e = P = {{P}e }d In other public key systems, you typically need two pairs of keys: one pair for privacy and the other pair for “signing” (authenticity). Coming up: Lessons 12 11 Lessons • RSA is the most widely used public key cryptosystem. • RSA is symmetric in the use of keys; most public key schemes are not. • A public key encryption can be used for authenticity or for privacy but not both at once. 13 12 Coming up: Cryptographic Hash Functions Cryptographic Hash Functions Dan Fleck CS 469: Security Engineering 14 13 Coming up: Hash Functions These slides are modified with permission from Bill Young (Univ of Texas) Hash Functions • A hash function is a function that converts variable-sized text into a small datum, usually a ﬁxed size integer. A cryptographic hash function has the additional qualities: • it is diﬃcult to construct a text that has a given hash, • it is diﬃcult to modify a given text without changing its hash, • it is unlikely that two diﬀerent messages will have the same hash. The hash value is sometimes called a message digest. Cryptographic hash functions are used to protect integrity. Coming up: Vocabulary 15 14 Vocabulary A function f is preimage resistant if, given h, it is hard to ﬁnd any m such that h = f (m). A function f is second preimage resistant if, given an input m1, it is hard to ﬁnd m2 ≠ m1 such that f (m1) = f (m2). This is sometimes called weak collision resistance. A function f is (strong) collision resistant if it is hard to ﬁnd two messages m1 and m2 such that f (m1) = f (m2). 16 15 Coming up: Birthday Attacks Birthday Attacks If a function f(x) yields any of H diﬀerent outputs with equal probability and H is suﬃciently large, then we expect to obtain a pair of diﬀerent arguments x1 and x2 with f (x1) = f (x2) after evaluating the function for about 1.25* H diﬀerent arguments on average. What does this mean for a hash value of 128 bits? for 160 bits? 17 16 Coming up: Cryptographic Hash Functions Cryptographic Hash Functions Hash functions usually are used for integrity, not conﬁdentiality. • In a document retrieval system containing legal records, it may be important to know that the copy retrieved is identical to that stored. • In a secure communications system, the correct transmission of messages may override conﬁdentiality concerns. A cryptographic hash function “binds” the bytes of a ﬁle together in a way that makes seal any alterations to the ﬁle apparent. We say that we the ﬁle to make it tamper-proof (actually tamper-resistant). Coming up: Using a Hash Function 18 17 Using a Hash Function The process is as follows: • Given a sensitive ﬁle f , compute the hash function h(f ) and store the result securely. • Each time the ﬁle is used or accessed, recompute the hash. • Compare it to the stored value. If the two values match, it is likely that no changes have occurred to the ﬁle. 19 18 Coming up: Common Hash Algorithms Common Hash Algorithms Two widely used cryptographic hash functions are: MD5: (Message Digest 5) invented by Ron Rivest and RSA Labs; SHA-1/SHA-2/SHS: (Secure Hash Algorithm or Standard) similar to MD5. MD5 hashes a message of any size to a 128-bit digest. SHA/SHS produce a 160-bit digest. 20 19 Coming up: Lessons Lessons • A cryptographic hash function takes an arbitrary text and produces a ﬁxed size bit string that depends on each value of the text. • It should be diﬃcult to ﬁnd collisions—values that hash to the same result. • A hash can be used to show with high probability that a text has not changed. 21 20 Coming up: Key Exchange Key Exchange Dan Fleck CS 469: Security Engineering 22 21 Coming up: The Key Exchange Problem These slides are modified with permission from Bill Young (Univ of Texas) The Key Exchange Problem Suppose you want to establish a secure communication channel with someone you don’t know. We call this a situation of mutual suspicion. This is extremely common. • You submit your income tax on-line. • You send your credit card information to a shopping website. • You wish to exchange encrypted email with another party. Once you agree on a shared secret (key) the communication can proceed. But how do you exchange the key? This is the key exchange problem. Coming up: Key Exchange: Attempt 1 23 22 Key Exchange: Attempt 1 Suppose both parties S and R have a public / private RSA key pair for asymmetric communication. Say S chooses a new symmetric key K and sends to R the following message: R can decrypt the message using S’s public key to retrieve K. What is wrong with this scheme? {K}K -1 s Answer: Any eavesdropper can intercept the message and decrypt it using S’s public key to retrieve K. Coming up: Key Exchange: Attempt 2 24 23 Key Exchange: Attempt 2 Instead, suppose S sends to R the following message: {K}K R Since only R can decrypt this message, conﬁdentiality is assured. What’s wrong this time? Now R doesn’t have any assurance that the message actually came from S. An intruder may be “spooﬁng” (pretending to be S) to obtain information that R intends only for S. Can we preserve both conﬁdentiality and authentication with one transaction? Coming up: Key Exchange (Cont.) 25 26 Key Exchange (Cont.) A third attempt is for S to send R the following: {{K}K -1 }KR S How does R extract K? What assurances does this provide? 1. Since, no one but R can decrypt the message, conﬁdentiality is assured. 2. No one but S could have performed the inner encryption, so authentication is accomplished. This notion of nested encryptions is very useful in a variety of cryptographic protocols. Could you have done the encryptions in the other order? Coming up: Lessons 26 24 Lessons • Public key cryptosystems can be used for key exchange, but you have to do it carefully. • Key exchange requires both conﬁdentiality and authentication. 27 25 Coming up: Diffie-Hellman Key Exchange Diffie-Hellman Key Exchange Dan Fleck CS 469: Security Engineering 28 27 Coming up: Diﬃe-Hellman Key Exchange These slides are modified with permission from Bill Young (Univ of Texas) Diﬃe-Hellman Key Exchange • The question of key exchange was one of the ﬁrst problems addressed by a cryptographic protocol. This was prior to the invention of public key cryptography. • The Diﬃe-Hellman key agreement protocol (1976) was the ﬁrst practical method for establishing a shared secret over an unsecured communication channel. • The point is to agree on a key that two parties can use for a symmetric encryption, in such a way that an eavesdropper cannot obtain the key. 29 28 Coming up: Diﬃe-Hellman Algorithm Diﬃe-Hellman Algorithm Steps in the algorithm: 1. Alice and Bob agree on a prime number p and a base g. 2. Alice chooses a secret number a, and sends Bob (ga mod p). 3. Bob chooses a secret number b, and sends Alice (gb mod p). 4. Alice computes ((gb mod p)a mod p). 5. Bob computes ((ga mod p)b mod p). Both Alice and Bob can use this number as their key. Notice that p and g need not be protected. Coming up: Diﬃe-Hellman Example 30 29 Diﬃe-Hellman Example 1. 2. 3. 4. 5. Alice and Bob agree on p = 23 and g = 5. Alice chooses a = 6 and sends 56 mod 23 = 8. Bob chooses b = 15 and sends 515 mod 23 = 19. Alice computes 196 mod 23 = 2. Bob computes 815 mod 23 = 2. Then 2 is the shared secret Clearly, much larger values of a, b, and p are required. An eavesdropper cannot discover this value even if she knows p and g and can obtain each of the messages. Coming up: Diﬃe-Hellman Security 31 30 Diﬃe-Hellman Security Suppose p is a prime of around 300 digits, and a and b at least 100 digits each. Discovering the shared secret given g, p, ga mod p and gb mod p would take longer than the lifetime of the universe, using the best known algorithm. This is called the discrete logarithm problem. 32 31 Coming up: Lessons Lessons • How can two parties agree on a secret value when all of their messages might be overheard by an eavesdropper? • The Diﬃe-Hellman algorithm accomplishes this, and is still widely used. • With suﬃciently large inputs, Diﬃe-Hellman is very secure. 33 32 End of presentation