# Network Security. Chapter 2 Basics 2.2 Public Key Cryptography. Public Key Cryptography. Public Key Cryptography

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1 Chair for Network Architectures and Services Department of Informatics TU München Prof. Carle Encryption/Decryption using Public Key Cryptography Network Security Chapter 2 Basics 2.2 Public Key Cryptography General Idea: encrypt with a publicly known key, but decryption only possible with a secret = private key Network Security, WS 2010/11, Chapter Public Key Cryptography Public Key Cryptography General idea: Use two different keys a private key K priv a public key K pub Given a ciphertext c = E(K pub, m) and K pub it should be infeasible to compute the corresponding plaintext without the private key K priv : m = D(K priv, c) = D(K priv, E(K pub, m)) It must also be infeasible to compute K priv when given K pub The key K priv is only known to the owner entity A called A s private key K priv-a The key K pub can be publicly known and is called A s public key K pub-a Applications: Encryption: If B encrypts a message with A s public key K pub-a, he can be sure that only A can decrypt it using K priv-a Integrity check and digital signatures: If B encrypts a message with his private key K priv-b, everyone knowing B s public key K pub-b can read the message and know that B has sent it. Important: If B wants to communicate with A, he needs to verify that he really knows A s public key and does not accidentally use the key of an adversary Known as the binding of a key to an identity Not a trivial problem so-called Public Key Infrastructures are one solution X.509 GnuPG Web of Trust Network Security, WS 2010/11, Chapter Network Security, WS 2010/11, Chapter 2.2 4

2 Public Key Cryptography Ingredients for a public key crypto system: One-way functions: It is believed that there are certain functions that are easy compute, while the inverse function is very hard to compute Real-world analogon: phone book When we speak of easy and hard, we refer to certain complexity classes more about that in crypto lectures and complexity theorey For us: Hard means infeasible on current hardware We know candidates, but have no proof for the existence of such functions Existence would imply P!= NP Special variant: Trap door functions Same as one-way functions, but if a second ( secret ) information is known, then the inverse is easy as well Blueprint: use a trap-door function in your crypto system Candidates: Factorization problem: basis of the RSA algorithm Complexity class unknown, but assumed to be outside P Discrete logarithm problem: basis of Diffie-Hellman and ElGamal No polynomial algorithms known, assumed to be outside P The RSA Public Key Algorithm The RSA algorithm was described in 1977 by R. Rivest, A. Shamir and L. Adleman [RSA78] Ron Rivest Adi Shamir Note: Clifford Cocks in the UK came up with the same scheme in 1973 but he worked for the government and it was treated classified and thus remained unknown to the scientific community. Leonard Adleman Network Security, WS 2010/11, Chapter Network Security, WS 2010/11, Chapter Some Mathematical Background The RSA Public Key Algorithm Definition: Euler s Φ Function: Let Φ(n) denote the number of positive integers m <n,such that m is relatively prime to n. m is relatively prime to n = the greatest common divisor (gcd) of m and n is one. Let p be prime, then {1,2,,p-1} are relatively prime to p, Φ(p) = p-1 Let p and q be distinct prime numbers and n = p q, then Φ(n) = (p-1) (q-1) RSA Key Generation: Randomly choose p, q distinct and large primes (really large: hundreds of bits = digits each) Compute n = p q, calculate Φ(n) = (p-1) (q-1) (Euler s Φ Function) Pick e Z such that 1 < e < Φ(n) and e is relatively prime to Φ(n), i.e. gcd(e,φ(n)) = 1 Use the extended Euclidean algorithm to compute d such that e d 1 MOD Φ(n) Euler s Theorem: Let n and a be positive and relatively prime integers, a Φ(n) 1 MOD n Proof: see [Niv80a] The public key is (n, e) The private key is d this is the trap door information Network Security, WS 2010/11, Chapter Network Security, WS 2010/11, Chapter 2.2 8

3 The RSA Public Key Algorithm The RSA Public Key Algorithm Definition: RSA function Let p and q be large primes; let n = p q. Let e N be relatively prime to Φ(n). Then RSA(e,n) := x x e MOD n Example: Let M be an integer that represents the message to be encrypted, with M positive, smaller than n. Example: Encode with <blank> = 99, A = 10, B = 11,..., Z = 35 So HELLO would be encoded as If necessary, break M into blocks of smaller messages: To encrypt, compute: C M e MOD n Why does RSA work: As d e 1MOD Φ(n) k Z: (d e) = 1 + k Φ( n) We sketch the proof for the case where M and n are relatively prime M C d MOD n (M e ) d MOD n M (e d) MOD n M (1 + k Φ( n)) MOD n M (M Φ ( n) ) k MOD n M 1 k MOD n (Euler s theorem*) M MOD n = M Decryption: To decrypt, compute: M C d MOD n In case where M and n are not relatively prime, Euler s theorem can not be applied. See [Niv80a] for the complete proof in that case. Network Security, WS 2010/11, Chapter Network Security, WS 2010/11, Chapter Using RSA On the Security of RSA All public-key crypto systems are much slower and more resourceconsuming than symmetric cryptography Thus, RSA is usually used in a hybrid way: Encrypt the actual message with symmetric cryptography Encrypt the symmetric key with RSA Using RSA requires some precautions Careful with choosing p and q: there are factorization algorithms for certain values that are very efficient Generally, one also needs a padding scheme to prevent certain types of attacks against RSA E.g. attack via Chinese remainder theorem: if the same clear text message is sent to e or more recipients in an encrypted way, and the receivers share the same exponent e, it is easy to decrypt the original clear text message Padding also works against a Meet-in-the-middle attack OAEP (from PKCS#1) is a well-known padding scheme for RSA The security of the RSA algorithm lies in the presumed difficulty of factoring n = p q It is known that computing the private key from the public key is as difficult as the factorization It is unknown if the private key is really needed for efficient decryption (there might be a way without, only no-one knows it yet) RSA is one of the most widely used and studied algorithms We need to increase key length regularly, as computers become more powerful 633 bit keys have already been factored Some claim 1024 bits may break in the near future (others disagree) Current recommendation is 2048 bit, should be on the safe side More is better, but slower Network Security, WS 2010/11, Chapter Network Security, WS 2010/11, Chapter

4 Alternatives to RSA Digital Signatures ElGamal (by Tahar El Gamal) Can be used for encryption and digital signatures ElGamal is based on another important difficult computational problem: Discrete logarithm (DLog) We discuss DLog soon We don t discuss ElGamal in detail here, but it has practical relevance: ElGamal is a default in GnuPG Digital Signature Algorithm (DSA) is based on ElGamal As such, ElGamal/DSA is also part of Digital Signature Standard (another NIST standard) Signing = adding a proof of who has created a message, and that it has not been altered on the way Who: authenticity Not altered: integrity Network Security, WS 2010/11, Chapter Network Security, WS 2010/11, Chapter Digital Signatures The Discrete Logarithm: DLog A wants to sign a message. General idea: A computes a cryptographic hash value of her message: h(m) Hashes are one-way functions, i.e. given h(m) it s infeasible to obtain m We ll discuss hash functions soon A encrypts h(m) with her private key K priv-a Sig = E K_priv (h(m)) Given m, everyone can now compute h(m) Decrypt signature: D(E(h(m))) = h(m) and check if hash values are the same If they match, A must have been the creator as only A knows the private key In the following, we will discuss another popular one-way / trap-door function: the discrete logarithm DLog is used in a number of ways Diffie-Hellman Key Agreement Protocol Can I agree on a key with someone else if the attacker can read my messages? ElGamal DLog problems can be transformed to Elliptic Curve Cryptography We ll discuss this later Now: more mathematics Network Security, WS 2010/11, Chapter Network Security, WS 2010/11, Chapter

5 Some Mathematical Background DLog: Some Mathematical Background Theorem/Definition: primitive root, generator Let p be prime. Then g {1,2,,p-1} such that {g a 1 a (p-1) } = {1,2,,p-1} if everything is computed MOD p i.e. by exponentiating g you can obtain all numbers between 1 and (p -1) For the proof see [Niv80a] g is called a primitive root (or generator) of {1,2,,p-1} Example: Let p = 7. Then 3 is a primitive root of {1,2,,p-1} MOD 7, MOD 7, MOD 7, MOD 7, MOD 7, MOD 7 Definition: discrete logarithm Let p be prime, g be a primitive root of {1,2,,p-1} and c be any element of {1,2,,p-1}. Then z such that: g z c MOD p z is called the discrete logarithm of c modulo p to the base g Example: 6 is the discrete logarithm of 1 modulo 7 to the base 3 as MOD 7 The calculation of the discrete logarithm z when given g, c, and p is a computationally difficult problem and the asymptotical runtime of the best known algorithms for this problem is exponential in the bit-length of p Network Security, WS 2010/11, Chapter Network Security, WS 2010/11, Chapter Diffie-Hellman Key Exchange (1) Diffie-Hellman Key Exchange (2) The Diffie-Hellman key exchange was first published in the landmark paper [DH76], which also introduced the fundamental idea of asymmetric cryptography The DH exchange in its basic form enables two parties A and B to agree upon a shared secret using a public channel: Public channel means, that a potential attacker can read all messages exchanged between A and B It is important that A and B can be sure that the attacker is not able to alter messages as in this case he might launch a man-in-the-middle attack The mathematical basis for the DH exchange is the problem of finding discrete logarithms in finite fields The DH exchange is not an encryption algorithm. Whitfield Diffie Generate random a < p Compute X = g a MOD p (p, g, X) Y Generate random b < p Compute Y = g b MOD p Compute K = X b MOD p Martin E. Hellman Compute K = Y a MOD p Network Security, WS 2010/11, Chapter Network Security, WS 2010/11, Chapter

6 Diffie-Hellman Key Exchange (3) If Alice (A) and Bob (B) want to agree on a shared secret K and their only means of communication is a public channel, they can proceed as follows: A chooses a prime p, a primitive root g of {1,2,,p-1} and a random number x A and B can agree upon the values p and g prior to any communication, or A can choose p and g and send them with his first message A chooses a random number a: A computes X = g a MOD p and sends X to B B chooses a random number b B computes Y = g b MOD p and sends Y to A Both sides compute the common secret: A computes K = Y a MOD p B computes K = X b MOD p As g (a.b) MOD p = g (b. a) MOD p, it holds: K = K An attacker Eve who is listening to the public channel can only compute the secret K, if she is able to compute either a or b which are the discrete logarithms of X and Y modulo p to the base g. In essence, A and B have agreed on a key without ever sending the key over the channel This does not work anymore if an attacker is on the channel and can replace the values with his own ones Elliptic Curve Cryptography (ECC) Motivation: RSA is probably the most widely implemented algorithm for Public Key Cryptography Does public key cryptography need long keys with bits? Also, it is good to think of alternatives due to the developments in the area of primality testing, factorization and computation of discrete logarithms Elliptic Curve Cryptocraphy (ECC) ECC is based on a finite field of points. Points are presented within a 2-dimensional coordinate system: (x,y) All points within the elliptic curve satisfy an equation of this type: y 2 = x 3 + ax + b Network Security, WS 2010/11, Chapter Network Security, WS 2010/11, Chapter Elliptic Curve Cryptography (ECC) Elliptic Curve Cryptography (ECC) Given this set of points an additive operator can be defined Any DLog-based algorithm can be turned into an ECC-based algorithm ECC problems are generally believed to be harder (though there is a lack of mathematic proofs) Allows us to have shorter key sizes good for storage and transmission over networks ECC is still a new thing but there are more implementations now A multiplication of a point P by a number n is simply the addition of P to itself n times Q = np = P + P + + P The problem of determining n, given P and Q, is called the elliptic curve s discrete logarithm problem (ECDLP) The ECDLP is believed to be hard in the general class obtained from the group of points on an elliptic curve over a finite field Network Security, WS 2010/11, Chapter Network Security, WS 2010/11, Chapter

8 Additional References [Bless05] R. Bless, S. Mink, E.-O. Blaß, M. Conrad, H.-J. Hof, K. Kutzner, M. Schöller: "Sichere Netzwerkkommunikation", Springer, 2005, ISBN: [Bre88a] D. M. Bressoud. Factorization and Primality Testing. Springer, [Cor90a] T. H. Cormen, C. E. Leiserson, R. L. Rivest. Introduction to Algorithms. The MIT Press, [DH76] W. Diffie, M. E. Hellman. New Directions in Cryptography. IEEE Transactions on Information Theory, IT-22, pp , [DSS] National Institute of Standards and Technology (NIST). FIPS , DRAFT Digital Signature Standard (DSS), March [ElG85a] T. ElGamal. A Public Key Cryptosystem and a Signature Scheme based on Discrete Logarithms. IEEE Transactions on Information Theory, Vol.31, Nr.4, pp , July [Ferg03] Niels Ferguson, B. Schneier: Practical Cryptography, Wiley, 1st edition, March 2003 [Kob87a] N. Koblitz. A Course in Number Theory and Cryptography. Springer, [Men93a] A. J. Menezes. Elliptic Curve Public Key Cryptosystems. Kluwer Academic Publishers, [Niv80a] I. Niven, H. Zuckerman. An Introduction to the Theory of Numbers. John Wiley & Sons, 4 th edition, [Resc00] Eric Rescorla, SSL and TLS: Designing and Building Secure Systems, Addison-Wesley, 2000 [RSA78] R. Rivest, A. Shamir und L. Adleman. A Method for Obtaining Digital Signatures and Public Key Cryptosystems. Communications of the ACM, February [Sham03] Adi Shamir, Eran Tromer, On the cost of factoring RSA-1024, RSA Cryptobytes vol. 6, 2003 Network Security, WS 2010/11, Chapter

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