# Public Key Cryptography. c Eli Biham - March 30, Public Key Cryptography

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1 Public Key Cryptography c Eli Biham - March 30, Public Key Cryptography

2 Key Exchange All the ciphers mentioned previously require keys known a-priori to all the users, before they can encrypt and decrypt. To communicate securely, they need to have a common key, known only to them. Possible key exchanges: 1. An a-priori meeting to exchange keys. 2. Sending the keys by a special courier. 3. Using an already existing common key, to encrypt additional keys. c Eli Biham - March 30, Public Key Cryptography

3 Key Exchange (cont.) The problem: A meeting is required between the users (or their couriers), in order to be able to communicate securely. Conclusion: Two users who have never met cannot communicate securely! c Eli Biham - March 30, Public Key Cryptography

4 Example Two armies during a war. Unit 1 Unit 2 Enemy The enemy wins, since it found all the keys that our army use for communication. Our army observes that the enemy found the keys. Our army must choose new keys, immediately, but it is impossible to send couriers through the enemies lines. If our army does not agree on new common secret keys, we will certainly loose. c Eli Biham - March 30, Public Key Cryptography

5 Public Key Cryptography - Background The model: A network of N users. In order to let any user to communicate securely to any other user, ( N 2) = O(N 2 ) keys should be distributed in advance in a secure way. Possible solution: Trusted center. Each user sets in advance a secret key with the trusted center. When user A wishes to communicate with user B, he chooses a new common key and sends it (encrypted under A s key) to the center, who forwards it (encrypted under B s key) to B. c Eli Biham - March 30, Public Key Cryptography

6 Public Key Cryptography - Background (cont.) Drawbacks: 1. There should be somebody that all the users trust. 2. All the users should have a common key with the trusted center in advance. 3. The center can always understand all the users messages (and can fake messages). c Eli Biham - March 30, Public Key Cryptography

7 Requirements for Key Exchange We prefer a solution that allows two users who have never met, and do not have any a-priori common information, to decide on a common key known only to them, and unknown to anybody else, even to passive eavesdroppers who listen to their communication (but cannot modify it). c Eli Biham - March 30, Public Key Cryptography

8 Public Key Cryptography 1 See: Diffie and Hellman, New Directions in Cryptography, IEEE Transactions on Information Theory, Vol. IT-22, No. 6, Nov c Eli Biham - March 30, Public Key Cryptography 1

9 Trapdoor Problems Basing the solution on the complexity of problems, which are easy to solve for the legal users, but are very difficult to the eavesdroppers. Such problems are called trapdoor problems. They allow to exchange secure common keys using insecure channels! c Eli Biham - March 30, Public Key Cryptography 1

10 Diffie-Hellman Key Exchange Protocol Based on number theory assumptions. The basic idea: 1. It is easy to calculate for any a, x and q. a x modq 2. There is no efficient algorithm which computes x given a, q, and a x modq. This is the discrete logarithm (DLOG) problem. c Eli Biham - March 30, Public Key Cryptography 1

11 Diffie-Hellman Key Exchange Protocol (cont.) Notations: Denote x in binary representation as x = x n 1 x n 2...x 1 x 0, where x = n 1 i=0 x i2 i. Let q be a large prime number. All the multiplications from now on are modulo q. c Eli Biham - March 30, Public Key Cryptography 1

12 Diffie-Hellman Key Exchange Protocol (cont.) Preparations: System parameters common to all users: Let q be a large prime number (q > ). Let a an integer 1 < a < q. Public and private keys: Each user U: chooses a random value X U (1 < X U < q) and keeps it secret. publishes Y U = a X U modq. c Eli Biham - March 30, Public Key Cryptography 1

13 Diffie-Hellman Key Exchange Protocol (cont.) The key exchange: Two users A,B who wish to have a common key, known only to them: A calculates K = (Y B ) X Amodq. B calculates K = (Y A ) X Bmodq. A and B result with the same common key K: (Y B ) X A (a X B ) X A a X BX A a X AX B (a X A ) X B (Y A ) X B (mod q). c Eli Biham - March 30, Public Key Cryptography 1

14 Diffie-Hellman Key Exchange Protocol (cont.) Security: 1. The secret keys are secure: if one can compute the secret key X A of A from Y A = a X Amodq, he solved the DLOG problem, and we assume it is difficult. 2. Can somebody compute the common key of A and B from their published keys (without computing the secret keys)? The problem of computing a X AX B modq from a, a X Amodq and a X Bmodq is assumed to be as difficult as DLOG. c Eli Biham - March 30, Public Key Cryptography 1

15 Public Key Cryptography Solution: Each user chooses two keys: A public key K E which he publishes. This key is publicly known. The public key is used for encryption. A secret keyk D which he keeps secret(also called private key). The secret key is used for decryption. c Eli Biham - March 30, Public Key Cryptography 1

16 Public Key Cryptography (cont.) Everybody (B) who knows A s public key can encrypt messages to A by but only A can decrypt it by C = E KE (M) M = D KD (C). Even B cannot decrypt messages he encrypted under A s public key(unless he keeps records of the messages he encrypted). c Eli Biham - March 30, Public Key Cryptography 1

17 Required properties: Public Key Cryptography (cont.) 1. the encryption and decryption functions E, D are publicly known and easy to compute. 2. It is possible to generate pairs of keys K E and K D which satisfy M : D KD (E KE (M)) = M. 3. Without the knowledge of K D, it is difficult to decrypt C, given only the public key K E (even though encryption is easy). Result: It is difficult compute K D from K E (even if the attacker have also many encrypted messages). c Eli Biham - March 30, Public Key Cryptography 1

18 The Key Generation It is difficult to calculate K D from K E. In many cases it is also difficult to calculate K E from K D. We need a trapdoor function E KE : easy to calculate, but difficult to invert. We should use an efficient function G(X) which takes a random X and generates both keys simultaneously. c Eli Biham - March 30, Public Key Cryptography 1

19 The Key Generation (cont.) Usage: Each user U generates a pair of random keys (K E,K D ) = G(random X), and publishes K E (in a public file). K D is kept secret. When another user A wishes to send a message M to U, he requests U s public key K E (from the public file), computes and sends C to U. U decrypts by C = E KE (M), M = D KD (C). c Eli Biham - March 30, Public Key Cryptography 1

20 The Key Generation (cont.) Properties: 1. Everybody can send messages to U, without the need to distribute a common secret key in advance. 2. Only U can decrypt. 3. The center (maintaining the public file) cannot decrypt (if he is only trusted to send U s real key to A). 4. There is no need to set common secret keys in advance. A and B can communicate securely after they request each others key from the center. The communication with the center does not have to be encrypted. c Eli Biham - March 30, Public Key Cryptography 1

21 The Key Generation (cont.) 5. Two users who have never met, can communicate securely even without a trusted center. However, they cannot authenticate each other without a trusted center. 6. The center can generate certificates for the users: he signs the users identity together with their public key. The users can then receive the certificates directly from the receivers, rather than asking the center for the public keys of the receivers. Then, they verify with the center s wellknown public key. c Eli Biham - March 30, Public Key Cryptography 1

22 Shortened Notation After the user U chooses his pair of keys K E and K D, and publishes his public key K E, we denote his encryption function (known to everybody) by E U ( ) = E KE ( ) and his decryption function (whose key is secret) by D U ( ) = D KD ( ). ForeveryuserU,E U canbecomputedbyalltheusers,butd U canbecomputed only by U. c Eli Biham - March 30, Public Key Cryptography 1

23 Remark Remark: Diffie and Hellman did notsuggestagood implementationofapublic key cryptosystem. Only after they published their paper, several public key cryptosystems were suggested, such as Merkle-Hellman s knapsack cryptosystem (broken later) and RSA. They predicted that public key cryptosystems will be based on the following problems (as was the case later in the listed systems) 1. Knapsack (an NP-complete problem): such as Merkle-Hellman. 2. Factoring: RSA, etc. 3. Discrete logarithm: ElGamal, DSS, etc. c Eli Biham - March 30, Public Key Cryptography 1

24 Public Key Signatures The encryption function E U is 1-1. If it is also onto, E U : M M, it can be used for signatures as well. U signs a message M by S = D U (H(M)), where H is a collision free hash function. Everybody can verify the originality of the signature S by checking whether E U (S)? = H(M). c Eli Biham - March 30, Public Key Cryptography 1

25 Public Key Signatures (cont.) Claim: E U (D U (X)) = X for every X. Proof: Let X be some value. From the definition, D U (E U (Y)) = Y for every Y, and in particular for Y = D U (X). Therefore, D U (E U (D U (X))) = D U (E U (Y)) = Y = D U (X). Since E U is 1-1, then D U is 1-1, and E U (D U (X)) = X. QED Secret signatures: If U wishes to keep the signature (sent to B) secret, he sends E B (S) = E B (D U (H(M))). B will decrypt and get S, and then will be able to verify it as before. c Eli Biham - March 30, Public Key Cryptography 1

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