June 6, 2016 this week Public key cryptography Pieter van den - PDF document

June 6, 2016 this week Public key cryptography Pieter van den Hombergh RSA and friends Crypto, PPK Public key cryptography Public and private Keys Information exchange Shared secret Authentication/signing Signing Pieter van den Hombergh Authenticity Public Key Fontys Hogeschool voor Techniek en Logistiek Infrastructure Securely (trusted) distribution of keys Current technology of the June 6, 2016 ww-web SSL/TLS Pieter van den Hombergh/FHTenL Public key cryptography June 6, 2016 1/27 Public key Scheme use and implementation cryptography Pieter van den Hombergh In 1976 1 Whitfield Diffie and Martin Hellman RSA and friends described how a digital signature scheme could be used. Crypto, PPK The idea of key exchange Public and private Keys Information exchange Ronald Rivest , Aid Shamir and Len Aldeman provided Shared secret Authentication/signing an implementaion for such a scheme in the RSA Signing algortihm in 1977. Authenticity Public Key Infrastructure Securely (trusted) distribution of keys Current technology of the ww-web SSL/TLS 1 quite recent compared to other, symmetric schemes Pieter van den Hombergh/FHTenL Public key cryptography June 6, 2016 2/27 Simplified explanation of RSA Public key cryptography Pieter van den Hombergh The concept heavily relies on number theory, some of RSA and friends which goes back a long time, and modulo arithmetic Crypto, PPK with big numbers. Public and private Keys Information exchange The strength lies in the fact that factoring large Shared secret Authentication/signing numbers is a slow process. Signing Authenticity The central model is Public Key Infrastructure ( m e ) d ≡ m (mod n ) Securely (trusted) distribution of keys Current technology of the where e , d and n are very large 2 positive integer numbers. ww-web SSL/TLS All computations, including the exponentiation, are done using modulo n arithmetic. Note that also in this case, ( m e ) d = ( m d ) e ≡ m ( mod n ), which shows the equivalence of the exponents d and e and is also essential in the approach. 2 think hundreds of bits Pieter van den Hombergh/FHTenL Public key cryptography June 6, 2016 3/27 Public key Example use cryptography Pieter van den Hombergh A lice creates key pair, of which ( n , e ) is the public key and RSA and friends ( n , d ) is the private key. She shares the public key with Crypto, PPK B ob. Public and private Keys B ob encrypts the message M. He therefor turns M into a Information exchange Shared secret large integer m , such that 0 ≤ m < m and gcd( m , n ) = 1 Authentication/signing Signing by using a padding scheme. From that he computes the Authenticity ciphertext c where Public Key Infrastructure c ≡ m e (mod n ) Securely (trusted) distribution of keys Current technology of the ww-web which he transmits to A lice. SSL/TLS A lice uses exponent d from her private key (n,d) to compute m ≡ c d (mod n ) producing the same m as B ob used. She than applies the reverse of the padding scheme, finding the original message. Pieter van den Hombergh/FHTenL Public key cryptography June 6, 2016 4/27 1

June 6, 2016 this week Key generation Public key cryptography Pieter van den Hombergh 1 Choose two distinct large primes p and q . These should RSA and friends be chosen random with a true random generator. (As Crypto, PPK in: unpredictable). Public and private Keys Information exchange 2 Compute n = p × q This n is the modulus in both public Shared secret Authentication/signing and private key. Its length, in bits, is the key length. Signing 3 Compute the Eulers totient function ϕ ( n ), which for the Authenticity primes p and q is simply n − ( p + q − 1). This is kept Public Key Infrastructure private. Securely (trusted) distribution of keys 4 Choose e such that 1 < e < ϕ ( n ) and gcd( e , ϕ ( n )) = 1, Current technology of the ww-web saying e and ϕ ( n ) are coprime . SSL/TLS 5 Determine d from d ≡ e − 1(mod ϕ ( n )) The e and n are combined into the public key, the d and n make up the private key. source: RSA Pieter van den Hombergh/FHTenL Public key cryptography June 6, 2016 5/27 Public key Public-key cryptography cryptography Pieter van den Hombergh Alice RSA and friends Crypto, PPK Public and private Keys Information exchange 0110101010 Large Shared secret 1101110100 Random Authentication/signing Signing 0011011010 Number Authenticity Key Public Key Infrastructure Generation Securely (trusted) distribution of keys Program Current technology of the ww-web SSL/TLS A Public A Private The key pair, simplified. Pieter van den Hombergh/FHTenL Public key cryptography June 6, 2016 6/27 Encrypt-decrypt Public key cryptography Pieter van den Hombergh Bob RSA and friends Crypto, PPK Hello Encrypt Public and private Keys Alice! Information exchange Shared secret Alice's Authentication/signing public key Signing 6EB69570 Authenticity 08E03CE4 Public Key Infrastructure Securely (trusted) distribution of keys Current technology of the Alice ww-web SSL/TLS Hello Decrypt Alice! Alice's private key The use of the keys Pieter van den Hombergh/FHTenL Public key cryptography June 6, 2016 7/27 Public key Shared secret cryptography Pieter van den Hombergh Alice RSA and friends Crypto, PPK Bob's Combine 751A696C Public and private Keys public key keys 24D97009 Information exchange Shared secret Alice and Bob's Authentication/signing shared secret Signing Alice's Authenticity private key Public Key Infrastructure Bob Securely (trusted) distribution of keys Current technology of the ww-web Alice's Combine 751A696C SSL/TLS public key keys 24D97009 Alice and Bob's shared secret Bob's private key A shared secret can be use as key for a symmetric encryption. (Diffy-Hellman) Pieter van den Hombergh/FHTenL Public key cryptography June 6, 2016 8/27 2

June 6, 2016 this week Signing Public key cryptography Pieter van den Hombergh To sign a message, a signature can be added, by RSA and friends encrypting with the public key. (Signed and sent by Crypto, PPK Alice to Bob). Public and private Keys Information exchange Shared secret Authentication/signing X = encrypt (pubkey Bob , encrypt (privkey Alice , M )) Signing Authenticity where M is the message and X the result of encrypting, Public Key then signing it. Infrastructure Securely (trusted) The encrypted X is sent along an insecure path. distribution of keys Current technology of the ww-web Decryption and verification is done with the reverse use SSL/TLS of opposite keys. M = decrypt (pubkey Alice , decrypt (privkey Bob , X )) Note that the intermediate product is still encrypted (with Alice’s pubkey). Remember the evaluation order of the function calls! Pieter van den Hombergh/FHTenL Public key cryptography June 6, 2016 9/27 Public key Signing with public key technology cryptography Pieter van den Hombergh We have seen that encryption uses the public key of the RSA and friends recipient to encrypt, and the private key of same to Crypto, PPK decrypt. Public and private Keys Information exchange The Owner of the key has the private key and is (or Shared secret Authentication/signing should be the only one that can decrypt. Signing Authenticity In signing we do the (almost) reverse: Public Key From the message to be signed, we compute a hash h . Infrastructure Securely (trusted) The sender encrypts this hash with his private key, distribution of keys Current technology of the ww-web appends it to the message. SSL/TLS some steps left out... (quiz which one?) The receiver decrypts the crypted hash, and if successfull, knows that the sender must have encrypted it, because he is the only one with access to the private key. Pieter van den Hombergh/FHTenL Public key cryptography June 6, 2016 10/27 Signing in a picture Public key cryptography Pieter van den Hombergh RSA and friends Crypto, PPK Public and private Keys Information exchange Shared secret Authentication/signing Signing Authenticity Public Key Infrastructure Securely (trusted) distribution of keys Current technology of the ww-web SSL/TLS In place of the wiggly vertical, we could add encryption to keep the information confidential. Quiz: What will the Pieter van den Hombergh/FHTenL Public key cryptography June 6, 2016 11/27 picture then look like? Public key Man in the Middle attack cryptography Pieter van den Hombergh RSA and friends Crypto, PPK Public and private Keys Information exchange Shared secret Authentication/signing Signing Authenticity Public Key Infrastructure Securely (trusted) distribution of keys Current technology of the ww-web SSL/TLS If the MITM can make the end user trust HIS certificate, he can decrypt the communication from the end user to him, and because he communicates to the server “on behalf” of the end user, also the traffic from the server intended for the end user. He can also modify the information. Pieter van den Hombergh/FHTenL Public key cryptography June 6, 2016 12/27 3