Discrete logarithms within computer and network security Prof Bill Buchanan, Edinburgh Napier

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1 Discrete logarithms within computer and network security Prof Bill Buchanan, Edinburgh Napier Introduction. Encryption: Public/Private Key. Key Exchange. Authentication. Signatures. ElGamal. Alice Bob Eve Trent Author: Prof Bill Buchanan

2 Discrete logarithms within computer and network security Prof Bill Buchanan, Edinburgh Napier Introduction. Encryption: Public/Private Key. Key Exchange. Authentication. Signatures. ElGamal. Alice Bob Introduction Eve Trent Author: Prof Bill Buchanan

3 Encryption Introduction Bruce Schneier Vincent Rijmen and Joan Daemen AES Twofish, Blowfish, Secrets and Lies Modern private key encryption Author: Prof Bill Buchanan

4 Encryption Introduction Intruder Eve Privacy (Private Key) Identity (Public Key) Integrity (Public/Private Key) Bob Alice Trent Trusted third party Author: Prof Bill Buchanan

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9 Discrete logarithms within computer and network security Prof Bill Buchanan, Edinburgh Napier Introduction. Encryption: Public/Private Key. Key Exchange. Authentication. Signatures. ElGamal. Alice Bob Encryption Eve Trent Author: Prof Bill Buchanan

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27 Encryption RSA Select two primes (p,q) Next, the n value is calculated. Thus: n = p x q = 11 x 3 = 33 Next PHI is calculated by: PHI = (p-1)(q-1) = 20 e selected so that GCD(e,PHI)=1 Public key: (n,e) Author: Prof Bill Buchanan

28 Discrete logarithms within computer and network security Prof Bill Buchanan, Edinburgh Napier Introduction. Encryption: Public/Private Key. Key Exchange. Authentication. Signatures. ElGamal. Alice Bob Eve Trent Authentication Author: Prof Bill Buchanan

29 Using Bob s private key to authenticate himself Message Message MD5 Encrypted MD5 The magic private key Bob s private key Bob Bob s public key Authentication Author: Prof Bill Buchanan

30 Bob encrypts the message/hash with Alice s public key Message Message MD5 Encrypted MD5 Bob The magic private key Bob s public key Bob s private key Encrypted Content Alice s public key Alice Authentication Alice s private key Author: Prof Bill Buchanan

31 Bob encrypts the message/hash with Alice s public key Message MD5 Message Encrypted MD5 Encrypted Content Bob Bob s private key The magic private key Authentication Bob s public key Encrypted Content Alice s public key Alice s private key Alice Author: Prof Bill Buchanan

32 Alice decrypts the message Message MD5 Message Encrypted MD5 Encrypted Content Bob Bob s private key The magic private key Bob s public key Alice Authentication Encrypted Content Message Encrypted MD5 Alice s public key Alice s private key Author: Prof Bill Buchanan

33 Alice decrypts the message Message MD5 Message Encrypted MD5 Encrypted Content Bob Bob s private key The magic private key Bob s public key Alice Authentication Encrypted Content Message Encrypted MD5 MD5 (message) MD5 (result) Alice compares the MD5 values. If they are the same Bob sent the message Author: Prof Bill Buchanan

34 Discrete logarithms within computer and network security Prof Bill Buchanan, Edinburgh Napier Introduction. Encryption: Public/Private Key. Key Exchange. Authentication. Signatures. ElGamal. Alice Bob ElGamal Eve Trent Author: Prof Bill Buchanan

35 Encryption ElGamal Bob Y = g x mod p Extremely difficult to the value of x, and there can be many solutions Eve Eve Y = 3 4 mod 17 -> 13 Alice Author: Prof Bill Buchanan

36 Encryption ElGamal Bob First Bob generates a prime number (p) and a number (g) which is between 1 and (p-1): P: G: p g Bob select a random number (x) which will be his private key: Bob selects a random number(x): He then calculates Y: x Bob sends g, p and Y to Alice. Y = g x mod p Alice Author: Prof Bill Buchanan

37 Alice p g y M (message) K (random) a=g k mod P b=y k M mod P a,b Bob Author: Prof Bill Buchanan

38 Encryption Conclusion Encryption Bob Eve Public key Encryption/ Decryption Typical application: Diffie-Hellman used to generate private-key. Public-key used for authentication. Private-key used for encryption. Encryption/ Decryption Key exchange (Diffie-Hellman) Secret key used to encrypt/decrypt (DES/3DES/AES) Communications Channel Used to authenticate (RSA) Encryption/ Decryption Alice Private key Private key Key exchange (Diffie-Hellman) Public key Author: Prof Bill Buchanan Secret key used to encrypt/decrypt (DES/3DES/AES) Public key Used to authenticate (RSA) Private key Private key RSA 2048 bits Replace by: ElGamal 160bits Public key Author: Prof Bill Buchanan

39 Discrete logarithms within computer and network security Prof Bill Buchanan, Edinburgh Napier Introduction. Encryption: Public/Private Key. Key Exchange. Authentication. Signatures. ElGamal. Alice Bob Eve Trent Author: Prof Bill Buchanan

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