Lukasz Pater CMMS Administrator and Developer

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1 Lukasz Pater CMMS Administrator and Developer EDMS

2 Agenda Introduction Why do we need asymmetric ciphers? One-way functions RSA Cipher Message Integrity Examples Secure Socket Layer Single Sign On Conclusion

3 Introduction Challenge: How perfect strangers can send each other encrypted messages? Powerful Idea: Public-key encryption! Diffie, Hellman, Merkle [1976] Rivest, Shamir, Adleman [1977]

4 Introduction Symmetric Cryptography Requires that sender and receiver know shared secret key Asymmetric Cryptography radically different approach Spring Campus yxs7d70k dy_mgh Spring Campus Sender and receiver do not share the same key Public encryption key known to all Private decryption key know only to the receiver

5 One-way functions Most functions are invertible; for any f(x) = y, there is an f -1 (y) = x (e.g. division, multiplication, DES) f(x) = 2! x, f -1 (x) = x / 2 A function which is easy to compute in one direction, but difficult to compute in the other, is called one-way function Polynomial factorization, hashing, modular arithmetic, One-way function that can be easily inverted with additional knowledge ( ) is known as trapdoor one-way function f(x) easy f -1 (x) hard Domain f -1 (x, ) easy Range

6 One-way functions and cryptography Public key encryption is based on the existence of trapdoor one-way functions Encryption, done with the public key ( ), is very easy Decryption is computationally hard (factorization of large numbers, discrete logarithm problem) Knowledge of the private key ( making inversion easy ) opens the trapdoor, Spring Campus yxs7d70k dy_mgh Spring Campus

7 One-way functions and cryptography Public key encryption is based on the existence of trapdoor one-way functions Encryption, done with the public key ( ), is very easy Decryption is computationally hard (factorization of large numbers, discrete logarithm problem) Knowledge of the private key ( making inversion easy ) opens the trapdoor, Spring Campus yxs7d70k dy_mgh Spring Campus

8 RSA (Rivest, Shamir, Adleman) RSA is the most well-known and common public key cryptosystem Basic notation: a key pair (e,d,n) contains two keys: (e,n) is the public key, (d,n) is the private key Let RSA be the encryption function and ABC plaintext message (e,n) (d,n) ABC 7#K ABC RSA(e, N, ABC ) RSA(d, N, 7#K )

9 RSA Algorithm RSA s security is based on modular arithmetic x mod n; remainder of x when divided by n (5 mod 2 = 1) p and q are relatively prime if their greatest common divisor is 1 p and q are multiplicative inverse when (p! q = 1 mod n) RSA algorithm Pick two large prime numbers p and q, let N = p! q Select small integer e that is relatively prime to (p-1)(q-1) Find d, the multiplicative inverse of e mod (p-1)(q-1) (e,n) is the public key, to encrypt compute C = M e mod N (d,n) is the private key, to decrypt compute M = C d mod N

10 RSA Algorithm / Example Pick two large prime numbers p and q, let N = p! q p = 3, q = 11, N = p! q = 33 Select small integer e that is relatively prime to (p-1)(q-1) (p-1)(q-1) = 20, e = 3 (prime to 20) Find d, the multiplicative inverse of e mod (p-1)(q-1) Find d, that d! 3 mod 20 = 1, d = 7 (e,n) is the public key, to encrypt compute M e mod N (3,33) is the public key, 2 3 mod 33 = 8 (d,n) is the private key, to decrypt compute M d mod N (7,33) is the private key, 8 7 mod 33 = 2

11 Strengths of RSA Why is RSA secure? Suppose you know the public key (e,n), N = p! q The security of RSA is dependent on the assumption that it s difficult to determine the private key d from (e,n) Essentially need to find factors of N without knowing p and q Factoring is thought to be computationally hard (no proof!) Some statistics for the fastest known factoring algorithm Key Size 512 bits 30,000 MIPS-years required to factor 1024 bits 300,000,000, bits 300,000,000,000,000,000,000

12 Message Integrity Allows communicating parties to verify that received messages are authentic Content of the message has not been altered Source of the message is who/what you think it is Message Digests A Hash function H() that takes as input an arbitrary length message and outputs a fixed-length string message digest

13 Message Integrity Allows communicating parties to verify that received messages are authentic Content of the message has not been altered Source of the message is who/what you think it is Message Digests A Hash function H() that takes as input an arbitrary length message and outputs a fixed-length string message digest

14 Digital Signatures Cryptographic technique analogous to hand-written signatures Sender digitally signs the digest of the document ( SHA ) with his/hers private key ( ), establishing that he is the owner SHA Gdgw490s SHA

15 Certification Authorities Certification Authority (CA entity, for example: to Bob Bob provides proof of identity to the CA ) binds public key to particular CA creates certificate binding Bob to his public key Certificate ( ) containing Bob s public key ( ) digitally signed by CA s private key ( ) says: This is Bob s public key

16 Secure Socket Layer (SSL) TCP provides reliable end-toend service TCP with SSL provides reliable and secure end-to-end service HTTPS: HTTP over SSL Subsequently became Internet standard know as TLS Provides message integrity (hash functions) and message confidentiality (symmetric encryption with shared secret key) Hello: SSL options Hello: SSL options, Certificate Session key Cipher Specification Cipher Specification Encrypted Traffic

17 Secure Socket Layer (SSL)

19 Asymmetric Cryptography Single Sign On (SSO) Multi sign on is troublesome Is it possible to just sign-on once to perform all the actions? Single sign-on can be used to answer that question SSO delegates the authentication to centralized service

20 Single Sign On / How does it work? Identity Provider (IP): the system that asserts information about a subject (by issuing a ticket ) Service Provider (SP): the system that relies on the information supplied to it by the Identity Provider (in the issued ticket) 1) User opens a web page 2) SP redirects to IP 3) User logs in 4) IP issues a ticket and redirects to SP 5) SP validates the ticket 1 5 Service Provider Identity Provider

21 Single Sign On (SAML) Ticket

22 SSO / Advantages and Disadvantages Advantages Reduces the chance of forgetting your password. By having your one-set master password Very reach API Less code to maintain for the web master Single Sign Out Disadvantages The SSO Identity Provider is highly-critical Bad for a multi-user computer, especially if the user stays logged in all the time

23 Conclusions Asymmetric encryption was introduced to complement the inherent problem of the need to share the same key in symmetric ciphers Symmetric encryption algorithms can be extremely fast Public key algorithms are orders of magnitude less efficient than symmetric key encryption Use public key cryptography just to exchange the key! Public key certificates allow the secure communication between services and client applications Single Sign On enables to access multiple related, but independent software systems

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Software Implementation of Gong-Harn Public-key Cryptosystem and Analysis by Susana Sin A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Master

[SMO-SFO-ICO-PE-046-GU-

Presentation This module contains all the SSL definitions. See also the SSL Security Guidance Introduction The package SSL is a static library which implements an API to use the dynamic SSL library. It