RSA Algorithm - Public Key Cryptography
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Public key cryptography uses two keys: a public key to encrypt messages and a private key to decrypt them. The RSA algorithm is based on the difficulty of factoring large prime numbers. It works by having users generate a public/private key pair and publishing their public key. To encrypt a message, the sender uses the recipient's public key. Only the recipient can decrypt with their private key. The security of RSA relies on the computational difficulty of factoring the modulus used to generate the keys.
RSA Algorithm - Public Key Cryptography
- 1. Public Key Cryptography and the RSAAlgorithm
- 2. Private-Key Cryptography • traditional private/secret/single key cryptography uses one key • Key is shared by both sender and receiver • if the key is disclosed communications are compromised • also known as symmetric, both parties are equal • hence does not protect sender from receiver forging a message & claiming is sent by sender
- 3. Public-Key Cryptography • probably most significant advance in the 3000 year history of cryptography • uses two keys – a public key and a private key • asymmetric since parties are not equal • uses clever application of number theory concepts to function • complements rather than replaces private key cryptography
- 4. Public-Key Cryptography • public-key/two-key/asymmetric cryptography involves the use of two keys: • a public-key, which may be known by anybody, and can be used to encrypt messages, and verify signatures • a private-key, known only to the recipient, used to decrypt messages, and sign (create) signatures • is asymmetric because • those who encrypt messages or verify signatures cannot decrypt messages or create signatures
- 6. Why Public-Key Cryptography? • developed to address two key issues: • key distribution – how to have secure communications in general without having to trust a KDC with your key • digital signatures – how to verify a message comes intact from the claimed sender • public invention due to Whitfield Diffie & Martin Hellman at Stanford U. in 1976 • known earlier in classified community
- 7. Public-Key Characteristics • Public-Key algorithms rely on two keys with the characteristics that it is: • computationally infeasible to find decryption key knowing only algorithm & encryption key • computationally easy to en/decrypt messages when the relevant (en/decrypt) key is known • either of the two related keys can be used for encryption, with the other used for decryption (in some schemes)
- 9. Public-Key Applications • can classify uses into 3 categories: • encryption/decryption (provide secrecy) • digital signatures (provide authentication) • key exchange (of session keys) • some algorithms are suitable for all uses, others are specific to one
- 10. Security of Public Key Schemes • like private key schemes brute force exhaustive search attack is always theoretically possible • but keys used are too large (>512bits) • security relies on a large enough difference in difficulty between easy (en/decrypt) and hard (cryptanalyse) problems • more generally the hard problem is known, its just made too hard to do in practise • requires the use of very large numbers • hence is slow compared to private key schemes
- 11. RSA • by Rivest, Shamir & Adleman of MIT in 1977 • best known & widely used public-key scheme • based on exponentiation in a finite (Galois) field over integers modulo a prime • nb. exponentiation takes O((log n)3 ) operations (easy) • uses large integers (eg. 1024 bits) • security due to cost of factoring large numbers • nb. factorization takes O(e lognloglogn ) operations (hard)
- 12. RSA Key Setup • each user generates a public/private key pair by: • selecting two large primes at random - p, q • computing their system modulus N=p.q • note ø(N)=(p-1)(q-1) • selecting at random the encryption key e • where 1<e<ø(N), gcd(e,ø(N))=1 • solve following equation to find decryption key d • e.d=1 mod ø(N) and 0≤d≤N • publish their public encryption key: KU={e,N} • keep secret private decryption key: KR={d,p,q}
- 13. RSA Use • to encrypt a message M the sender: • obtains public key of recipient KU={e,N} • computes: C=Me mod N, where 0≤M<N • to decrypt the ciphertext C the owner: • uses their private key KR={d,p,q} • computes: M=Cd mod N • note that the message M must be smaller than the modulus N (block if needed)
- 14. Why RSA Works • because of Euler's Theorem: • aø(n) mod N = 1 • where gcd(a,N)=1 • in RSA have: • N=p.q • ø(N)=(p-1)(q-1) • carefully chosen e & d to be inverses mod ø(N) • hence e.d=1+k.ø(N) for some k • hence : Cd = (Me )d = M1+k.ø(N) = M1 .(Mø(N) )q = M1 .(1)q = M1 = M mod N
- 15. RSA Example 1. Select primes: p=17 & q=11 2. Compute n = pq =17×11=187 3. Compute ø(n)=(p–1)(q-1)=16×10=160 4. Select e : gcd(e,160)=1; choose e=7 5. Determine d: de=1 mod 160 and d < 160 Value is d=23 since 23×7=161= 10×160+1 6. Publish public key KU={7,187} 7. Keep secret private key KR={23,17,11}
- 16. RSA Example cont • sample RSA encryption/decryption is: • given message M = 88 (nb. 88<187) • encryption: C = 887 mod 187 = 11 • decryption: M = 1123 mod 187 = 88
- 17. Exponentiation • can use the Square and Multiply Algorithm • a fast, efficient algorithm for exponentiation • concept is based on repeatedly squaring base • and multiplying in the ones that are needed to compute the result • look at binary representation of exponent • only takes O(log2 n) multiples for number n • eg. 75 = 74 .71 = 3.7 = 10 mod 11 • eg. 3129 = 3128 .31 = 5.3 = 4 mod 11
- 18. Exponentiation
- 19. RSA Key Generation • users of RSA must: • determine two primes at random - p, q • select either e or d and compute the other • primes p,q must not be easily derived from modulus N=p.q • means must be sufficiently large • typically guess and use probabilistic test • exponents e, d are inverses, so use Inverse algorithm to compute the other
- 20. RSA Security • three approaches to attacking RSA: • brute force key search (infeasible given size of numbers) • mathematical attacks (based on difficulty of computing ø(N), by factoring modulus N) • timing attacks (on running of decryption)
- 21. Factoring Problem • mathematical approach takes 3 forms: • factor N=p.q, hence find ø(N) and then d • determine ø(N) directly and find d • find d directly • currently believe all equivalent to factoring • have seen slow improvements over the years • as of Aug-99 best is 130 decimal digits (512) bit with GNFS • biggest improvement comes from improved algorithm • cf “Quadratic Sieve” to “Generalized Number Field Sieve” • barring dramatic breakthrough 1024+ bit RSA secure • ensure p, q of similar size and matching other constraints
- 22. Timing Attacks • developed in mid-1990’s • exploit timing variations in operations • eg. multiplying by small vs large number • or IF's varying which instructions executed • infer operand size based on time taken • RSA exploits time taken in exponentiation • countermeasures • use constant exponentiation time • add random delays • blind values used in calculations
- 23. Summary • have considered: • principles of public-key cryptography • RSA algorithm, implementation, security
Editor's Notes
- #3: So far all the cryptosystems discussed have been private/secret/single key (symmetric) systems. All classical, and modern block and stream ciphers are of this form.
- #4: Will now discuss the radically different public key systems, in which two keys are used. Anyone knowing the public key can encrypt messages or verify signatures, but cannot decrypt messages or create signatures, counter-intuitive though this may seem. It works by the clever use of number theory problems that are easy one way but hard the other. Note that public key schemes are neither more secure than private key (security depends on the key size for both), nor do they replace private key schemes (they are too slow to do so), rather they complement them.
- #6: Stallings Fig 9-1.
- #7: The idea of public key schemes, and the first practical scheme, which was for key distribution only, was published in 1977 by Diffie & Hellman. The concept had been previously described in a classified report in 1970 by James Ellis (UK CESG) - and subsequently declassified in 1987. See History of Non-secret Encryption (at CESG). Its interesting to note that they discovered RSA first, then Diffie-Hellman, opposite to the order of public discovery!
- #8: Public key schemes utilise problems that are easy (P type) one way but hard (NP type) the other way, eg exponentiation vs logs, multiplication vs factoring. Consider the following analogy using padlocked boxes: traditional schemes involve the sender putting a message in a box and locking it, sending that to the receiver, and somehow securely also sending them the key to unlock the box. The radical advance in public key schemes was to turn this around, the receiver sends an unlocked box to the sender, who puts the message in the box and locks it (easy - and having locked it cannot get at the message), and sends the locked box to the receiver who can unlock it (also easy), having the key. An attacker would have to pick the lock on the box (hard).
- #9: Stallings Fig 9-4. Here see various components of public-key schemes used for both secrecy and authentication. Note that separate key pairs are used for each of these – receiver owns and creates secrecy keys, sender owns and creates authentication keys.
- #11: Public key schemes are no more or less secure than private key schemes - in both cases the size of the key determines the security. Note also that you can&apos;t compare key sizes - a 64-bit private key scheme has very roughly similar security to a 512-bit RSA - both could be broken given sufficient resources. But with public key schemes at least there&apos;s usually a firmer theoretical basis for determining the security since its based on well-known and well studied number theory problems.
- #12: RSA is the best known, and by far the most widely used general public key encryption algorithm.
- #13: This key setup is done once (rarely) when a user establishes (or replaces) their public key. The exponent e is usually fairly small, just must be relatively prime to ø(N). Need to compute its inverse to find d. It is critically important that the private key KR={d,p,q} is kept secret, since if any part becomes known, the system can be broken. Note that different users will have different moduli N.
- #15: Can show that RSA works as a direct consequence of Euler’s Theorem.
- #16: Here walk through example using “trivial” sized numbers. Selecting primes requires the use of primality tests. Finding d as inverse of e mod ø(n) requires use of Inverse algorithm (see Ch4)
- #17: Rather than having to laborious repeatedly multiply, can use the &quot;square and multiply&quot; algorithm with modulo reductions to implement all exponentiations quickly and efficiently (see next).
- #20: Both the prime generation and the derivation of a suitable pair of inverse exponents may involve trying a number of alternatives, but theory shows the number is not large.
- #22: See Stallings Table 9-3 for progress in factoring. Best current algorithm is the “Generalized Number Field Sieve” (GNFS), which replaced the earlier “Quadratic Sieve” in mid-1990’s. Do have an even more powerful and faster algorithm - the “Special Number Field Sieve” (SNFS) which currently only works with numbers of a particular form (not RSA like). However expect it may in future be used with all forms. Numbers of size 1024+ bits look reasonable at present, and the factors should be of similar size