1. Public Key Cryptography
Modular Arithmetic Tables
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2. + 1 2 3 4 5 6 7 8 9
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3. × 1 2 3 4 5 6 7 8 9
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4. 1 2 3 4 5 6 7 8 9 10 11 12
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5. Terminology Asymmetric cryptography Public key (known to entire world)
Private key (not secret key)
Encryption process (P to C with public key)
Decryption Process (C to P with private key)
Digital signature (P signed with private key)
Only holder of private key can sign, so can’t be forged
But, can be recognized!
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6. Uses
Orders of magnitude slower than symmetric key crypto, so usually used to initiate symmetric key session
Much easier to configure, so used widely in network protocols to establish temporary shared key that is used to transmit secret (symmetric) key
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7. Uses Transmitting over insecure channel
Alice <puA, prA> , Bob <puB, PrB>
Alice to Bob encrypt m with puB
Bob to alice cncrypt m with puA
Accurately knowing public key of other person is one of biggest challenges of using public key crypto.
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8. Uses Secure storage on insecure media
Encrypt not whole file, but a randomly generated secret key with public key. Then encrypt file using secret key.
Note if lose private key, you’re out of luck. To backup, encrypt secret key with public key of a trusted friend (lawyer).
Important advantage: Alice can enrypt a message for Bob without knowing Bob’s decryption key
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9. Uses
Authentication
If Bob wants to prove his identity with secret key crypto, he needs a different secret key shared with each potential correspondent (otherwise friends can impersonate him)
Alice can verify she’s talking to Bob (assuming she knows his public key) by sending a message r to Bob encrypted with Bob’s public key. Bob sends back the cleartext message r (which only he could have decrypted).
Note Alice need not keep any secret information in order to verify Bob. (Unlike secret key crypto, in which a backup tape with a copy of a secret key might be used to impersonate Bob)
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10. Uses
Digital Signatures: prove message generated by particular individual
“Forged in USA (engraved on screwdriver claiming to be of brand Craftsman)
If Bob encrypts a message with his private key, this proves both
Bob generated the message
The message has not been modified (if so, the signature will no longer match!)
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11. Uses
Digital Signatures: prove message generated by particular individual
Non-repudiation: Bob cannot deny having generated the message, since Alice could not have generated the proper signature without knowledge of Bob’s private key.
Note that this can’t be done with symmetric key. If Bob tries to claim he didn’t send the message, Alice would know he’s lying (because no one but herself and Bob would have the secret key), but Alice could not prove this to anyone else (since she herself could have generated the authentication code).
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12. Modular Arithmetic Addition Multiplication
Can be used as scheme to encrypt digits, since it maps each digit to different digit in a reversible way (decryption is addition by additive inverse)
Actually a Caeser cipher (and not good)
Multiplication
Look at mod 10. Multiplication by 1,3,7, or 9 works, but not any of the others. Decryption done by multiplying by multiplicative inverse.
Multiplicative inverses can be found by using Euclid’s Algorithm (don’t sweat the details). Given x and n, it finds y such that xy = 1 mod n (if there is such a y)
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13. Modular Arithmetic
Why 1,3,7,9? These are the numbers that are relatively prime to 10. All numbers that are relatively prime to 10 will have inverses, others won’t (so we can use these as ciphers, though not good ones).
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14. Totient Function Allegedly from total and quotient
How many numbers less than n are relatively prime to n?
Totient function, φ(n) gives this.
If n is prime, φ(n) = n-1 (1,2,…n-1)
If p and q are prime, φ(pq) = (p-1)(q-1)
1p, 2p, … (q-1)p 1q, 2q, … (p-1)q and 0 so have
pq – ((p-1) + (q-1) + 1) = (p-1)(q-1)
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15. Modular Exponentiation
Note exponentiation by 3 acts as encryption of digits. Is there an inverse to this operation? Sometimes.
Fact:
Not true for all n, but for all any square free n (any n that doesn’t have p^2 as a factor for any prime p)
Note that if y = 1 mod φ(n), then x^y mod n = x mod n.
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16. RSA Key length variable (usually around 512 or now 1024 bits)
Plaintext block must be smaller than key length
Ciphertext block will be length of key
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17. RSA
Choose two large primes (around 256 bits each) p and q. Let n = pq (impossible to factor)
Choose number e that is relatively prime to φ(n). Can do this since you know p and q and thus φ(pq) (and from the derivation know exactly which numbers are relatively prime!
Public key is <e, n>
To make private key, find d that is the multiplicative inverse of e mod φ(n) (so ed = 1 mod φ(n)) (use Euclid’s algorithm)
Private key is <d,n>
To encrypt a number m, compute C = m^e mod n.
To decrypt: m = C^d mod n.
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18. Questions Why does it work? Why is it secure?
Are operations sufficiently efficient?
How do we find big primes?
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19. Why Does It Work?
We chose d and e so that de = 1 mod φ(n), so for any x, x^(ed) mod n = x^(ed mod φ(n)) mod n = x^1 mod n = x mod n. And (x^e)^d = x^(ed)
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20. Why Is It Secure? We’re not sure it is, but it seems to be
Based on premise that factoring a big number is difficult. Best known algorithm takes 30,000 MIPS years to factor a 512 bit number.
If you can factor n, you’re golden:
Problem is one of finding modular log (i.e. exponentiative inverse.) Why? Adversary knows <e,n>. So for message m, knowns ciphertext is c = m^e mod n. Also knows that key is value x that satisfies c^x = m, so if you can solve this, can find d.
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21. Why Is It Secure?
How did we originally find this exponentiative inverse? By knowing φ(n). And this is difficult to know if you can’t factor n. If you can, then you’re golden.
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22. Private-Key Cryptography
traditional private/secret/single key cryptography uses one key
shared by both sender and receiver
if this key is disclosed communications are compromised
also is symmetric, parties are equal
hence does not protect sender from receiver forging a message & claiming is sent by sender
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.
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23. Public-Key Cryptography
probably most significant advance in the 3000 year history of cryptography
uses two keys – a public & a private key
asymmetric since parties are not equal
uses clever application of number theoretic concepts
Complements, but does not replace private key crypto
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.
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24. 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
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25. Public-Key Cryptography
This configuration
provides privacy,
but not authentication
Stallings Fig 9.1
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26. Public-Key Cryptography
This configuration
provides authentication,
but not privacy
Stallings Fig 9.1
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27. 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 University in 1976
known earlier in classified community
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 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!
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28. 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)
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29. Public-Key Characteristics
Public key schemes utilize 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:
Symmetric Key: involves 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.
Public Key: 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).
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30. Public Key Privacy
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31. Public Key Authentication
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32. Public-Key Cryptosystems
This configuration
provides both authentication
and privacy
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.
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33. 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
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34. Security of Public Key Schemes
like private key schemes brute force exhaustive search attack is always theoretically possible
but keys used are too large (>512 bits)
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 practice
requires the use of very large numbers
hence is slow compared to private key schemes
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'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's usually a firmer theoretical basis for determining the security since its based on well-known and well studied number theory problems.
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35. 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 bits)
security due to cost of factoring large numbers
nb. factorization takes O(e log n log log n) operations (hard)
RSA is the best known, and by far the most widely used general public key encryption algorithm.
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36. Description of RSA Algorithm
Plaintext encrypted in blocks, each block having a binary value less than some number n
I.e. block size  log2(n)
In practice, block size k bits where 2k < n  2k+1
Let M be plaintext, C ciphertext. Then
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37. Description of RSA Algorithm
Both sender and receiver know n
Only sender knows e, only receiver knows d
Thus:
Private key is {d,n}
Public key is {e,n}
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38. Description of RSA Algorithm
To make this work, require
It is possible to find values of e,d, and n such that
It is relatively easy to calculate Me and Cd for all values of M<n
It is infeasible to determine d given e and n
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39. Description of RSA Algorithm
Recall the corollary to Euler’s Theorem: If p, q prime, n=pq, m such that 0<m<n, then for any integer k,
Thus we have (1) from previous slide satisfied if we let
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40. Description of RSA Algorithm
By rules of modular arithmetic, this can only occur if e and d are relatively prime to
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41. RSA Ingredients
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42. 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}
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.
Note: all
must remain
private!
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43. RSA Example Select primes: p=17 & q=11 Compute n = pq =17×11=187
Select e : gcd(e,160)=1; choose e=7
Determine d: de=1 mod 160 and d < 160 Value is d=23 since 23×7=161= 10×160+1
Publish public key KU={7,187}
Keep secret private key KR={23,17,11}
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)
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44. 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
Rather than having to laborious repeatedly multiply, can use the "square and multiply" algorithm with modulo reductions to implement all exponentiations quickly and efficiently (see next).
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45. 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 = = 3.7 = 10 mod 11
eg = = 5.3 = 4 mod 11
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46. Exponentiation
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47. 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
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.
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48. 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)
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49. 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 bit RSA secure
ensure p, q of similar size and matching other constraints
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 bits look reasonable at present, and the factors should be of similar size
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50. Progress in Factorization
In 1977, RSA inventors dare Scientific American readers to decode a cipher printed in Martin Gardner’s column.
Reward of $100
Predicted it would take 40 quadrillion years
Challenge used a public key size of 129 decimal digits (about 428 bits)
In 1994, a group working over the Internet solved the problem in 8 months.
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51. Progress In Factorization
Factoring is a hard problem, but not as hard as it used to be!
MIPS year is a 1-MIPS machine running for a year
For reference: a 1-GHz Pentium is about a 250-MIPS machine
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53. 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
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54. Summary have considered: principles of public-key cryptography
RSA algorithm, implementation, security
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