The RSA Algorithm : The RSA Algorithm JooSeok Song
2007. 11. 13. Tue
Private-Key Cryptography : 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
Public-Key Cryptography : 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 to function
complements rather than replaces private key crypto
Public-Key Cryptography : 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
Public-Key Cryptography : Public-Key Cryptography
Why Public-Key Cryptography? : 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 Uni in 1976
known earlier in classified community
Public-Key Characteristics : 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)
Public-Key Cryptosystems : Public-Key Cryptosystems
Public-Key Applications : 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
Security of Public Key Schemes : 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
RSA : 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 log n log log n) operations (hard)
RSA Key Setup : 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}
RSA Use : 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)
Prime Numbers : Prime Numbers prime numbers only have divisors of 1 and self
they cannot be written as a product of other numbers
note: 1 is prime, but is generally not of interest
eg. 2,3,5,7 are prime, 4,6,8,9,10 are not
prime numbers are central to number theory
list of prime number less than 200 is:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199
Prime Factorisation : Prime Factorisation to factor a number n is to write it as a product of other numbers: n=a × b × c
note that factoring a number is relatively hard compared to multiplying the factors together to generate the number
the prime factorisation of a number n is when its written as a product of primes
eg. 91=7×13 ; 3600=24×32×52
Relatively Prime Numbers & GCD : Relatively Prime Numbers & GCD two numbers a, b are relatively prime if have no common divisors apart from 1
eg. 8 & 15 are relatively prime since factors of 8 are 1,2,4,8 and of 15 are 1,3,5,15 and 1 is the only common factor
conversely can determine the greatest common divisor by comparing their prime factorizations and using least powers
eg. 300=21×31×52 18=21×32 hence GCD(18,300)=21×31×50=6
Fermat's Theorem : Fermat's Theorem ap-1 mod p = 1
where p is prime and gcd(a,p)=1
also known as Fermat’s Little Theorem
useful in public key and primality testing
Euler Totient Function ø(n) : Euler Totient Function ø(n) when doing arithmetic modulo n
complete set of residues is: 0..n-1
reduced set of residues is those numbers (residues) which are relatively prime to n
eg for n=10,
complete set of residues is {0,1,2,3,4,5,6,7,8,9}
reduced set of residues is {1,3,7,9}
number of elements in reduced set of residues is called the Euler Totient Function ø(n)
Euler Totient Function ø(n) : Euler Totient Function ø(n) to compute ø(n) need to count number of elements to be excluded
in general need prime factorization, but
for p (p prime) ø(p) = p-1
for p.q (p,q prime) ø(p.q) = (p-1)(q-1)
eg.
ø(37) = 36
ø(21) = (3–1)×(7–1) = 2×6 = 12
Euler's Theorem : Euler's Theorem a generalisation of Fermat's Theorem
aø(n)mod N = 1
where gcd(a,N)=1
eg.
a=3;n=10; ø(10)=4;
hence 34 = 81 = 1 mod 10
a=2;n=11; ø(11)=10;
hence 210 = 1024 = 1 mod 11
Why RSA Works : 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
RSA Example : RSA Example Select primes: p=17 & q=11
Compute n = pq =17×11=187
Compute ø(n)=(p–1)(q-1)=16×10=160
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}
RSA Example cont : 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
Exponentiation : 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
Exponentiation : Exponentiation
RSA Key Generation : 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
RSA Security : 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)
Factoring Problem : 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
Timing Attacks : 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
Summary : Summary have considered:
prime numbers
Fermat’s and Euler’s Theorems
Primality Testing
Chinese Remainder Theorem
Discrete Logarithms
principles of public-key cryptography
RSA algorithm, implementation, security
Assignments : Assignments Perform encryption and decryption using RSA algorithm, as in Figure 1, for the following:
p = 3; q = 11, e = 7; M = 5
p = 5; q = 11, e = 3; M = 9
In a public-key system using RSA, you intercept the ciphertext C = 10 sent to a user whose public key is e = 5, n = 35. What is the plaintext M? 31