The rsa algorithm JooSeok Song

The RSA Algorithm

JooSeok Song
2007. 11. 13. Tue

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

CCLAB

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

CCLAB

 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

CCLAB


CCLAB

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

CCLAB

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)

CCLAB

Public-Key Cryptosystems

CCLAB

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

CCLAB

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
CCLAB

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)

CCLAB

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}
CCLAB

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)

CCLAB

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

CCLAB

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

CCLAB

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

CCLAB

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

CCLAB

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)

CCLAB

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

CCLAB

Euler's Theorem
 a generalisation of Fermat's Theorem
 aø(n)mod N = 1
– where gcd(a,N)=1

 eg.
–
–
–
–

CCLAB

a=3;n=10;
hence 34 =
a=2;n=11;
hence 210 =

ø(10)=4;
81 = 1 mod 10
ø(11)=10;
1024 = 1 mod 11

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

CCLAB

RSA Example
1.
2.
3.
4.
5.

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
6. Publish public key KU={7,187}
7. Keep secret private key KR={23,17,11}

CCLAB

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

CCLAB

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
CCLAB

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

CCLAB

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)

CCLAB

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

CCLAB

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

CCLAB

Summary
 have considered:
–
–
–
–
–
–
–

CCLAB

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
1. Perform encryption and decryption using RSA
algorithm, as in Figure 1, for the following:
a. p = 3; q = 11, e = 7; M = 5
b. p = 5; q = 11, e = 3; M = 9
Encryption
Plaintext
88

887 mod 187 = 11

Decryption
Ciphertext
11

1123 mod 187 = 88

KU = 7, 187
KR = 23, 187
Figure 1. Example of RSA Algorithm

Plaintext
88

2. 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?
CCLAB

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The rsa algorithm JooSeok Song

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