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How Secure is RSA: Mathematical Approach By: Rana Khalil, Undergraduate
Supervisor: Dr. Monica Nevins, Professor and Chair University of Ottawa, Faculty of Science
Rana Khalil University of Ottawa, Undergraduate Research Opportunity Program (UROP) rkhal101@uottawa.ca
Contact References: 1. Hoffstein, J. , Pipher, J. , Silverman, J. H. , (2008). An Introduction to Mathematical Cryptography. New York, NY: Springer Science+Business
Media. 2. Mironov, I. (2012, May 15) . Factoring RSA Moduli. Part I. . Retrieved from windowsontheory.org/2012/05/15/979/ 3. Mironov, I. (2012, May 17) . Factoring RSA Moduli. Part II. . Retrieved from windowsontheory.org/2012/05/17/factoring-rsa-moduli-part-ii/ 4. Lenstra, A. K., Hughes J. P., Augier, M., Bos, J. W., Kleinjung T. & Wacher, C. (2012). Ron was wrong, Whit is right, 064. Retrieved from
eprint.iacr.org/2012/064.pdf
5. Survivability and Information Assurance (n.d.). Retrieved March 16, 2014, from http://people.ubuntu.com/~duanedesign/SurvivabilityandInformationAssuranceCurriculum/02everything/02everything.html#AEN379
References
Introduction
Cryptography is the art and science of writing in secret
code. With the advent of the Computer Age,
cryptography has become essential in today's economy.
Public key cryptography is based on mathematical
problems that currently admit to no efficient solution. This
in turn has allowed individuals to share secret
communication over insecure channels, such as the
internet, which begs the question: How secure are
public key algorithms? Recently, it was revealed that
the National Security Agency (NSA) has obtained access
to RSA protected information by exploiting some
unknown flaws. In this research we analyze a paper by
Lenstra et al. called "Ron was wrong, Whit is right" which
identifies some of these potential flaws. Our project is to
understand the mathematical theory behind two major
algorithms: RSA and Diffie Hellman, with the probability
of these flaws occurring.
Chart 1. Distributions of discrete logarithms for g = 641 modulo p = 941
10941738641570527421809707322040357612003732945449205990913842131476349984288934784717997257891267332497625752899781833797076537244027146743531593354333897
102639592829741105772054196573991675900716567808038066803341933521790711307779
106603488380168454820927220360012878679207958575989291522270608237193062808643
Prime Numbers
There are infinitely many prime numbers which are
randomly distributed. The prime number theorem states
that for a randomly chosen number N, the probability that
it is prime is 1/ ln (N), i.e. prime numbers are “easy” to
find.
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Methods and Materials
(I) Symmetric Cryptography
(II) Asymmetric Cryptography
Results
Flaw: In the Lenstra et al. paper it was mentioned that
among 4.7 million 1024-bit RSA moduli collected, more
than 12500 have a single prime factor in common! This
flaw allows for easy factorization of n.
Analysis: By the prime number theorem, there are
approximately 2504 primes that are less than 2510.
Therefore, the odds that two people randomly choose at
least one prime factor in common is
P = 1 −2504−2
2504 2
= 1 − 1 − 1
2503
2 ≈
1
2502
However, if we sample M ( = 4.7 million) different N, then
the odds that two of these N’s share at least one common
factor is approximately,
1 − 1 − 𝑃𝑀 𝑀 −1
2 ≈𝑀 𝑀 −1
2P ≈
243
2502 ≈ 1
2459 ≈ 0
Symmetric encryption
algorithms work on one
basic principle- the
same key that is used to
encrypt the plaintext is
used to decrypt the ciphertext.
Asymmetric encryption
and decryption uses
separate keys.
Decryption is done
using the private
(secret) key, while
encryption is done using
the public key.
Table 1. Sieve of Eratosthenes from 1 to 100
Table 2. Diffie-Hellman key exchange
Table 3. RSA key creation, encryption, and decryption
Table 4. Key lengths and Security Levels
Symmetric RSA, DL Comments
64 Bit 70 Bit Short term security
80 Bit 1024 Bit Medium security
256 Bit 3072 Bit High security
0
100
200
300
400
500
600
700
800
900
0 50 100 150 200 250 300Powers 641i mod 941 for i = 1, 2,......300
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
Figure 1. DLP example
Public Parameter Creation
A trusted party chooses and publishes a (large) prime p and an integer g (which is a primitive root for Z/pZ) having large prime order in Z/pZ*.
Private Computations Alice Bob
Choose a secret integer a. Compute A ≡ ga (mod p).
Choose a secret integer b. Compute B ≡ gb (mod p).
Public Exchange of Values Alice sends A to Bob A B Bob sends B to Alice
Further Private Computations Alice Bob
Compute the number Ba (mod p).
Compute the number Ab (mod p).
The shared secret key value is Ba ≡ (gb)a ≡ gab ≡ (ga)b ≡ Ab (mod p).
Alice Bob
Key Creation
Choose secret primes p and q.
Choose encryption exponent e with
gcd( e, (p – 1)(q – 1)) = 1
Publish N (= pq) and e.
Encryption
Choose plaintext m.
Use Alice's public key (N, e) to
compute c ≡ me (mod N)
Send ciphertext c to Bob.
Decryption
Compute d satisfying
ed ≡ 1 (mod (p – 1)(q -1)).
Compute m' ≡ cd (mod N).
Then m' equals the plaintext m.
Discrete Logarithm Problem Let g and h ∈ Z/pZ. The Discrete Logarithm Problem (DLP) is the problem of finding an exponent x such that gx ≡ h (mod p) The number x exists if g is a primitive root for Z/pZ.
Note: Exponentiation mod p is random; see Chart 1 for an example with p=941.
Conclusion
In conclusion, the mathematics behind RSA is still very solid.
The major flaw detected by Lenstra et al. seems to be an
implementation error not a mathematical error. For further research refer to [2], [3] & [4].
Figure 2. Symmetric Key Cryptography
Figure 3. Asymmetric Key Cryptography
First and foremost, I would like to thank God for giving me the
opportunity to participate in this research. I would also like to
thank UROP for providing me with the funding and resources
to complete my research. Lastly, special thanks to Dr. Monica
Nevins for taking time out of her very busy schedule to teach
me about the fascinating world of cryptography!