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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!