A Creative Way of Breaking RSA

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A Creative Way of Breaking RSA

• Azeem Jiva

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Overview

• What is RSA?
• Public Key Algorithm
• Is it secure?
• Ways to break RSA
• Discover the Public Key
• Number Field Sieve

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What is RSA?

• RSA is named after its Inventors
• Public Key Algorithm
• Variable Key Length
• 512bits, 1024bits are most common
• 2048bits, 4096bits are extreme
• Variable Plaintext Length
• Ciphertext Length same as Key Length
• RSA is slow, used mostly to encrypt second key

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Definitions

• Relative Prime
• No common divisors except for 1
• Sometimes called Coprime and Strangers 2
• e.g. 3 and 5
• Multiplicative Inverse
• Number which multiplied by another number gives
  you one 3

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How RSA Works

• Need Public and Private Key
• Don't tell anyone the Private Key!
• To create Public Key need two large primes
• P and Q 256bits each
• Multiply them together, result is N
• ø(N)(P-1)(Q-1)
• Find E relative prime to ø(N)
• Public Key is (E,N)

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Private Key and Encryption

• Need to find multiplicative inverse of
• D E mod ø(N)
• Private Key is (D, N)
• Now you can encrypt
• Ciphertext c ME mod N
• And decrypt
• M CD mod N
• And sign messages
• S MD mod N

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Is RSA Secure?

• Fundamental Tenent of Cryptography
• Difficult to factor, but there are other ways
• Wait/Attack
• RSA-155 (512bits) was broken in 7 months 1
• Took 8700 MIPS years
• Estimated time till factored 1
• 768bits 2010
• 1024bits 2018

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Breaking RSA

• Quadratic Sieve
• Previous way
• Number Field Sieve
• Currently the fastest way
• Oexpc(log n)1/3 (log log n)2/3
• Can be parallelized on multiple machines
• Interesting work by Dan Bernstein
• http//cr.yp.to/papers/nfscircuit.ps

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Number Field Sieve

• Four main steps
• Polynomial Selection
• Sieving
• Linear Algebra
• Square Root
• The Sieving is the most time consuming

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Polynomial Selection

• Select two irreducible polynomials (Primes)
• F1(x)
• F2(x)
• They have a common root
• M mod N

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Sieving

• Finds two numbers that are relatively prime
• GCD(a,b) 1
• Both numbers are smooth over factor bases
• B deg(f1) f1 (a/b)
• B deg(f2) f2 (a/b)
• These two numbers are relations
• Find as many of these relations so that several
  subsets S with property X2 Y2
  (mod n) can be found

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Linear Algebra

• Filters the results from Sieving
• Remove duplicates
• Remove relations that do not occur anywhere else
• Certain relations are merged
• Eliminate primes and prime ideals which occur
  exactly k times in k situations
• Use a Lanczos Algorithm
• Most time consuming of Linear Algebra Step

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Square Root

• Computes Square Roots
• a is the root of a polynomial f1(x), f2(x)
• a ba have smooth norms
• Cardinality of S is in the millions

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Projects

• NFS Net
• http//www.iaeste.dk/henrik/projects/nfsnet.html
• Lattice Siever
• http//www.lehigh.edu/bad0/nfs2-137.html

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Factoring Records
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References

• Factorization of a 512-bit Modulus, Cavallar, etc
• mathworld.wolfram.com/RelativelyPrime.html
• www.mathnstuff.com/math/spoken/here/1words/m/m31.h
  tm
• www.cs.sjsu.edu/stamp/SecurityEngineering/chapter
  5/knapsack.html