Elliptic Curve Cryptography

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Elliptic Curve Cryptography

• majority of public-key crypto (RSA, D-H) use
  either integer or polynomial arithmetic with very
  large numbers/polynomials
• Depend on difficulty of reversing exponentiation
  (discrete logs) or multiplication (modular
  factors)
• imposes a significant load in storing and
  processing keys and messages
• an alternative is to use elliptic curves
• offers same security with smaller key sizes

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Real Elliptic Curves

• an elliptic curve is defined by an equation in
  two variables x y, with coefficients
• consider a cubic elliptic curve of form
• y2 x3 ax b
• where x,y,a,b are all real numbers
• also define zero point O
• have addition operation for elliptic curve
• geometrically sum of QR is reflection of
  intersection R

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Real Elliptic Curve Example
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Finite Elliptic Curves

• Elliptic curve cryptography uses curves whose
  variables coefficients are finite
• have two families commonly used
• prime curves Ep(a,b) defined over Zp
• use integers modulo a prime
• best in software
• binary curves E2m(a,b) defined over GF(2n)
• use polynomials with binary coefficients
• best in hardware

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Ep(a,b) Math
The key equation is
A point on the curve E11(1,2) is given by (9, 5).
To verify, 25 mod 11 (729 9 2) mod
11 25 mod 11 740 mod 11 3 3
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Ep(a,b) Math

• Addition Rules
• if x1 n.e. x2,
  where

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Ep(a,b) Math

• Addition Rules (cont)
• If y1 n.e. 0,
• where

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Ep(a,b) Math

• Multiplication just do sequence of adds

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Elliptic Curve Cryptography

• ECC addition is analog of modulo multiply
• ECC repeated addition is analog of modulo
  exponentiation
• need hard problem equiv to discrete log
• QkP, where Q,P belong to a prime curve
• is easy to compute Q given k,P
• but hard to find k given Q,P
• known as the elliptic curve logarithm problem

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ECC Diffie-Hellman

• can do key exchange analogous to D-H
• users select a suitable curve Ep(a,b)
• select base point P(x1,y1)
• Alice select dA (secret) and compute QAdAP
  (public key)
• Bob select dB (secret) and compute QBdBP
  (public key)
• Public info p, a, b, P, QA, QB

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ECC Diffie-Hellman

• Bob computes sB dBQA dBdAP
• Alice computes sA dAQB dAdBP
• But dBdAP dAdBP and sB sA

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ECC Encryption/Decryption

• several alternatives, will consider simplest
• must first encode any message M as a point on the
  elliptic curve Pm
• select suitable curve point G as in D-H
• each user chooses private key nAltn
• and computes public key PAnAG
• to encrypt Pm CmkG, Pmk Pb, k random
• decrypt Cm compute
• PmkPbnB(kG) Pmk(nBG)nB(kG) Pm

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ECC Security

• relies on elliptic curve logarithm problem
• fastest method is Pollard rho method
• compared to factoring, can use much smaller key
  sizes than with RSA etc
• for equivalent key lengths computations are
  roughly equivalent
• hence for similar security ECC offers significant
  computational advantages

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ECC Key Size vs. RSA