Elliptic curve arithmetic and applications to cryptography

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Elliptic curve arithmetic and applications to
cryptography

• By Uros Abaz
• Supervised by Dr. Shaun Cooper and Dr. Andre
  Barczak

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Structure of the talk

• Introduction to cryptography
• Motivation for studying elliptic curve
  cryptography (ECC)
• Theory of elliptic curves
• Very brief about arithmetic on the elliptic curve
• ElGamal Public Key Encryption algorithm
• Tour of the computer program

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Introduction to cryptography

• Cryptography dates back to Julius Caesar
• Introduction of computers drives development in
  cryptography
• Two main types of cryptosystems
• Private key same key used for encryption and
  decryption
• Public key public key used for encryption and
  private key used for decryption

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Public key cryptosystem
Bob
Alice

• 1977 RSA algorithm (R. Rivest, A. Shamir, L.
  Adleman)
• 1985 ECC algorithm (N. Koblitz and V. Miller)

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Motivation for the study

• Advantages of ECC over RSA
• 108-bit ECC Vs 512-bit RSA using 9500 machines
• To understand theory behind elliptic curves
• To improve efficiency of Maple code
• Provide user friendly application using Visual
  C and M.I.R.A.C.L

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Elliptic curves

• Defined by equation y2 x3 Ax B
• y2 x3 4x is equivalent to y v(x3-4x)
• If x and y are complex gt elliptic curve is torus
  in C2
• There are other interesting shapes of elliptic
  curves

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Arithmetic on the elliptic curve

• Elliptic curves over finite field Fq
• Marios webpage - www.mariospage.com
• Addition of two points on an elliptic curve
• Geometrically
• Algebraically
• Multiplication of a point and a scalar on a curve

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Description of ElGamal algorithm

• Alice wants to send message to Bob
• Bob establishes his public key as following
• Chooses an elliptic curve E over Fq
• Chooses a point P on E
• Chooses integer s and computes BsP
• E, Fq, P and B are Bobs public key
• Integer s is Bobs private key

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Description of ElGamal algorithm

• Alice does following to encrypt a message
• Downloads Bobs public key
• Expresses her message as a point M on the curve E
• Chooses a secret random integer k and computes
  M1kP and M2MkB
• Sends M1 and M2 to Bob

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Description of ElGamal algorithm

• Bob decrypts the message by calculating
• M2 sM1 (M kB) s(kP) M ksP skP M
• Eavesdropper, Eve, knows Bobs public key and
  point M1 and M2
• She needs to solve discrete logarithm problem
  (hard!)

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Tour of the program