The Elliptic Curve in https
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1 The Elliptic Curve in https Marco Streng Universiteit Leiden 25 November 2014 Marco Streng (Universiteit Leiden) The Elliptic Curve in https
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6 The s in HyperText Transfer Protocol Secure Uses algebra, number theory, and algebraic geometry to verify the identity of the web page (no fake cave) make sure communication is encrypted (no eavesdroppers) Main players: RSA (see Algebra 1) Elliptic Curve Cryptography Marco Streng (Universiteit Leiden) The Elliptic Curve in https
7 Examples of use of ECC in the real world Marco Streng (Universiteit Leiden) The Elliptic Curve in https
8 Encryption (protection against eavesdroppers) Alice Bob key e key d M E (encrypt) C D (decrypt) M Symmetric cryptography: one key d = e = k is shared Example: Caesar code E shifts every letter forward by k in the alphabet D shifts every letter backward by k in the alphabet k = 3: DELFT GHOIW DELFT More modern encryption (AES) is safe and fast, but... need to share a key k first! Marco Streng (Universiteit Leiden) The Elliptic Curve in https
9 Asymmetric cryptography Alice Bob key e key d M E (encrypt) C D (decrypt) M Symmetric cryptography: one key d = e = k is shared Asymmetric cryptography: d e; e is made public Examples: everyone can encrypt messages that only Bob can read; Bob makes signatures that everyone can verify. One way functions. Impossible? Marco Streng (Universiteit Leiden) The Elliptic Curve in https
10 The integers modulo 7 with the rules F 7 = {0, 1, 2, 3, 4, 5, 6} x + y = z, where z is remainder of x + y on division by 7 x y = z, where z is remainder of x y on division by 7 Examples: 2 4 = = = 2 because = 9 = = 6 because 5 4 = 20 = Satisfies many rules of arithmetic such as x ab = (x a ) b. (It is a ring.) Marco Streng (Universiteit Leiden) The Elliptic Curve in https
11 Exponentiation modulo Examples of multiplication in F = F \ {0}: = = = = = = = ( ) 2 = = ( ) 2 = =???? = , so g = g (213 ) g (2 10 ) g (2 9 ) g (2 8 ) g (2 4 ). Takes not 9999 multiplications, but only 17. Marco Streng (Universiteit Leiden) The Elliptic Curve in https
12 Exponentiation modulo p = Let g F p, a Z with 77 digits, say p = , g = , a = Computing h = g a takes not multiplications in F p, but only 377. Phew! So: easy to compute the map a g a. Marco Streng (Universiteit Leiden) The Elliptic Curve in https
13 Exponentiation modulo p = Let g F p, a Z with 77 digits, p = , g = , a = Computing h = g a takes not multiplications in F p, but only 377. Phew! So: easy to compute the map a g a. Inverse? Given g and h = g a, find a. (Discrete log problem (DLP) in F p) Naive algorithm: try g 1 = h?, g 2 = h?, g 3 = h?,... Need tries to find a. Takes impossibly long! We have our one way map: a g a. Marco Streng (Universiteit Leiden) The Elliptic Curve in https
14 Exponentiation modulo p = Given g and h = g a, find a. (Discrete log problem (DLP) in F p) Naive algorithm: try g 1 = h?, g 2 = h?, g 3 = h?,... Need tries to find a. Takes impossibly long! We have our one way map: a g a. There exist encryption and digital signatures based on the DLP. Instead, I will explain a simpler cryptographic protocol. Marco Streng (Universiteit Leiden) The Elliptic Curve in https
15 Diffie-Hellman key exchange (1976) Goal: agree on a key for symmetric encryption Public parameters: p a prime number, g F p. Alice (communication channel) Bob random a Z compute g a g a g b random b Z compute g b compute g (ab) = (g b ) a and use as key compute g (ab) = (g a ) b and use as key Given only p, g, g a, g b, it is believed to be hard to find g (ab). Finding a is the discrete logarithm problem in F p, and breaks the scheme. Marco Streng (Universiteit Leiden) The Elliptic Curve in https
16 Elliptic curves Next: what are elliptic curves, and why are they better? Marco Streng (Universiteit Leiden) The Elliptic Curve in https
17 Elliptic curves Consider the curve in the (x, y)-plane R 2 given by the equation y 2 = x 3 3x. Marco Streng (Universiteit Leiden) The Elliptic Curve in https
18 Elliptic curves Consider the curve in the (x, y)-plane R 2 given by the equation y 2 = x 3 3x. Q P Marco Streng (Universiteit Leiden) The Elliptic Curve in https
19 Elliptic curves Consider the curve in the (x, y)-plane R 2 given by the equation y 2 = x 3 3x. Q P P Q Marco Streng (Universiteit Leiden) The Elliptic Curve in https
20 Elliptic curves Consider the curve in the (x, y)-plane R 2 given by the equation y 2 = x 3 3x. Q P P Q We get an addition law + on the set E(R) of points, including one extra point 0 at infinity. + satisfies standard rules of arithmetic such as (P + Q) + R = P + (Q + R) (E(R) is a group ). np = P + P + + P Marco Streng (Universiteit Leiden) The Elliptic Curve in https
21 Formulas (up to typos) Given P = (x 1, y 1 ), Q = (x 2, y 2 ) E(R). If x 1 = x 2 and y 1 = y 2, then P + Q = 0. Otherwise, P + Q = (x 3, y 3 ), where: If x 1 x 2, then ( ) y2 y 2 ( ) 1 y2 y 1 x 3 = x 1 x 2, y 3 = x 3 y 1x 2 y 2 x 1 ; x 2 x 1 x 2 x 1 x 2 x 1 If x 1 = x 2, then x 3 = (x 2 + 3) 2 4x(x 2 3), y 3 = 3x 3(x 2 1) + x 3 + 3x 2y 1. So defining E and + only uses arithmetic. In particular, we can define elliptic curves over F p and add points on them! Marco Streng (Universiteit Leiden) The Elliptic Curve in https
22 Elliptic curves over F p Defining E and + only uses arithmetic. In particular, we can define elliptic curves over F p and add points on them! Example: y 2 = x 3 3x over F 17. G = (3, 1) E(F 17 ) (as 1 2 = ) Marco Streng (Universiteit Leiden) The Elliptic Curve in https
23 The elliptic curve discrete logarithm problem Example: y 2 = x 3 3x over F 17. G = (3, 1) E(F 17 ) (as = 1 2 ) 2G = (2, 11) 4G = 2(2G) = (4, 16) 8G = 2(4G) = (1, 10) 16G = 2(8G) = (15, 7) So 18G = 16G + 2G = (1, 7) takes not 17 but only 6 additions. Now suppose p Computing ag with a takes not additions but less than 500. The elliptic curve discrete log problem asks: given p, E, G E(F p ), H = ag, compute a. Marco Streng (Universiteit Leiden) The Elliptic Curve in https
24 Diffie-Hellman key exchange (1976) Goal: agree on a key for symmetric encryption Public parameters: p a prime number, g F p. Alice (communication channel) Bob random a Z compute g a g a g b random b Z compute g b compute g (ab) = (g b ) a and use as key compute g (ab) = (g a ) b and use as key Given only p, g, g a, g b, it is believed to be hard to find g (ab). Finding a is the discrete logarithm problem in F p, and breaks the scheme. Marco Streng (Universiteit Leiden) The Elliptic Curve in https
25 Elliptic Curve cryptography, Koblitz and Miller (1985) Replace F p by E(F p ) in any discrete log cryptosystem, e.g.: Public parameters: p a prime number, E elliptic curve, G E(F p ). Alice (communication channel) Bob random a Z compute ag ag bg random b Z compute bg compute (ab)g = a(bg) and use as key compute (ab)g = b(ag) and use as key Given only E, G, ag, bg, it is believed to be hard to find (ab)g. Finding a is the discrete logarithm problem in E(F p ), and breaks the scheme. Marco Streng (Universiteit Leiden) The Elliptic Curve in https
26 Fast attacks So far, only mentioned naive attack: try a = 1, a = 2, a = 3,... There exist much faster methods for solving the discrete log problem in F p based on algebraic geometry and algebraic number theory. To make these attacks infeasible, for F p-cryptography use not p (77 digits), but use p (924 digits). For EC discrete logs, no fast attacks are known (in spite of much effort!), and working with 77 digits is safe. This makes ECC much more efficient at the same security level! Marco Streng (Universiteit Leiden) The Elliptic Curve in https
27 Example: Gmail (screenshots of Firefox) Marco Streng (Universiteit Leiden) The Elliptic Curve in https
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31 Example: Facebook (screenshots of Chrome) Marco Streng (Universiteit Leiden) The Elliptic Curve in https
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36 The ECC standard curve secp256r1 p = E given by where y 2 = x 3 3x + b, b = G =( , ) Marco Streng (Universiteit Leiden) The Elliptic Curve in https
37 Conclusions Geometry exists not only over R, but also over F p (integers modulo p). https always uses algebra, geometry, number theory for setting up the connection (RSA, Diffie-Hellman, ECC). Elliptic curves are used by Google Mail and Facebook. Marco Streng (Universiteit Leiden) The Elliptic Curve in https
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