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Cryptography: Helping Number Theorists Bring Home the Bacon Since 1977

Cryptography: Helping Number Theorists Bring Home the Bacon Since 1977. Dan Shumow SDE Windows Core Security dshumow@microsoft.com. Outline. Introduction Symmetric Key Encryption Key Distribution: Diffie-Hellman Key Generation Elliptic Curve Cryptography. Introduction.

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Cryptography: Helping Number Theorists Bring Home the Bacon Since 1977

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  1. Cryptography:Helping Number Theorists Bring Home the Bacon Since 1977 Dan Shumow SDE Windows Core Security dshumow@microsoft.com 1

  2. Outline • Introduction • Symmetric Key Encryption • Key Distribution:Diffie-Hellman Key Generation • Elliptic Curve Cryptography 2

  3. Introduction • Cryptography, what is it and why should we care? • Cryptography is the science of communicating secretly. • Today so much communication is done over the internet and radio waves, and these media are very prone to eavesdropping. Cryptography allows people to communicate securely across these media. 3

  4. Cryptography Allows Alice to communicate with Bob without being overheard by Eavesdropper Eve. Eve Bob Alice 4

  5. Symmetric KeyEncryption • Alice and Bob share a key K. • They use an encryption function c=Ek(p). • p is the plaintext and c is the ciphertext. • It has to be reversible: p=Dk(c). • If Alice wants to send Bob a message m she computes c = EK(m) and sends Bob c. • Bob computes m = DK(c). 5

  6. Symmetric KeyEncryption • Want it to be hard to compute p given c. So if Eve doesn’t know K it is hard for her to compute m even if she intercepts c. • Want Ekand Dk to be easy to compute. So there is little overhead to communication • Want K to be hard to calculate given p and c. Otherwise if Eve can guess parts of the message she can recover the key. 6

  7. Symmetric KeyEncryption Examples: • Substitution Ciphers: Substitute each letter in the alphabet for another one. • One Time Pads: A key that is the same length as the message, used only once. • Modern Ciphers • Stream Ciphers: RC4 • Block Ciphers: DES, AES 7

  8. Symmetric KeyEncryption Attacks on Encryption Algorithms: • Substitution Ciphers: Frequency Attacks • One Time Pads are provably secure. • Modern Attacks: • Linear Cryptanalysis looks for a linear relationship between plaintext and ciphertext. (Known Plaintext Attack.) • Differential Cryptanalysis looks at how differences in plaintext cause differences in ciphertext. (Chosen Plaintext Attack.) 8

  9. Symmetric KeyEncryption Modern Encryption Algorithm Design Techniques • Confusion and Diffusion • Diffusion means many bits of the plaintext (possibly all) affect each bit of the ciphertext. • Confusion means there is a low statistical bias of bits in the ciphertext. • Non-Linearity: The encryption function is not linear (represented by a small matrix) • Prevents Linear Cryptanalysis. 9

  10. Symmetric KeyEncryption Problem: Key Distribution • Can’t keep using same key, Eve will eventually recover K. • Need to establish shared secret key: • Could agree to physically meet and establish keys. • But what if you want to communicate with someone on the other side of the world? Key distribution is a big problem. 10

  11. Diffie-HellmanKey Generation Basic Idea: • Alice and Bob agree on an integer g. • (a) Alice secretly chooses integer x, computes X = gx and sends it to Bob.(b) Bob secretly chooses integer y, computes Y = gy and sends it to Alice. • (a) Alice computes Yx=(gy)x=gxy.(b)Bob computes Xy=(gx)y=gxy. • Alice and Bob both share gxywhich they can use to create a secret key. 11

  12. Diffie-HellmanKey Generation Wait!! It’s not secure. If Eve overhears what g,X, and Y are she can compute: x = loggX and y = loggY And use this information to calculate gxy. To make this secure Alice and Bob pick a large prime number P and reduce everything mod P (take the remainder after division by P) 12

  13. Diffie-HellmanKey Generation New and Improved Idea: • Alice and Bob agree on an integer g and prime P. • (a) Alice secretly chooses integer x, computesX = gx mod P and sends it to Bob.(b) Bob secretly chooses integer y, computes Y = gy mod P and sends it to Alice. • (a) Alice computesYx mod P=(gy)x mod P =gxy mod P.(b)Bob computesXy mod P=(gx)y mod P =gxy mod P. • Alice and Bob both share the value gxymod P which they can use to create a secret key. 13

  14. Diffie-HellmanKey Generation By adding the prime P into the equation we now need to make sure that g is a “generator” of P. This means that for every integer x in {1,2,3,…,P-1}there exists an integer d such that: x = gdmod P. d is called the “discrete log” of g mod P. 14

  15. Diffie-HellmanKey Generation Why Does This Work? • Because the positive integers less than P form a multiplicative, cyclic group with generator g. • It is hard to compute the discrete log of a generator mod P. Given these two things: • This algorithm works. • It is hard for Eve to calculate gxymod P. 15

  16. Groups • A group is a set G with a binary operation ·:G×G→Gwith the following properties: • Associativity: a(bc)=(ab)c • Identity Element: there exists e in G, such that for all a in Gea=ae=a. • Inverses: for all a in G there exists an element a-1 in G such that aa-1 =a-1a = e 16

  17. Special Groups • Abelian Groups are groups that have a fourth axiom • Commutative: for all a and b in Gab = ba • Cyclic Groups are groups that have a generator g. Where g is an element of G such that for all a in G:a = gxwhere x is a positive integer.Note that all Cyclic groups are Abelian.Can you see why? 17

  18. Special Groups • Multiplicative Groups are groups where the operation is called multiplication. Example: the group of n×n invertible matrices. • Additive Groups are groups where the operation is called addition. Additive Groups are abelian. Example: the integers. 18

  19. Diffie-Hellman Key Generation What does this all mean for Diffie-Hellman Key Generation? Answer: It means that Diffie-Hellman will work as a key exchange algorithm in any cyclic group where computing discrete logarithms is hard. 19

  20. Elliptic CurveCryptography • Elliptic Curves are a way of modifying existing crypto systems like DH to make them “stronger.” • “Stronger” means the expected time of an attack is longer with equal key sizes. • This allows us to use smaller key sizes and therefore speed up the whole process. • This makes ECC very useful for small devices like phones or other embedded systems. 20

  21. Elliptic Curves • An Elliptic Curve is such an alternate cyclic group. The group consists of all points of the form: y2 = x3 + ax + b. Where x, y, a, and b are all elements of a field F. 21

  22. Fields • A field is a set that has mathematical operations multiplication and addition that behave in nice ways. • Basically a field is any set that you can do everything from your high school algebra class in. 22

  23. Fields A field F is a set S along with two binary operations (+,·) that have the following properties: • S contains two distinct elements 0 and 1 • (S-{0},·) is a multiplicative group, with identity 1. • (S,+) is an additive group, with identity 0. • Multiplication is distributive on the left and the right:a·(b+c) = a·b+a·c(a+b)·c = a·c+b·c 23

  24. Elliptic Curves Group operation: Let P = (xP,yP) and Q = (xQ,yQ) be points on the an Elliptic Curve E. Then: R = P + Q = (xR,yR) is defined by:xR= s2-xP-xQyR=-yP+s(xP-xR) where:s = (yP-yQ)/(xP-xQ) if xP≠xQors = (3xP2+a)/(2yP2) if xP=xQ Identity: A “point at infinity” is added to the set of points on the curve. This point is infinitely far along the y access. 24

  25. Elliptic Curves Intuition: If you have 2 points on this curve, they define a line that intersects the curve at 1 other point. Addition is derived from this. Inverses are reflections about the x access. 25

  26. Elliptic CurveCryptography Newer and more Improved Idea: • Alice and Bob agree on an Elliptic Curve E (specified by the field F and parameters a,b) and a base point g on E. • (a) Alice secretly chooses integer x, computesX = xg and sends it to Bob.(b) Bob secretly chooses integer y, computes Y = yg and sends it to Alice. • (a) Alice computes: xY= x(yg)=xyg.(b)Bob computes: yX= y(xg)=yxg=xyg. • Alice and Bob both share the point xyg which they can use to create a secret key. 26

  27. Elliptic CurveCryptography • In the preceding example all math is done in the group defined by E. Exponentiation is taken to be iterative addition. • Because Elliptic Curves are groups we are guaranteed that we can perform all these operations. • Computing logarithms in elliptic curves is difficult, so Eve can not recover the secret values and determine the shared value xyg. 27

  28. References • Eric W. Weisstein. "Elliptic Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/EllipticCurve.html • Eric W. Weisstein et al. "Group." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Group.html • Eric W. Weisstein. "Field." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Field.html • http://en.wikipedia.org/wiki/Group_%28mathematics%29 • http://en.wikipedia.org/wiki/Field_(mathematics) • http://en.wikipedia.org/wiki/Elliptic_curves 28

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