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Cryptography: Helping Number Theorists Bring Home the Bacon Since 1977

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Cryptography:Helping Number Theorists Bring Home the Bacon Since 1977

Dan Shumow SDE

Windows Core Security

dshumow@microsoft.com

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- Introduction
- Symmetric Key Encryption
- Key Distribution:Diffie-Hellman Key Generation
- Elliptic Curve Cryptography

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- 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.

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

Eve

Bob

Alice

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- 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).

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- 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.

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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

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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.)

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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.

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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.

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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.

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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)

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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.

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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.

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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.

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- 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

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- 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?

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- 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.

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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.

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- 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.

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- 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.

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- 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.

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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

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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.

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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.

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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.

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- 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.

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- 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

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