Asymmetric Key Signatures David Evans and Samee Zahur CS4501, Fall 2015.

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Presentation transcript:

Asymmetric Key Signatures David Evans and Samee Zahur CS4501, Fall 2015

Please pay $1000 to my employee --TheBoss You have money!

Real-life Signatures Easy to verify Bank has your signature Forging unlikely Legal consequences of forging Checkbooks are well-guarded Copying it requires physical access Hard to repudiate Bank keeps a copy for few months

Digital Signatures

Topics Asymmetric cryptography Digital signatures Elliptic curve cryptography Implementation pitfalls

Ordinary (or symmetric) crypto Message key

Whitfield Diffie Martin Hellman New Directions in Cryptography, 1976

Diffie-Hellman Key Exchange

Discrete Logarithm Problem

Random element out of …?

Mod 5 Exponentiation … … … … … … Order 1 Order 2

Exponent Modulus

Public-key Cryptography

Man-in-the-Middle (MITM)

Digital Signature

Recall

Discrete-log based signature

ElGamal Signature Scheme

Bitcoin Payment Sign it like a check!

Recap 1.We want to sign transactions digitally on the bitcoin network, such that they are: a)Easy to verify b)Hard to forge c)Hard to repudiate 2.Discrete exponentiation is easy, logarithm is hard 3.We used it to make asymmetric (aka. public) key crypto 4.Same principle used for digital signatures

Avoiding (overly) long numbers

Informal Requirements

Group

Additional Cryptographic Properties Discrete logarithm should be hard Group operation should be efficient Implies small key sizes

Elliptic Curve Cryptography (ECC) Group elements: points on the curve, P, Q, and R Point “addition”: using “geometry”. P+Q=R P Q R

Elliptic “Curve” Image from:

Elliptic Curve Digital Signature Algorithm (ECDSA) ECDSA Inputs: message, private key 1.Pick random k 2.Compute a), let 3.Send with message Verification If, check

Please pay $1000 to my employee --TheBoss You have money! Jason Benjamin

Logistics Next class: hash functions and Bitcoin consensus Checkup 1 on Monday. Includes everything till today