The Blockchain Identity - faculty.fuqua.duke.educharvey/Teaching/697_2019/4c_DSA/... · Definition...
Transcript of The Blockchain Identity - faculty.fuqua.duke.educharvey/Teaching/697_2019/4c_DSA/... · Definition...
Digital SignaturesCampbell R. Harvey
Duke University and NBER
January 26, 2019
Innovation and Cryptoventures
Campbell R. Harvey 2019 2
DefinitionCryptography is the science of communication in the presence of an adversary. Part of the field of cryptology.
3Campbell R. Harvey 2019
Goals of Adversary• Alice sends message to Bob• Eve is the adversary
4Campbell R. Harvey 2019
Goals of Adversary
Eve’s goals could be:1. Eavesdrop2. Steal secret key so that all future messages can be intercepted3. Change Alice’s message to Bob4. Masquerade as Alice in communicating to Bob
5Campbell R. Harvey 2019
Symmetric Keys
Early algorithms were based on symmetric keys.• This meant a common key encrypted and decrypted the message• You needed to share the common key and this proved difficult
6Campbell R. Harvey 2019
Secret Keys
Symmetric key• DES (Data Encryption Standard)
was a popular symmetric key method, initially used in SET (first on-line credit card protocol)
• DES has been replaced by AES (Advanced Encryption Standard)
11Campbell R. Harvey 2019
Diffie-Hellman Key Exchange
• Breakthrough in 1976 with Diffie-Hellman-Merkle key exchange• There is public information that everyone can see. Each person, say Alice and
Bob, have secret information.• The public and secret information is combined in a way to reveal a single
secret key that only they know
12https://www.youtube.com/watch?v=YEBfamv-_do
Campbell R. Harvey 2019
Diffie-Hellman Key Exchange
• Will use prime numbers and modulo arithmetic• We already encountered one example of modular arithmetic simple ciphers
(also the SHA-256 which uses mod=232 or 4,294,967,296)
13https://www.youtube.com/watch?v=YEBfamv-_do
Campbell R. Harvey 2019
Symmetric Key Exchange
Numerical example• “5 mod 2” = 1• Divide 5 by 2 the maximum number of times (2) • 2 is the modulus• The remainder is 1• Remainders never larger than (mod-1) so for mod 12 (clock) you would
never see remainders greater than 11.• EXCEL function = mod(number, divisor) e.g., mod(329, 17) = 6
14
“mod”
Campbell R. Harvey 2019
Symmetric Key Exchange
Alice and Bob decide on two public pieces for information• A modulus (say 17)• A generator (or the base for an exponent) (say 3)
• Alice has a private key (15)• Bob has a private key (13)
• Is it possible for them to share a common secret that is unlikely to be intercepted?
15https://www.khanacademy.org/computing/computer-science/cryptography/modern-crypt/v/diffie-hellman-key-exchange-part-2
Campbell R. Harvey 2019
Symmetric Key Exchange
Alice: Calculates 315 mod 17 = 6 (i.e., =mod(3^(15), 17))
• Alice send the message “6” to Bob
16Campbell R. Harvey 2019
Symmetric Key Exchange
Alice: Calculates 315 mod 17 = 6 (i.e., =mod(3^(15), 17))
• Alice send the message “6” to Bob• Eve intercepts the message!
17Campbell R. Harvey 2019
Symmetric Key Exchange
Bob: Calculates 313 mod 17 = 12 (i.e., =mod(3^(13), 17))
• Bob send the message “12” to Alice
18Campbell R. Harvey 2019
Symmetric Key Exchange
Bob: Calculates 313 mod 17 = 12 (i.e., =mod(3^(13), 17))
• Bob send the message “12” to Alice• Eve intercepts the message! Now Eve has the 6 and the 12.
19Campbell R. Harvey 2019
Symmetric Key Exchange
Alice: She takes Bob’s message of 12 and raises it to the power of her private key. Calculates 1215 mod 17 = 10 (i.e., =mod(12^(15), 17))*
• This is their common secret
20*EXCEL only does 15 digits so this will not work Campbell R. Harvey 2019
Symmetric Key Exchange
Bob: He takes Alice’s message of 6 and raises it to the power of his private key. Calculates 613 mod 17 = 10 (i.e., =mod(6^(13), 17))
• This is their common secret
21Campbell R. Harvey 2019
Symmetric Key Exchange
Eve She has intercepted their message. However, without the common secret key, there is little chance she can recover the shared secret.
22Campbell R. Harvey 2019
Symmetric Key Exchange
Common secret• Alice can now encrypt a message with the common secret and Bob can
decrypt it with the common secret. • Notice this is a common secret. • Next we will talk private/public keys. That is, both and Alice have separate
public keys and separate private keys.
23Campbell R. Harvey 2019
Asymmetric Keys: RSA - High Level Overview
RSA • RSA stands for Rivest, Shamir and Adleman. Discovered earlier by UK
Communications-Electronics Security Group (CESG) – but kept secret.• Receiver generates two public pieces of information and a private key
• One piece of public information is just the product of two prime numbers, N=p*q(called “max”)
• The other is the public key, e, is just another prime that is greater than 2 and less than the product, N
• The prime numbers, p and q, that are used are huge. The private key is mathematically linked to public keys.
• Sender encrypts with the two public keys, e and N• Receiver can easily decrypt
28Campbell R. Harvey 2019
Asymmetric Keys: RSA - High Level Overview
See my Cryptography 101 (linked) deck for much more detail.• Two prime numbers are chosen and they are secret (say 7 and 13, p, q).• Multiply them together. The product (N=91) is public but people don’t know
the prime numbers used to get it.• A public key, e, is chosen (say 5).• Given the two prime numbers, 7 and 13, and the public key, 5, we can derive
the private key, which is 29.
29Campbell R. Harvey 2019
Asymmetric Keys: RSA - High Level OverviewIssues with RSA• RSA relies on factoring• N is public (our example was 91) as is e• If you can guess the factors, p, q, then you
can discover the private key
30Campbell R. Harvey 2019
Asymmetric Keys: RSA - High Level OverviewIssues with RSA• Factoring algorithms have become very efficient• To make things worse, the algorithms become more efficient as the size of
the N increases• Hence, larger and larger numbers are needed for N (moving to 2,048 bits)• This creates issues for mobile and low power devices that lack the
computational power
31http://www.slate.com/articles/health_and_science/science/2016/01/the_world_s_largest_prime_number_has_22_338_618_digits_here_s_why_you_should.html
Campbell R. Harvey 2019
Elliptic Curve Cryptography
Mathematics of elliptic curves• Does not rely on factoring• Curve takes the form of
y2 = x3 + ax + b
32
Note: 4a3 + 27b2 ≠ 0
Campbell R. Harvey 2019
Bitcoin uses a=0 and b=7
Note that diagram is “continuous” but wewill be using discrete versions of this arithmetic
Elliptic Curve Cryptography
Properties• Symmetric in x-axis• Any non-vertical line between two points intersects in three points• Algebraic representation
33Campbell R. Harvey 2019
Elliptic Curve Cryptography
Properties: Addition
34
P Q
R
P+Q
Define a system of “addition”. To add “P” and“Q” pass a line through and intersect at third point“R”. Drop a vertical line down to symmetric part.This defines P+Q (usually denoted 𝑃𝑃 ⊕𝑄𝑄)
Denote Elliptic Curveas E Campbell R. Harvey 2019
Elliptic Curve Cryptography Properties: Doubling
35
P
2P
Define a system of “addition”. To add “P” and“P” use a tangent line and intersect at third point.Drop a vertical line down to symmetric part. This definite 2P (usually denoted 𝑃𝑃 ⊕ 𝑃𝑃)
Denote Elliptic Curveas E Campbell R. Harvey 2019
Elliptic Curve Cryptography (Optional slide)
Properties: Other
36
(a) P + O = O + P = P for all P ∈ E.(existence of identity)
(b) P + (−P) = O for all P ∈ E.(existence of inverse)
(c) P + (Q + R) = (P + Q) + R for all P, Q, R ∈ E.(associative)
(d) P + Q = Q + P for all P, Q ∈ E(communativity)
Denote Elliptic Curveas E Campbell R. Harvey 2019
Elliptic Curve Cryptography
Why use in cryptography?• Suggested by Koblitz and Miller in 1985• Implemented in 2005• Key insight:
• Adding and doubling on the elliptic curve is easy but undoing the adding is very difficult• 256 bit ECC public key provides about the same security as 3,072 bit RSA public key
• Bitcoin uses a particular type of ECC known as secp256k1
37Campbell R. Harvey 2019http://www.nicolascourtois.com/bitcoin/groups_ECC_7B.pdf
ECDSA
• Private key is a number called “signing key” (SK). It is secret.• Public key is the “verification key” and is mathematically linked to
the private key
49Campbell R. Harvey 2019
SK EC VK
Private key:(number)
Elliptic curve operations: Need base point, modulus, order
Public key:coordinate (x, y)
Note: Easy to generate a public key with a private key. Not easy to go the other way.
ECDSA• Digital signature
50Campbell R. Harvey 2019
SK
EC DS
Private key:(number)
Elliptic curve operations: Need base point, modulus, order (n)
Digital signature:coordinate (r, s)
Message
Nonce
Nonce:(random number)
ECDSA• Verification
51Campbell R. Harvey 2019
VK
EC (x’, y’)
Public key:(x, y)
Elliptic curve operations: Need base point, order (n)
Derive new pointon elliptic curve
Message
r
DScoordinates
sr = x’ mod n ?
Yes (verified)
No(rejected)
Check x coordinateof new point and DS
Note r not used until verification step
How DSAs Work
Notice• Proves that the person with the private key (that generated the public
key) signed the message. • Interestingly, digital signature is different from a usual signature in that
it depends on the message, i.e., the signature is different for each different message.
• In practice, we do not sign the message, we sign a cryptographic hash of the message. This means that the size of the input is the same no matter how long the message is.
52Campbell R. Harvey 2019
ECDSA in Action
53Campbell R. Harvey 2019https://kjur.github.io/jsrsasign/sample/sample-ecdsa.html
ECDSA in Action
54Campbell R. Harvey 2019
OP_CHECKSIG uses Public Key + Digital Signature + Hash of Transaction
Verifies whether this transaction has been signed by the owner of the Private Key
https://www.youtube.com/watch?v=ir4dDCJhdB4 (advanced by Matt Thomas)
Application: PGP EmailMy public key for secure email• You can encrypt an email
to me with my public key and only I can decrypt with my private key.
55Campbell R. Harvey 2019
Application: PGP Email
Steps1. Message compressed2. Random session key (based on mouse
movements and keystrokes) is generated.3. Message encrypted with session key4. Session key is encrypted with receiver’s public key5. Encrypted message + encrypted session key sent via email6. Recipient uses their private key to decrypt the session key7. Session key is used to decrypt the message8. Message decompressed
56Campbell R. Harvey 2019http://www.pgpi.org/doc/pgpintro/
References
• The Math Behind Bitcoin [recommended]
• Elliptic Curve Digital Signature Algorithm (Bitcoin)
• What does the curve used in Bitcoin, secp256k1, look like?
• Elliptic Curve Digital Signature Algorithm (Wikipedia)
• Elliptic Curve Cryptography (UCSB)
• Elliptic Curve Cryptography and Digital Rights Management (Purdue)
• Zero to ECC in 30 minutes (Entrust)
• The Elliptic Curve Cryptosystem
• Goldwasser, Shaffi and Mihir Bellare, 2008, Lecture Notes on Cryptography
• Dan Boneh, Stanford University, Introduction to Cryptography
• Dan Boneh, Stanford University, Cryptography II
• https://arstechnica.com/security/2013/10/a-relatively-easy-to-understand-primer-on-elliptic-curve-cryptography/
57Campbell R. Harvey 2019