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Cryptography 101 &Secure ComputationCenter for Security and Emerging Technologyat Georgetown University, Washington, D.C.
Tjerand Silde
Department of Mathematical Sciences,NTNU Trondheim
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Overview
– Cryptography 101– Symmetric-Key Encryption– Public-Key Encryption– Hash Functions– Digital Signatures
– Secure Computation– Secret Sharing– Multi-Party Computation– Fully Homomorphic Encryption– Secure computation in practice
– Cryptography & Machine Learning
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Cryptography 101
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Symmetric-Key Encryption I
A symmetric-key encryption-scheme is an encryption-system for two parties withtwo functions Enc(·, ·) and Dec(·, ·) together with a shared key k :
— Enc takes as input a key k and a message m,and outputs a ciphertext c: Enc(k ,m) = c.
— Dec takes as input a key k and a ciphertext c,and outputs a message m: Dec(k , c) = m.
Only the two parties that know the key can encrypt and decrypt messages.
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Symmetric-Key Encryption II
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Symmetric-Key Encryption III
Properties of a symmetric-key encryption-scheme:
— Both parties need to know the key k in advance
— Usually have keys and blocks of size 128 or 256 bits
— Super fast (only using XOR, AND, vectors etc.)
— Length of ciphertext ≈ Length of plaintext
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Symmetric-Key Encryption IV
Security of symmetric-key encryption:
— The best way to break it is to guess the key
— It does not hide the length of the encrypted data
— Might be vulnerable to side-channel attacks
— Might be vulnerable to attacks by quantum computers
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Public-Key Encryption I
A public-key encryption-scheme is an encryption-system for two parties with twofunctions Enc(·, ·) and Dec(·, ·) together with a public key kp and a secret key ks:
— Enc takes as input a public key kp and a message m,and outputs a ciphertext c: Enc(kp,m) = c.
— Dec takes as input a secret key ks and a ciphertext c,and outputs a message m: Dec(ks, c) = m.
Everyone can encrypt a message, but only one party can decrypt.
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Public-Key Encryption II
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Public-Key Encryption III
Properties of a public-key encryption-scheme:
— Only one party knows the private key, everyone knows the public key
— Pretty slow, working with big numbers thousands of bits long
— Length of ciphertext >> Length of plaintext
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Public-Key Encryption IV
The RSA cryptosystem:
— Choose two prime numbers p and q and let n = p · q
— Choose a number e and find a special number d depending on p and q
— The public key is the two numbers (e,n)
— The private key is the two numbers (d ,n)
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Public-Key Encryption V
The RSA cryptosystem:
— Encryption of message m: Enc((e,n),m) = me mod n = c
— Decryption of ciphertext c: Dec((d ,n), c) = cd mod n = m
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Public-Key Encryption VI
Security of RSA public-key encryption:
— The best way to break RSA is to factorize n into p and q
— The cryptosystem must be randomized to be secure
— Might be vulnerable to padding attacks
— Might be vulnerable to side-channel attacks
— Will be broken by attacks by quantum computers
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Hash Functions I
A hash function H is a deterministic function that takes input of arbitrary length,and produce a fixed length output. A hash function has the three followingproperties:
— Collision resistance,
— Pre-image resistance,
— Second pre-image resistance.
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Hash Functions II
Collision resistance:It should be difficult to find two different messagesm1 and m2 such that H(m1) = H(m2).
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Hash Functions III
Pre-image resistance:Given a hash value h it should be difficult tofind any message m such that h = H(m).
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Hash Functions IV
Second pre-image resistance:Given an input m1, it should be difficult to finda different input m2 such that H(m1) = H(m2).
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Hash Functions V
Security of hash-functions:
— The best way to break a hash-function is to guess inputs
— The best way to get collisions is the birthday attack
— Commonly used hash-functions have outputs of length 256 and 512 bits
— Might be vulnerable to attacks by quantum computers
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Digital Signatures I
A digital signature-scheme is a signature-system for two parties with two functionsSign(·, ·) and Verify(·, ·, ·) together with hash function H, a secret key ks and apublic key kp:
— Sign takes as input a secret key ks and a message m,and outputs a signature σ: Sign(ks,m) = σ.
— Verify takes as input a public key kp, a signature σ and a message m,and outputs a message True or False: Verify(kp, σ,m) = True or False.
Only one party can sign a message, but everyone can verify a signature.
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Digital Signatures II
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Digital Signatures III
The RSA digital signature-system:
— Choose two prime numbers p and q and let n = p · q
— Choose a number v and find a special number s depending on p and q
— The public key is the two numbers (v ,n)
— The private key is the two numbers (s,n)
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Digital Signatures IV
The RSA digital signature-system:
— Signature of message m: Sign((s,n),m) = (H(m))s mod n = σ
— Verification of signature σ: Verify((v ,n), σ,m) = σv mod n ?= H(m)
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Digital Signatures V
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Secure Computation
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Secure Computation
Goal:
being able to do computations on encrypted data, and thereforecompute valuable information without violating privacy.
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Secret Sharing I
Goal:
share a secret with a group in a way so that none of the peoplecan reconstruct the secret on their own, but some threshold ofmembers can reconstruct the secret if they work together.
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Secret Sharing II
First; some geometry:
— How many points in the plane do you need to define a straight line?
— How many points in the plane do you need to define a parabola?
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Secret Sharing III
y = ax + b
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Secret Sharing IV
y = ax2 + bx + c
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Secret Sharing V
We have a secret number, say, b = 4.
We want to share this secret with three friends in a way sothat none of them knows our secret, but if at least two of themwork together, then they are able to reconstruct the secret.
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Secret Sharing VI
We want to construct a polynomial of degree one: p(x) = ax + b,and give one point p(1),p(2),p(3) to each of our three friends.
Assume that we can only chose coefficient from the numbers 0 to 4.Our secret number will be b = 4, and we randomly choose, say, a = 2.
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Secret Sharing VII
Then p(0) = 4 is the secret we want to share.
We share the following with our friends:
— p(1) = 2 · 1 + 4 = 6 mod 5 = 1 with player A,
— p(2) = 2 · 2 + 4 = 8 mod 5 = 3 with player B,
— p(3) = 2 · 3 + 4 = 10 mod 5 = 0 with player C.
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Secret Sharing VIII
Individually,
— player A has the value p(1) = 1,
— player B has the value p(2) = 3,
— player C has the value p(3) = 0,
but none of them know what the secret is, because theyneed two points to be able to reconstruct the whole line.
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Multi-Party Computation I
Goal:
allow a group to compute a public function,without having to reveal their individual input.
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Multi-Party Computation II
Procedure:
— Everyone secret-share their input with everyone
— Addition and scalar multiplication is done locally
— Multiplications require sending a new secret-share to every party
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Multi-Party Computation III
Security:
— Secret sharing with threshold n − 1 of n parties
— Parties can provide "false" input
— Can notice and abort if someone is cheating
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Multi-Party Computation IV
Advantages:
— Computation is really fast
— Post-quantum security
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Multi-Party Computation V
Disadvantages:
— Require a lot of communication
— Everyone must be "online" at all time
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Fully Homomorphic Encryption I
Goal:
allow a third party to compute a function,without knowing the input nor the output.
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Fully Homomorphic Encryption II
"Noisy encryption":
— The encryption of a message contains some extra "noise"
— As long as the "noise" is small, you can still decrypt
— Adding ciphertexts also adds the "noise"
— Multiplying ciphertexts also multiplies the "noise"
— When the "noise" gets larger, you can "bootstrap" to make it smaller
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Fully Homomorphic Encryption III
Procedure:
— The user encrypts their data and sends it over
— The third party does the computations on the encrypted data
— The user gets the encrypted data back and then decrypts
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Fully Homomorphic Encryption IV
Advantages:
— No communication while computing
— Small functions are fast to compute
— Post-quantum security
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Fully Homomorphic Encryption V
Disadvantages:
— Require a lot of computation
— "Bootstrapping" is expensive
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Secure computation in practice I
Multi-Party Computation:
— Pre-computation independent on data and function using FHE
— Secret share information as normal
— Fast MPC using the pre-computed data
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Secure computation in practice II
Fully Homomorphic Encryption:
— Agree on function before choosing parameters
— Choose parameters large enough to handle some "noise"
— Do as little "bootstrapping" as possible
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Secure computation in practice III
Some applications in use:
— Journalist can share sensitive information in case anything happens...
— Private companies can analyze medical information without violating privacy
— Competing companies can work together without sharing data
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Cryptography & Machine Learning
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Cryptography & Machine Learning I
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Cryptography & Machine Learning II
Cryptography for machine learning:
— Training neural networks on encrypted data
— Prove correctness of machine learning algorithms
— Protect machine learning algorithms against abuse
— Training neural networks while keeping the training model secret
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Cryptography & Machine Learning III
Machine learning for security:
— Detection of vulnerabilities and attacks
— Crypto- and malware-analysis
— Machine learning for attacking cryptography
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Cryptography & Machine Learning IV
Resources:— Rivest, Asiacrypt 1991. “Cryptography and machine learning”.
• https://people.csail.mit.edu/rivest/pubs/Riv91.pdf
— Goldwasser, Crypto 2018. "From Idea to Impact, the Crypto Story"• https://www.youtube.com/watch?v=culuNbMPP0k
— Alani, 2019. "Applications of Machine Learning in Cryptography: A Survey."• https://dl.acm.org/citation.cfm?doid=3309074.3309092
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