Cryptography Today. Ali El Kaafarani. Mathematical Institute Oxford University. 1 of 44
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1 Cryptography Today Ali El Kaafarani Mathematical Institute Oxford University 1 of 44
2 About the Course Regular classes with worksheets so you can work with some concrete examples (every Friday at 1pm). (Every week) Write a short summary ( 500 words) about one research paper (suggested in the further reading sections in the slides). You hand in your worksheets/summaries every Wednesday by noon. First week, you solve sheet-0 with the tutor and investigate some useful crypto-tools. One class (Friday 11th of Nov) to give presentations (in groups) about a chosen research paper (not graded). One implementation class using SageMath. 2 of 44
3 About the Course Mini project. Reading research papers! Weekly cryptography seminar, https: // (Almost) every Wednesday during terms. You are encouraged to attend and take notes, the best notes will be published on the cryptography web page. It is a good exercise to learn L A T E X as well! 3 of 44
4 Outline 1 Cryptography Usage in the Real World: Do we use it? Where? 2 Modern Cryptography Provable Security Symmetric Key Cryptosystems Public Key Cryptosystems Hash Functions Digital Signatures Advanced Cryptographic Tools/Schemes 3 Classical Vs Post-Quantum Cryptography 4 of 44
5 Web Browsers 5 of 44
6 Some Recent Cryptanalysis 6 of 44
7 E-voting 7 of 44
8 Mobile Applications 8 of 44
9 Bitcoin 9 of 44
10 Cryptography Usage in the Real World: a Summary 10 of 44
11 Cryptography Usage in the Real World: a Summary On-line banking, e-commerce ( 10 of 44
12 Cryptography Usage in the Real World: a Summary On-line banking, e-commerce ( SSH: to remotely login and to transfer files. 10 of 44
13 Cryptography Usage in the Real World: a Summary On-line banking, e-commerce ( SSH: to remotely login and to transfer files. Bitcoin: a decentralized digital currency 10 of 44
14 Cryptography Usage in the Real World: a Summary On-line banking, e-commerce ( SSH: to remotely login and to transfer files. Bitcoin: a decentralized digital currency Electronic Voting. 10 of 44
15 Cryptography Usage in the Real World: a Summary On-line banking, e-commerce ( SSH: to remotely login and to transfer files. Bitcoin: a decentralized digital currency Electronic Voting. s, cloud computing, etc. 10 of 44
16 Cryptography Usage in the Real World: a Summary On-line banking, e-commerce ( SSH: to remotely login and to transfer files. Bitcoin: a decentralized digital currency Electronic Voting. s, cloud computing, etc. Mobile applications. 10 of 44
17 Cryptography Usage in the Real World: a Summary On-line banking, e-commerce ( SSH: to remotely login and to transfer files. Bitcoin: a decentralized digital currency Electronic Voting. s, cloud computing, etc. Mobile applications. ATM machines, etc. 10 of 44
18 Outline 1 Cryptography Usage in the Real World: Do we use it? Where? 2 Modern Cryptography Provable Security Symmetric Key Cryptosystems Public Key Cryptosystems Hash Functions Digital Signatures Advanced Cryptographic Tools/Schemes 3 Classical Vs Post-Quantum Cryptography 11 of 44
19 Hardness Assumptions Some mathematical problems are believed to be computationally hard (to different extents): Integer Factorization: given a composite number n, compute its (unique) factorization n = p e i i where p i are prime numbers. It is believed to be hard if n = pq for well-chosen p q. 12 of 44
20 Hardness Assumptions Some mathematical problems are believed to be computationally hard (to different extents): Integer Factorization: given a composite number n, compute its (unique) factorization n = p e i i where p i are prime numbers. It is believed to be hard if n = pq for well-chosen p q. Discrete Logarithm: given a cyclic group (G = g, ), h G, compute k Z G such that g k = h Dlog is believed to be hard in G = F p and even harder in groups of points on (well-chosen) elliptic/hyperelliptic curves. 12 of 44
21 Hardness Assumptions Short Vector Problem in Lattices (SVP) A lattice is a discrete version of a vector subspace, more formally; Given n linearly independent vectors b 1,..., b n R m, the lattice generated by them is defined as { n } L( b 1,..., b n ) def = x i b i x i Z SVP: it is hard to determine the smallest non-zero vector in an arbitrary lattice (easy in low dimensions). i=1 13 of 44
22 Hardness Assumptions b 2 b 2 b 1 b 1 (a) The lattice Z 2 with B = {(0, 1), (1, 0)} (b) The lattice Z 2 with a B = {(1, 1), (2, 1)} 14 of 44
23 Hardness Assumptions: Average vs. Worst Cases 1 Do we have the same confidence in different cryptosystems that are based on different hardness assumptions? 1 Post-Quantum Cryptography, Daniel Bernstein et al. 15 of 44
24 Hardness Assumptions: Average vs. Worst Cases 1 Do we have the same confidence in different cryptosystems that are based on different hardness assumptions? Breaking a lattice-based cryptographic scheme is at least as hard as solving several hard lattice problems in the worst case. 1 Post-Quantum Cryptography, Daniel Bernstein et al. 15 of 44
25 Hardness Assumptions: Average vs. Worst Cases 1 Do we have the same confidence in different cryptosystems that are based on different hardness assumptions? Breaking a lattice-based cryptographic scheme is at least as hard as solving several hard lattice problems in the worst case. Breaking a cryptographic scheme that is based on factoring might imply the ability to factor some numbers chosen according to a certain distribution, but not the ability to factor all numbers! 1 Post-Quantum Cryptography, Daniel Bernstein et al. 15 of 44
26 Hardness Assumptions: Average vs. Worst Cases 1 Do we have the same confidence in different cryptosystems that are based on different hardness assumptions? Breaking a lattice-based cryptographic scheme is at least as hard as solving several hard lattice problems in the worst case. Breaking a cryptographic scheme that is based on factoring might imply the ability to factor some numbers chosen according to a certain distribution, but not the ability to factor all numbers! How can we prove the security of our cryptosystems? Proofs by reduction! 1 Post-Quantum Cryptography, Daniel Bernstein et al. 15 of 44
27 Security Games: Proofs by Reduction Challenger Adversary 16 of 44
28 Challenger Adversary 17 of 44
29 Public parameters Challenger Adversary 18 of 44
30 Public parameters Queries Challenger Adversary 19 of 44
31 Challenger Public parameters Queries Answers Adversary 20 of 44
32 Challenger Public parameters Queries Answers Challenge Adversary 21 of 44
33 Challenger Public parameters Queries Answers Challenge More queries Adversary 22 of 44
34 Challenger Public parameters Queries Answers Challenge More queries Answers Adversary 23 of 44
35 Challenger Public parameters Queries Answers Challenge More queries Answers Guess Adversary 24 of 44
36 Challenger Public parameters Queries Answers Challenge More queries Answers Guess Adversary 25 of 44
37 Outline 1 Cryptography Usage in the Real World: Do we use it? Where? 2 Modern Cryptography Provable Security Symmetric Key Cryptosystems Public Key Cryptosystems Hash Functions Digital Signatures Advanced Cryptographic Tools/Schemes 3 Classical Vs Post-Quantum Cryptography 26 of 44
38 Symmetric Key Cryptosystems A symmetric encryption scheme consists of three algorithms that are (KeyGen, Enc, Dec); Let M be message space whereas the key space is K. Below are the descriptions of the algorithms: KeyGen(n): 2 is a randomized algorithm that, given the security parameter n, returns a key SK K. Enc(SK, m): is a randomized algorithm that on input a key SK K and a plaintext m M, outputs a ciphertext c. Dec(SK, c): is a deterministic algorithm that on input a key SK and a ciphertext c outputs a message m M. Correctness: m M, Pr[SK KeyGen(n) : Dec(SK, Enc(SK, m)) = m] = 1 2 You often equivalently see KeyGen(1 n ), which emphasizes that the algorithm runs in time polynomial in the length of its input. 27 of 44
39 Outline 1 Cryptography Usage in the Real World: Do we use it? Where? 2 Modern Cryptography Provable Security Symmetric Key Cryptosystems Public Key Cryptosystems Hash Functions Digital Signatures Advanced Cryptographic Tools/Schemes 3 Classical Vs Post-Quantum Cryptography 28 of 44
40 Public Key Cryptosystems An asymmetric encryption scheme consists of the following algorithms: KeyGen(n): is a randomized algorithm that takes the security parameters as input and returns a pair of keys (PK, SK), the public key PK and its matching secret key SK, respectively. Enc(PK, m): A randomized algorithm that takes a public key PK, a plaintext m and returns a ciphertext c. Dec(SK, c): A deterministic algorithm that takes the secret key SK and a ciphertext c, and returns a message m M. Correctness: m M, Pr[(SK, PK) KeyGen(n) : Dec(Enc(PK, m), SK) = m] = 1 29 of 44
41 Outline 1 Cryptography Usage in the Real World: Do we use it? Where? 2 Modern Cryptography Provable Security Symmetric Key Cryptosystems Public Key Cryptosystems Hash Functions Digital Signatures Advanced Cryptographic Tools/Schemes 3 Classical Vs Post-Quantum Cryptography 30 of 44
42 Hash Functions Informally speaking, hash functions take a long input string and output a shorter string of a fixed length called a digest. They are used to achieve integrity (or authenticity) in the private-key setting. They are used almost everywhere in Cryptography, e.g. HMAC, commitment schemes, saved passwords, etc. If you imagine that hash functions are truly random (modelled as random oracle model), then proving the security of some cryptographic schemes becomes achievable (e.g. RSA-OAEP). A debate/controversy over the soundness of the random oracle model. Cryptographic hash functions are much harder to design than those used to build hash tables in data structures. 31 of 44
43 Outline 1 Cryptography Usage in the Real World: Do we use it? Where? 2 Modern Cryptography Provable Security Symmetric Key Cryptosystems Public Key Cryptosystems Hash Functions Digital Signatures Advanced Cryptographic Tools/Schemes 3 Classical Vs Post-Quantum Cryptography 32 of 44
44 Digital Signatures Are used to achieve integrity (or authenticity) in the public key setting. 33 of 44
45 Digital Signatures Are used to achieve integrity (or authenticity) in the public key setting. If a signature σ on a message m is verified correctly against a given public key PK, it ensures that the message was indeed sent by the owner of this public key (already known to potential verifiers) and the message was NOT modified in transit. 33 of 44
46 Digital Signatures Are used to achieve integrity (or authenticity) in the public key setting. If a signature σ on a message m is verified correctly against a given public key PK, it ensures that the message was indeed sent by the owner of this public key (already known to potential verifiers) and the message was NOT modified in transit. More importantly, signers cannot deny having signed a message, also known as non-repudiation. 33 of 44
47 Digital Signatures Are used to achieve integrity (or authenticity) in the public key setting. If a signature σ on a message m is verified correctly against a given public key PK, it ensures that the message was indeed sent by the owner of this public key (already known to potential verifiers) and the message was NOT modified in transit. More importantly, signers cannot deny having signed a message, also known as non-repudiation. This is how you verify that the update sent by a certain company is authentic. 33 of 44
48 Digital Signatures Are used to achieve integrity (or authenticity) in the public key setting. If a signature σ on a message m is verified correctly against a given public key PK, it ensures that the message was indeed sent by the owner of this public key (already known to potential verifiers) and the message was NOT modified in transit. More importantly, signers cannot deny having signed a message, also known as non-repudiation. This is how you verify that the update sent by a certain company is authentic. In comparison to message authentication codes (MAC); Key distribution and management is hugely simplified. Signatures are publicly verifiable! 33 of 44
49 Outline 1 Cryptography Usage in the Real World: Do we use it? Where? 2 Modern Cryptography Provable Security Symmetric Key Cryptosystems Public Key Cryptosystems Hash Functions Digital Signatures Advanced Cryptographic Tools/Schemes 3 Classical Vs Post-Quantum Cryptography 34 of 44
50 Secret Sharing Lagrange Interpolating Polynomial: given n points (x 1, y 1 ),, (x n, y n ), one can construct the polynomial P(x) of degree (n 1) that passes through them as follows: P j (x) = y j n k=1 k j (x x k ) (x j x k ) Online demo: 35 of 44
51 Shamir Secret Sharing Shamir Secret Sharing works in two phases as follows: Distribute the shares: first pick a random polynomial Q(x) F p [x] of degree l < n (where n is the number of participants) s.t. Q(0) = s. Then, compute the shares S i = Q(i) mod p for i = 1,, n and send them over to the participants A 1,, A n. Reconstruct the secret: According to Lagrange interpolation, any l + 1 participants can together compute Q(0) mod p which is the secret s. 36 of 44
52 Multi-Party Computation Suppose that we have n parties P 1,, P n, each has a secret input s i. They all want to evaluate a public function f on inputs (s 1,, s n ) to learn the output and yet keep their inputs hidden from each other. 37 of 44
53 Multi-Party Computation Suppose that we have n parties P 1,, P n, each has a secret input s i. They all want to evaluate a public function f on inputs (s 1,, s n ) to learn the output and yet keep their inputs hidden from each other. Secure Multi-Party Computation is the solution! 37 of 44
54 Multi-Party Computation: an Application 38 of 44
55 Zero-Knowledge Proofs Completeness: Prover can always convince a verifier that a given statement is true 39 of 44
56 Zero-Knowledge Proofs Completeness: Prover can always convince a verifier that a given statement is true (A valid vote will always be accepted). 39 of 44
57 Zero-Knowledge Proofs Completeness: Prover can always convince a verifier that a given statement is true (A valid vote will always be accepted). Soundness: Prover cannot convince the verifier if the statement is false 39 of 44
58 Zero-Knowledge Proofs Completeness: Prover can always convince a verifier that a given statement is true (A valid vote will always be accepted). Soundness: Prover cannot convince the verifier if the statement is false (You cannot fool the verifier and vote with -1000). 39 of 44
59 Zero-Knowledge Proofs Completeness: Prover can always convince a verifier that a given statement is true (A valid vote will always be accepted). Soundness: Prover cannot convince the verifier if the statement is false (You cannot fool the verifier and vote with -1000). Zero-Knowledge: The proof doesn t reveal any extra information beyond the validity of the statement 39 of 44
60 Zero-Knowledge Proofs Completeness: Prover can always convince a verifier that a given statement is true (A valid vote will always be accepted). Soundness: Prover cannot convince the verifier if the statement is false (You cannot fool the verifier and vote with -1000). Zero-Knowledge: The proof doesn t reveal any extra information beyond the validity of the statement (The vote is still secret!). 39 of 44
61 Zero-Knowledge Proofs Blog: http: //blog.cryptographyengineering.com/2014/11/ zero-knowledge-proofs-illustrated-primer.html Online demo: http: //web.mit.edu/~ezyang/public/graph/svg.html 40 of 44
62 Fully Homomorphic Encryption Cloud computing is a hot topic nowadays! Companies want to store their huge data on the clouds and let the cloud companies do the computation on their data. 41 of 44
63 Fully Homomorphic Encryption Cloud computing is a hot topic nowadays! Companies want to store their huge data on the clouds and let the cloud companies do the computation on their data. But they want to preserve data confidentiality, so they decide to encrypt their data (and not give away the encryption keys!) 41 of 44
64 Fully Homomorphic Encryption Cloud computing is a hot topic nowadays! Companies want to store their huge data on the clouds and let the cloud companies do the computation on their data. But they want to preserve data confidentiality, so they decide to encrypt their data (and not give away the encryption keys!) How can the cloud companies do computation on encrypted data and give back the result in an encrypted format! 41 of 44
65 Fully Homomorphic Encryption Some encryption schemes are naturally partially homomorphic, i.e., Enc(A) Enc(B) = Enc(A B). Fully homomorphic encryption allows for arbitrary computation on ciphertexts. You can write a program of any functionality and run it on a given ciphertext to get the desirable result in an encrypted format! In theory, this was proven possible in In practice, it is still far away from being practical! 42 of 44
66 Classical Vs Post-Quantum Cryptography What would happen to Dlog and Factorisation based Cryptosystems if quantum computers existed? 43 of 44
67 Classical Vs Post-Quantum Cryptography What would happen to Dlog and Factorisation based Cryptosystems if quantum computers existed? Shor s algorithm! (using quantum Fourier transform). 43 of 44
68 Classical Vs Post-Quantum Cryptography What would happen to Dlog and Factorisation based Cryptosystems if quantum computers existed? Shor s algorithm! (using quantum Fourier transform). Any alternatives? 43 of 44
69 Classical Vs Post-Quantum Cryptography What would happen to Dlog and Factorisation based Cryptosystems if quantum computers existed? Shor s algorithm! (using quantum Fourier transform). Any alternatives? Lattice-Based Cryptography (e.g. fully homomorphic encryption) 43 of 44
70 Classical Vs Post-Quantum Cryptography What would happen to Dlog and Factorisation based Cryptosystems if quantum computers existed? Shor s algorithm! (using quantum Fourier transform). Any alternatives? Lattice-Based Cryptography (e.g. fully homomorphic encryption) Code-Based Cryptography (e.g. McEliece cryptosystem) 43 of 44
71 Classical Vs Post-Quantum Cryptography What would happen to Dlog and Factorisation based Cryptosystems if quantum computers existed? Shor s algorithm! (using quantum Fourier transform). Any alternatives? Lattice-Based Cryptography (e.g. fully homomorphic encryption) Code-Based Cryptography (e.g. McEliece cryptosystem) Hash-Based Cryptography (e.g. Merkle signature) 43 of 44
72 Classical Vs Post-Quantum Cryptography What would happen to Dlog and Factorisation based Cryptosystems if quantum computers existed? Shor s algorithm! (using quantum Fourier transform). Any alternatives? Lattice-Based Cryptography (e.g. fully homomorphic encryption) Code-Based Cryptography (e.g. McEliece cryptosystem) Hash-Based Cryptography (e.g. Merkle signature) Hot research topic- Ideas? 43 of 44
73 Further Reading (1) Jean-Jacques Quisquater, Myriam Quisquater, Muriel Quisquater, Michaël Quisquater, Louis Guillou, Marie Guillou, Gaïd Guillou, Anna Guillou, Gwenolé Guillou, and Soazig Guillou. How to explain zero-knowledge protocols to your children. In Advances in Cryptology CRYPTO 89 Proceedings, pages Springer, of 44
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