Math4Security Julio López Fenner Departamento Ciencias de Computación e Informática (DCI), Universidad de La Frontera, Temuco, Chile January 2020 English Week 2019, IUT2, Université Grenoble Alpes, Grenoble, France Outline What is Security? Math 4 Crypto Computational Number Theory & Algebra Probability & Statistics Elliptic Curves Emerging Topic: IoT 1 What is Security? Shutdown or the most secure system 2 Figure 1: https://techlawforum.files.wordpress.com/2017/03/internet-shutdowns.jpg Security 3 Digital Footprints and social engineering Figure 2: https://www.teachthought.com/the-future-of-learning/11-tips-for-students-tomanage-their-digital-footprints/ 4 Social Engineering: OSINT Figure 3: https://www.pcwdld.com/osint-tools-and-software 5 OSINT https://www.pcwdld.com/osint-tools-and-software • Recon-ng: A web reconnaissance and OSINT framework written in Python. It can automate the process of information-gathering by thoroughly and quickly exploring the open-source information on the web. • Shodan: Search engine for interconnected or IoT devices. It can also be used to monitor databases to see if they have data leaks on public sites, and it can even find hidden video game servers within corporate networks. • Maltego: Computer forensics, GHDB (Google Hacking Database), etc. 6 10 Steps for Cybersecurity Figure 4: https://i.pinimg.com/originals/11/f9/ca/11f9caf74854ecba52b9d6067a7e6238.png 7 Information Security Figure 5: https://www.techopedia.com/definition/10282/information-security-is 8 The Parkerian Hexad Figure 6: https://i.pinimg.com/originals/11/f9/ca/11f9caf74854ecba52b9d6067a7e6238.png 9 The CIA Triad 10 Figure 7: https://i.pinimg.com/originals/11/f9/ca/11f9caf74854ecba52b9d6067a7e6238.png Parker 1998 Figure 8: https://i.pinimg.com/originals/11/f9/ca/11f9caf74854ecba52b9d6067a7e6238.png 11 Vulnerabilities of Computing Systems Figure 9: https://books.google.co.in/books/about/Security_in_Computing.html?id=O3VB-zspJo4C 12 Find the X! 13 Figure 10: https://www.tshirtsandallstore.com/63-thickbox_default/why-do-we-need-math.jpg Responsive or Preemptive? Figure 11: http://news.mit.edu/2016/ai-system-predicts-85-percent-cyber-attacks-using-input-human-experts-0418 14 Math for Cybersecurity Entry level cyber security careers generally only require basic math concepts that are used in binary, cryptography or programming tasks, https://startacybercareer.com/do-i-need-math-for-cyber-security/ Contents • Probability theory • Data Analysis & Statistics • Analysis of algorithms • Graph Theory • Complexity theory • Game Theory • Number theory • Visualization & String Analysis • Group theory • Linear Algebra 15 Math 4 Crypto Math 4 Crypto Contents • Classical cryptographical constructions: Diffie Hellman Key exchange, discrete logarithm, RSA cryptosystems, digital signatures • Mathematical tools: primality testing, factorization algorithms, probability theory, information theory, collission algorithms • Innovations: Elliptic curves, latttice based cryptography, NTRU cryptosystems 16 The integers 17 Solving linear congruences Divisibility Primality a divides b is a divisor of c is a Fundamental theorem of multiple of d is divisible by e with arithmetic: Every non-zero integer remainder r and integer division k. is a product of primes: e1 e2 er b = k ·a+r; r 2 f0; 1;::: a−1g: n = ±p1 p2 ::: pr : The mod operator a mod b := r () a = q · b + r; 0 ≤ r < b 18 Solving linear congruences a · z = b mod n 19 Chinese remainder theorem 20 ∗ ∗ Residue Classes Zn and Euler’s phi function '(n) := jZnj 21 ∗ ∗ Residue Classes Zn and Euler’s phi function '(n) := jZnj Zn = f[0]; [1];::: [n − 1]:[i] = i + nZg the set of residue classes modulo n. ∗ −1 Zn = fα 2 Zn : 9α () gcd(α; n) = 1g the set of elements of Zn that have a multiplicative inverse. n is prime n composite ∗ ∗ Zn = Zn n f[0]g Zn (Zn n f[0]g 22 Some properties of ' with primes • '(pe ) = pe − pe−1 = pe−1(p − 1) e1 e2 er Qr • n = p1 · p2 ··· pr , then '(n) = n i=1(1 − 1=pi ) ∗ '(n) • Euler’s theorem: α 2 Zn, then α = 1. p • Fermat’s little theorem: p prime, then for all α 2 Zp: α = α. • Wilson’s theorem: p 6= 2 prime number: (p − 1)! = −1 mod p. 23 Long story short: What can we do with integers? Caesar’s encryption or affine encryption x 7! x + k mod 26; k 2 Z26 2 x 7! a · x + b mod 26 k = (a; b) 2 Z26 Block (stream) encryption Example: Blocks of 10 bits https://www.usna.edu/Users/cs/wcbrown/courses/S18SI335/notes/03/notes.html 24 Long story short: What can we do with integers? Figure 12: https://cdn.ttgtmedia.com/rms/onlineImages/block_stream_cipher_01_mobile.jpg 25 RSA or Public Key Figure 13: https://www.isites.info/PastConferences/ISITES2015/ISITES2015/papers/B7-ISITES2015ID28.pdf 26 RSA Key Gen 27 Kerkhoff’s principles 28 Secrecy of the Key not the Method! Diffie-Hellman Key exchange Figure 14: https://www.practicalnetworking.net/wp-content/uploads/2015/11/dh-revised.png 29 Other Applications Signatures Figure 15: https://www.tutorialspoint.com/cryptography/images/public_key_cryptography.jpg 30 Other Applications Hash functions Figure 16: http://i.stack.imgur.com/eCCob.png 31 Hash functions for Crypto Figure 17: commons.wikimedia.com 32 Birthday Paradox Figure 18: https://demonstrations.wolfram.com/BirthdayParadoxProbabilityEstimates/ 33 Salting the Hash Figure 19: https://crackstation.net/hashing-security.htm Needs Random Numbers (Cryptographically secure) Figure 20: https://laughingsquid.com/cloudflare-wall-of-lava-lamps/ 34 Elliptic curve cryptosystems (ECC) ECC Procedure • Discovered 1985 by Miller & • y 2 = x 3 + ax + b Koblitz • Horizontal symmetry • Base upon logarithms in finite • Any vertical line intersect the fields curve at three points at most • Provides equivalent security as RSA with shorter Key lengths, Figure 21: 35 https://www.allaboutcircuits.com/technical-articles/elliptic-curve-cryptography-in-embedded-systems/ The Diffie-Hellman Elliptic-Curve Key Exchange (DHEC) • Alice and Bob first agree to use the same curve and a few other parameters, and then they pick a random point G on the curve. • Alice choose secret α, Bob choose secret β and each determine αG and βG which they interchange publicly. • The secret is S = α(βG) = β(αG) Figure 22: https://www.allaboutcircuits.com/technical-articles/elliptic-curve-cryptography-in-embedded-systems/ 36 DHEC DHEC uses a publicly known equation with large coefficients and modulus, for example, curve1559, which might very well be securing your browser right now. Figure 23: https://www.allaboutcircuits.com/technical-articles/elliptic-curve-cryptography-in-embedded-systems/ Elliptic-curve Diffie-Hellman allows microprocessors to securely determine a shared secret key while making it very difficult for a bad actor to determine that same shared key. 37 The NSA-Cryptography controversy of 2012 Figure 24: https://itsfoss.com/nsas-encryption-algorithm-in-linux-kernel-is-creating-unease-in-the-community/comment-page-5/ 38 NSA Backdoors? • Random number generators: The Dual-EC-DBRG generator was based on an elliptic curve cryptosystem, ... was proposed as a standard by NIST (with the technical support of NSA) in 2006, and became a U.S. standard (NIST Special Publication 800-90A) in 2007. • Parameters for Dual-EC were specified as certain constants with no explanation or justification for those numbers. • “Based on public concerns and an evaluation of the algorithm, NIST is proposing the removal of the Dual Elliptic Curve Deterministic Random Bit Generator.” https://www.nist.gov/news-events/news/2015/06/ nist-revises-key-computer-security-publication-random-number-generation • ISO blocks NSA’s latest IoT encryption systems amid murky tales of backdoors and bullying. Experts complain of shoddy tech specs and personal attacks https://www.theregister.co.uk/2018/04/25/nsa_iot_encryption/ 39 NSA 2020? Figure 25: https://searchsecurity.techtarget.com/news/252476828/NSA-reports-flaw-in-Windows-cryptography-core Affects: HTTPS connections, signed files and emails and signed executable code. Figure 26: https://media.defense.gov/2020/Jan/14/2002234275/-1/-1/0/CSA-WINDOWS-10-CRYPT-LIB-20190114.PDF 40 Splitting the bits: SMPC Shamir’s secret sharing Figure 27: https://en.wikipedia.org/wiki/Shamir’s_Secret_Sharing and https://image.cagle.com/195595/750/195595.png 41 Splitting the bits: SMPC An example Figure 28: https://en.wikipedia.org/wiki/Shamir’s_Secret_Sharing 42 SMPC Figure 29: https://en.wikipedia.org/wiki/Secure_multi-party_computation 43 SMPC Figure 30: https://www.reddit.com/r/instar/comments/8k6yir/secure_multiparty_computation_smpc_in_a_nutshell/ 44 Millionnaire’s problem Figure 31: https://www.reddit.com/r/instar/comments/8k6yir/secure_multiparty_computation_smpc_in_a_nutshell/ 45 Protocols • Zero knowledge proof’s: Prove to other parties that [something] is true without revealing anything about that [something] • (Shamir’s) (Threshold) Secret Sharing: A pre-configured amount of parties have to agree (threshold) to decrypt the answer before the answer can be decrypted. • Oblivious Transfer: Fragmented data is sent without revealing what and when part of the data is sent. 46 ZKP’s 47 Figure 32: https://101blockchains.com/zero-knowledge-proof/ ZKP’s (continued) Figure 33: https://www.cryptologie.net/article/193/schnorrs-signature-and-non-interactive-protocols/ 48 ZKP’s (continued) Figure 34: https://www.cryptologie.net/article/193/schnorrs-signature-and-non-interactive-protocols/ 49 Oblivious Transfer Oblivious Transfer (OT): ”... a type of protocol in which a sender transfers one of potentially many pieces of information to a receiver, but remains oblivious as to what piece (if any) has been transferred. Claude Crépeau showed that Rabin’s oblivious transfer is equivalent to 1-2 oblivious transfer”. • https://www.semanticscholar.org/topic/ Oblivious-transfer/333513 • https://en.wikipedia.org/wiki/Oblivious_transfer https://crypto.stanford.edu/~dabo/courses/cs355_ spring14/syllabus.html • https://www.cis.upenn.edu/~nadiah/courses/ cis800-02-f13/hemenway-otsmc.pdf (No lo veremos ahora) • https://github.com/mayank0403/ Oblivious-Transfer-and-Zero-Knowledge-Proof 50 1-out of 2 OT • Alice has two messages, m0, m11 and wants to send exactly one of them to Bob.