The document demonstrates breaking a 768-bit RSA encryption by factorizing the public key's modulus into its prime factors. It begins with an overview of RSA and integer factorization, then shows the encryption of a sample plaintext under a 768-bit public key. Finally, it programs and runs the decryption using the pre-computed prime factors of the modulus, successfully recovering the original plaintext in under a second. The document concludes that RSA security relies on the computational difficulty of integer factorization and recommends using key sizes of 1024 bits or more.
Cryptography is the practice and study of techniques for conveying information security.
The goal of Cryptography is to allow the intended recipients of the message to receive the message securely.
The most famous algorithm used today is RSA algorithm
A detailed description about Cryptography explaining the topic from the very basics. Explaining how it all started, and how is it currently being applied in the real world. Mostly useful for students in engineering and mathematics.
This presentation is based on the paper :
"A Method for Obtaining Digital Signatures and Public-Key Cryptosystems" by R.L. Rivest, A. Shamir, and L. Adleman
Cryptography is the practice and study of techniques for conveying information security.
The goal of Cryptography is to allow the intended recipients of the message to receive the message securely.
The most famous algorithm used today is RSA algorithm
A detailed description about Cryptography explaining the topic from the very basics. Explaining how it all started, and how is it currently being applied in the real world. Mostly useful for students in engineering and mathematics.
This presentation is based on the paper :
"A Method for Obtaining Digital Signatures and Public-Key Cryptosystems" by R.L. Rivest, A. Shamir, and L. Adleman
Information and network security 38 birthday attacks and security of hash fun...Vaibhav Khanna
Birthday attack can be used in communication abusage between two or more parties. ... The mathematics behind this problem led to a well-known cryptographic attack called the birthday attack, which uses this probabilistic model to reduce the complexity of cracking a hash function
This presentation contains the contents pertaining to the undergraduate course on Cryptography and Network Security (UITC203) at Sri Ramakrishna Institute of Technology. This covers the ElGamal Cryptosystem.
We will discuss the following: RSA Key generation , RSA Encryption , RSA Decryption , A Real World Example, RSA Security.
https://www.youtube.com/watch?v=x7QWJ13dgGs&list=PLKYmvyjH53q13_6aS4VwgXU0Nb_4sjwuf&index=7
This presentation will show you the basics of cryptography.
Main topics like basic terminology,goals of cryptography,threats,types of cryptography,algorithms of cryptography,etc. are covered in this presentation.If you like this presentation please do hit the like.
cyber Security and Cryptography Elgamal Encryption Algorithm, Not-petya Case study all in one.
ElGamal encryption is a public-key cryptosystem
ElGamal Algo. uses asymmetric key encryption for communicating between two parties and encrypting the message.
This cryptosystem is based on the difficulty of finding discrete logarithm in a cyclic group
It is based on the Diffie–Hellman key exchange And It was described by Taher Elgamal in 1985.
Receiver Generates public and private keys.
Select Large Prime No. (P)
Select Decryption key/ private Key (D)
gcd(D,P)=1
Select Second part of Encryption key or public key (E1) & gcd(E1,P)=1
Third part of the encryption key or public key (E2)
E2 = E1D mod P
Public Key=(E1, E2, P) & Private key=D
In 2017 Maersk was impacted by Not-Petya ransomware attack and their network was down for a whole 9 days.
A total of 49,000 PCs and 7,000 servers were encrypted by Not-petya. Other companies that were impacted by the same attack are Merck, TNT express etc.
The tools used in Notpetya were EternalBlue and Mimikatz and hence the attack was very fast and devastating for victims.
It was The Most Devastating Cyber attack in History that’s
How a single piece of code crashed the world.
In cryptography, a block cipher is a deterministic algorithm operating on ... Systems as a means to effectively improve security by combining simple operations such as .... Finally, the cipher should be easily cryptanalyzable, such that it can be ...
Results of some basic experiments with the Diffie-Hellman Key Exchange System. I analyse the key-exchange algorithm using brute-force as well using the Baby-step Giant-step algorithm.
A SURVEY ON ELLIPTIC CURVE DIGITAL SIGNATURE ALGORITHM AND ITS VARIANTScsandit
The Elliptic Curve Digital Signature Algorithm (ECDSA) is an elliptic curve variant of the
Digital Signature Algorithm (DSA). It gives cryptographically strong digital signatures making
use of Elliptic curve discrete logarithmic problem. It uses arithmetic with much smaller
numbers 160/256 bits instead of 1024/2048 bits in RSA and DSA and provides the same level of
security. The ECDSA was accepted in 1999 as an ANSI standard, and was accepted in 2000 as
IEEE and NIST standards. It was also accepted in 1998 as an ISO standard. Many cryptologist
have studied security aspects of ECDSA and proposed different variants. In this paper, we
discuss a detailed analysis of the original ECDSA and all its available variants in terms of the
security level and execution time of all the phases. To the best of our knowledge, this is a unique
attempt to juxtapose and compare the ECDSA with all of its variants.
Public Key Cryptography and RSA algorithmIndra97065
Public Key Cryptography and RSA algorithm.Explanation and proof of RSA algorithm in details.it also describer the mathematics behind the RSA. Few mathematics theorem are given which are use in the RSA algorithm.
HASH FUNCTIONS AND DIGITAL SIGNATURES
Authentication requirement – Authentication function – MAC – Hash function – Security of hash function and MAC –MD5 – SHA – HMAC – CMAC – Digital signature and authentication protocols – DSS – EI Gamal – Schnorr.
A short introduction to cryptography. What is public and private key cryptography? What is a Caesar Cipher and how do we decrypt it? How does RSA work?
The Cryptography puzzle discussed here is part of an online challenge. I demonstrate how I broke RSA when random prime numbers were common among a set of keys. I discuss basic metrics as well as implementation/design of my exploit scripts, too.
We study the behavior of the RSA trapdoor function by repeatedly encrypting the ciphertext sent over the public channel. We discuss the problem of finding a cycle in order to reverse the plaintext from the given ciphertext. Simple demos and algorithms/python programs are also presented. While the attack is not necessarily practical, it is educational to learn how the RSA trapdoor function behaves.
Information and network security 38 birthday attacks and security of hash fun...Vaibhav Khanna
Birthday attack can be used in communication abusage between two or more parties. ... The mathematics behind this problem led to a well-known cryptographic attack called the birthday attack, which uses this probabilistic model to reduce the complexity of cracking a hash function
This presentation contains the contents pertaining to the undergraduate course on Cryptography and Network Security (UITC203) at Sri Ramakrishna Institute of Technology. This covers the ElGamal Cryptosystem.
We will discuss the following: RSA Key generation , RSA Encryption , RSA Decryption , A Real World Example, RSA Security.
https://www.youtube.com/watch?v=x7QWJ13dgGs&list=PLKYmvyjH53q13_6aS4VwgXU0Nb_4sjwuf&index=7
This presentation will show you the basics of cryptography.
Main topics like basic terminology,goals of cryptography,threats,types of cryptography,algorithms of cryptography,etc. are covered in this presentation.If you like this presentation please do hit the like.
cyber Security and Cryptography Elgamal Encryption Algorithm, Not-petya Case study all in one.
ElGamal encryption is a public-key cryptosystem
ElGamal Algo. uses asymmetric key encryption for communicating between two parties and encrypting the message.
This cryptosystem is based on the difficulty of finding discrete logarithm in a cyclic group
It is based on the Diffie–Hellman key exchange And It was described by Taher Elgamal in 1985.
Receiver Generates public and private keys.
Select Large Prime No. (P)
Select Decryption key/ private Key (D)
gcd(D,P)=1
Select Second part of Encryption key or public key (E1) & gcd(E1,P)=1
Third part of the encryption key or public key (E2)
E2 = E1D mod P
Public Key=(E1, E2, P) & Private key=D
In 2017 Maersk was impacted by Not-Petya ransomware attack and their network was down for a whole 9 days.
A total of 49,000 PCs and 7,000 servers were encrypted by Not-petya. Other companies that were impacted by the same attack are Merck, TNT express etc.
The tools used in Notpetya were EternalBlue and Mimikatz and hence the attack was very fast and devastating for victims.
It was The Most Devastating Cyber attack in History that’s
How a single piece of code crashed the world.
In cryptography, a block cipher is a deterministic algorithm operating on ... Systems as a means to effectively improve security by combining simple operations such as .... Finally, the cipher should be easily cryptanalyzable, such that it can be ...
Results of some basic experiments with the Diffie-Hellman Key Exchange System. I analyse the key-exchange algorithm using brute-force as well using the Baby-step Giant-step algorithm.
A SURVEY ON ELLIPTIC CURVE DIGITAL SIGNATURE ALGORITHM AND ITS VARIANTScsandit
The Elliptic Curve Digital Signature Algorithm (ECDSA) is an elliptic curve variant of the
Digital Signature Algorithm (DSA). It gives cryptographically strong digital signatures making
use of Elliptic curve discrete logarithmic problem. It uses arithmetic with much smaller
numbers 160/256 bits instead of 1024/2048 bits in RSA and DSA and provides the same level of
security. The ECDSA was accepted in 1999 as an ANSI standard, and was accepted in 2000 as
IEEE and NIST standards. It was also accepted in 1998 as an ISO standard. Many cryptologist
have studied security aspects of ECDSA and proposed different variants. In this paper, we
discuss a detailed analysis of the original ECDSA and all its available variants in terms of the
security level and execution time of all the phases. To the best of our knowledge, this is a unique
attempt to juxtapose and compare the ECDSA with all of its variants.
Public Key Cryptography and RSA algorithmIndra97065
Public Key Cryptography and RSA algorithm.Explanation and proof of RSA algorithm in details.it also describer the mathematics behind the RSA. Few mathematics theorem are given which are use in the RSA algorithm.
HASH FUNCTIONS AND DIGITAL SIGNATURES
Authentication requirement – Authentication function – MAC – Hash function – Security of hash function and MAC –MD5 – SHA – HMAC – CMAC – Digital signature and authentication protocols – DSS – EI Gamal – Schnorr.
A short introduction to cryptography. What is public and private key cryptography? What is a Caesar Cipher and how do we decrypt it? How does RSA work?
The Cryptography puzzle discussed here is part of an online challenge. I demonstrate how I broke RSA when random prime numbers were common among a set of keys. I discuss basic metrics as well as implementation/design of my exploit scripts, too.
We study the behavior of the RSA trapdoor function by repeatedly encrypting the ciphertext sent over the public channel. We discuss the problem of finding a cycle in order to reverse the plaintext from the given ciphertext. Simple demos and algorithms/python programs are also presented. While the attack is not necessarily practical, it is educational to learn how the RSA trapdoor function behaves.
We look into the nitty-gritty details of the RSA key generation algorithm. We study how RSA can be exploited when the public exponent e is not chosen carefully. We examine why many digital certificates use e=65537. We also experiment with Hastad's broadcast attack for short RSA exponents in particular.
Slides demonstrate how to break RSA when no padding is applied. I replicated the meet-in-the-middle attack discussed in the existing Crypto literature.
The slides demonstrate how to break RSA when used incorrectly without integrity checks. The man-in-the-middle is allowed to edit the RSA public exponent e in such a way that the Extended Euclidean Algorithm can be employed to reconstruct the plaintexts from the given ciphertexts.
The slides demonstrate how to reverse the plaintext from the RSA encrypted ciphertext using an oracle that answers the question: is the last bit of the message 0 or 1?
An RSA private key is made of a few private variables. We analyze how these private variables are chained together. Further, we study if one of the private variables is leaked, can we derive the other private variables? Demos of the algorithms are also provided.
Slides present a demo of exploiting the homomorphic properties of raw RSA (i.e., without any padding) to reverse an RSA ciphertext, without the private key. We have two roles: Adversary and Challenger. The challenger presents a ciphertext to the adversary to break it. The adversary is allowed to ask for encryption/decryption of any text, except the decryption of the challenge ciphertext. The goal of the adversary is to break the ciphertext.
Can we reveal the RSA private exponent d from its public key <e, n>? We study this question for two specific cases: e = 3 and e = 65537. Using demos, we verify that RSA reveals the most significant half of the private exponent d when the public exponent e is small. For example, for 2048-bit RSA, the most significant 1024 bits are revealed!
We experiment with Wiener's attack to break RSA when the secret exponent is short, meaning it is smaller than one quarter of the public modulus size. We discuss cryptanalysis details and present demos of the attack. Our very minor extension of Wiener's attack is also discussed.
If we have an RSA 2048 bits configuration, but our private exponent d is only about 512 bits, then the above attack breaks RSA in a few seconds.
This work uses Continued Fractions to derive the private keys from the given public keys. It turned out that one can derive the private exponent d by approximating it as a ratio of e/n, both are public values.
In a default settings of standard RSA libaries, this attack and my minor extension are not relevant (to the best of our knowledge). However, if we configure our library to choose a very large public encryption exponent e, then our private decryption exponent d could be short enough to mount an attack.
Computing the Square Roots of Unity to break RSA using Quantum AlgorithmsDharmalingam Ganesan
We study the problem of finding the square roots of unity in a finite group in order to factor composite numbers used in RSA. We implemented Peter Shor’s algorithm to find the square root of unity. Experimental results showed that finding the square roots of unity in a finite group multiplicative group is “hard”.
Information and network security 33 rsa algorithmVaibhav Khanna
RSA algorithm is asymmetric cryptography algorithm. Asymmetric actually means that it works on two different keys i.e. Public Key and Private Key. As the name describes that the Public Key is given to everyone and Private key is kept private
We study the internal structure of the SRP key exchange protocol and experiment with it. SRP establishes a shared encryption key between communicating parties using passwords that were shared out-of-band. We perform basic cryptanalysis of SRP using open-source implementations. We present a demo of how SRP was compromised due to an implementation bug, allowing the attacker to login without the password. The author of the Go-SRP library promptly fixed the issue on the very same day we reported the vulnerability.
We allow Eve to modify DH parameters as well as public keys of Alice and Bob. This allows Eve to derive the secret key and break the DH crypto system. We demonstrate that the DH key exchange algorithm should not be used without digital signatures.
This was an invited talk at the Central Middle School, Maryland. Without going into a lot of math, I try to explain the fundamental key exchange problem. It was a blast. 8th graders enjoyed it as much as I enjoyed it.
IRSim implements an approach to establish traceability links among artifacts such as requirements, source code, and test cases. This presentation shows how we used IRSim on NASA software to establish traceability links for sofware analysis, program understanding, and quality improvement, etc.
Threat Modeling: Applied on a Publish-Subscribe Architectural StyleDharmalingam Ganesan
1. Introduction to threat modeling.
2. Applying threat modeling to identify security vulnerabilities and security threats on a simplified real-world system.
In software engineering, the right architecture is essential for robust, scalable platforms. Wix has undergone a pivotal shift from event sourcing to a CRUD-based model for its microservices. This talk will chart the course of this pivotal journey.
Event sourcing, which records state changes as immutable events, provided robust auditing and "time travel" debugging for Wix Stores' microservices. Despite its benefits, the complexity it introduced in state management slowed development. Wix responded by adopting a simpler, unified CRUD model. This talk will explore the challenges of event sourcing and the advantages of Wix's new "CRUD on steroids" approach, which streamlines API integration and domain event management while preserving data integrity and system resilience.
Participants will gain valuable insights into Wix's strategies for ensuring atomicity in database updates and event production, as well as caching, materialization, and performance optimization techniques within a distributed system.
Join us to discover how Wix has mastered the art of balancing simplicity and extensibility, and learn how the re-adoption of the modest CRUD has turbocharged their development velocity, resilience, and scalability in a high-growth environment.
In 2015, I used to write extensions for Joomla, WordPress, phpBB3, etc and I ...Juraj Vysvader
In 2015, I used to write extensions for Joomla, WordPress, phpBB3, etc and I didn't get rich from it but it did have 63K downloads (powered possible tens of thousands of websites).
Providing Globus Services to Users of JASMIN for Environmental Data AnalysisGlobus
JASMIN is the UK’s high-performance data analysis platform for environmental science, operated by STFC on behalf of the UK Natural Environment Research Council (NERC). In addition to its role in hosting the CEDA Archive (NERC’s long-term repository for climate, atmospheric science & Earth observation data in the UK), JASMIN provides a collaborative platform to a community of around 2,000 scientists in the UK and beyond, providing nearly 400 environmental science projects with working space, compute resources and tools to facilitate their work. High-performance data transfer into and out of JASMIN has always been a key feature, with many scientists bringing model outputs from supercomputers elsewhere in the UK, to analyse against observational or other model data in the CEDA Archive. A growing number of JASMIN users are now realising the benefits of using the Globus service to provide reliable and efficient data movement and other tasks in this and other contexts. Further use cases involve long-distance (intercontinental) transfers to and from JASMIN, and collecting results from a mobile atmospheric radar system, pushing data to JASMIN via a lightweight Globus deployment. We provide details of how Globus fits into our current infrastructure, our experience of the recent migration to GCSv5.4, and of our interest in developing use of the wider ecosystem of Globus services for the benefit of our user community.
Exploring Innovations in Data Repository Solutions - Insights from the U.S. G...Globus
The U.S. Geological Survey (USGS) has made substantial investments in meeting evolving scientific, technical, and policy driven demands on storing, managing, and delivering data. As these demands continue to grow in complexity and scale, the USGS must continue to explore innovative solutions to improve its management, curation, sharing, delivering, and preservation approaches for large-scale research data. Supporting these needs, the USGS has partnered with the University of Chicago-Globus to research and develop advanced repository components and workflows leveraging its current investment in Globus. The primary outcome of this partnership includes the development of a prototype enterprise repository, driven by USGS Data Release requirements, through exploration and implementation of the entire suite of the Globus platform offerings, including Globus Flow, Globus Auth, Globus Transfer, and Globus Search. This presentation will provide insights into this research partnership, introduce the unique requirements and challenges being addressed and provide relevant project progress.
Advanced Flow Concepts Every Developer Should KnowPeter Caitens
Tim Combridge from Sensible Giraffe and Salesforce Ben presents some important tips that all developers should know when dealing with Flows in Salesforce.
Globus Compute wth IRI Workflows - GlobusWorld 2024Globus
As part of the DOE Integrated Research Infrastructure (IRI) program, NERSC at Lawrence Berkeley National Lab and ALCF at Argonne National Lab are working closely with General Atomics on accelerating the computing requirements of the DIII-D experiment. As part of the work the team is investigating ways to speedup the time to solution for many different parts of the DIII-D workflow including how they run jobs on HPC systems. One of these routes is looking at Globus Compute as a way to replace the current method for managing tasks and we describe a brief proof of concept showing how Globus Compute could help to schedule jobs and be a tool to connect compute at different facilities.
SOCRadar Research Team: Latest Activities of IntelBrokerSOCRadar
The European Union Agency for Law Enforcement Cooperation (Europol) has suffered an alleged data breach after a notorious threat actor claimed to have exfiltrated data from its systems. Infamous data leaker IntelBroker posted on the even more infamous BreachForums hacking forum, saying that Europol suffered a data breach this month.
The alleged breach affected Europol agencies CCSE, EC3, Europol Platform for Experts, Law Enforcement Forum, and SIRIUS. Infiltration of these entities can disrupt ongoing investigations and compromise sensitive intelligence shared among international law enforcement agencies.
However, this is neither the first nor the last activity of IntekBroker. We have compiled for you what happened in the last few days. To track such hacker activities on dark web sources like hacker forums, private Telegram channels, and other hidden platforms where cyber threats often originate, you can check SOCRadar’s Dark Web News.
Stay Informed on Threat Actors’ Activity on the Dark Web with SOCRadar!
How to Position Your Globus Data Portal for Success Ten Good PracticesGlobus
Science gateways allow science and engineering communities to access shared data, software, computing services, and instruments. Science gateways have gained a lot of traction in the last twenty years, as evidenced by projects such as the Science Gateways Community Institute (SGCI) and the Center of Excellence on Science Gateways (SGX3) in the US, The Australian Research Data Commons (ARDC) and its platforms in Australia, and the projects around Virtual Research Environments in Europe. A few mature frameworks have evolved with their different strengths and foci and have been taken up by a larger community such as the Globus Data Portal, Hubzero, Tapis, and Galaxy. However, even when gateways are built on successful frameworks, they continue to face the challenges of ongoing maintenance costs and how to meet the ever-expanding needs of the community they serve with enhanced features. It is not uncommon that gateways with compelling use cases are nonetheless unable to get past the prototype phase and become a full production service, or if they do, they don't survive more than a couple of years. While there is no guaranteed pathway to success, it seems likely that for any gateway there is a need for a strong community and/or solid funding streams to create and sustain its success. With over twenty years of examples to draw from, this presentation goes into detail for ten factors common to successful and enduring gateways that effectively serve as best practices for any new or developing gateway.
Developing Distributed High-performance Computing Capabilities of an Open Sci...Globus
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Paketo Buildpacks : la meilleure façon de construire des images OCI? DevopsDa...Anthony Dahanne
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Venez le découvrir lors de cette session ignite
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Large Language Models (LLMs) are currently the center of attention in the tech world, particularly for their potential to advance research. In this presentation, we'll explore a straightforward and effective method for quickly initiating inference runs on supercomputers using the vLLM tool with Globus Compute, specifically on the Polaris system at ALCF. We'll begin by briefly discussing the popularity and applications of LLMs in various fields. Following this, we will introduce the vLLM tool, and explain how it integrates with Globus Compute to efficiently manage LLM operations on Polaris. Attendees will learn the practical aspects of setting up and remotely triggering LLMs from local machines, focusing on ease of use and efficiency. This talk is ideal for researchers and practitioners looking to leverage the power of LLMs in their work, offering a clear guide to harnessing supercomputing resources for quick and effective LLM inference.
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Multiple Your Crypto Portfolio with the Innovative Features of Advanced Crypt...Hivelance Technology
Cryptocurrency trading bots are computer programs designed to automate buying, selling, and managing cryptocurrency transactions. These bots utilize advanced algorithms and machine learning techniques to analyze market data, identify trading opportunities, and execute trades on behalf of their users. By automating the decision-making process, crypto trading bots can react to market changes faster than human traders
Hivelance, a leading provider of cryptocurrency trading bot development services, stands out as the premier choice for crypto traders and developers. Hivelance boasts a team of seasoned cryptocurrency experts and software engineers who deeply understand the crypto market and the latest trends in automated trading, Hivelance leverages the latest technologies and tools in the industry, including advanced AI and machine learning algorithms, to create highly efficient and adaptable crypto trading bots
Climate Science Flows: Enabling Petabyte-Scale Climate Analysis with the Eart...Globus
The Earth System Grid Federation (ESGF) is a global network of data servers that archives and distributes the planet’s largest collection of Earth system model output for thousands of climate and environmental scientists worldwide. Many of these petabyte-scale data archives are located in proximity to large high-performance computing (HPC) or cloud computing resources, but the primary workflow for data users consists of transferring data, and applying computations on a different system. As a part of the ESGF 2.0 US project (funded by the United States Department of Energy Office of Science), we developed pre-defined data workflows, which can be run on-demand, capable of applying many data reduction and data analysis to the large ESGF data archives, transferring only the resultant analysis (ex. visualizations, smaller data files). In this talk, we will showcase a few of these workflows, highlighting how Globus Flows can be used for petabyte-scale climate analysis.
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Unleash Unlimited Potential with One-Time Purchase
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1. Security of RSA and Integer
Factorization
Public Key Size Matters: Demo of decrypting RSA 768-bits ciphertext
Dr. Dharma Ganesan, Ph.D.,
2. Disclaimer
● The opinions expressed here are my own
○ But not the views of my employer
● The source code fragments and exploits shown here can be reused
○ But without any warranty nor accept any responsibility for failures
● Do not apply the exploit discussed here on other systems
○ Without obtaining authorization from owners
2
3. Agenda
● Brief overview of public key cryptography
● RSA formal definition
● Integer Factorization
● Demo - break RSA-768 bits encryption
● Discussion/Recommendation
● Slides are intended for newcomers to Cryptography
3
4. Goal
● Demonstrate that security of RSA is rooted in
computational hardness of integer factorization
● If the prime factors of a number are known, game over!
● Let’s see how to exploit 768-bits RSA modulus
4
5. Prerequisite
Some familiarity with the following topics will help to follow the rest of the slides
● Group Theory (Abstract Algebra/Discrete Math)
● Modular Arithmetic (Number Theory)
● Algorithms and Complexity Theory
● If not, it should still be possible to obtain a high-level overview
5
6. How can Alice send a message to Bob securely?
6
Public Key PuA
● Alice and Bob never met each other
● Bob will encrypt using Alice’s public key
○ Assume that public keys are known to the world
● Alice will decrypt using her private key
○ Private keys are secrets (never sent out)
● Bob can sign messages using his private key
○ Alice verifies message integrity using Bob’s public key
○ Not important for this presentation/attack
● Note: Alice and Bob need other evidence (e.g., passwords,
certificates) to prove their identity to each other
● Who are Alice, Bob, and Eve?
Private Key PrA
Public Key PuB
Private Key PrB
7. RSA Public Key Cryptography System
● Published in 1977 by Ron Rivest, Adi Shamir and Leonard Adleman
● Rooted in elegant mathematics - Group Theory and Number Theory
● Core idea: Anyone can encrypt a message using recipient's public key but
○ (as far as we know) no one can efficiently decrypt unless they got the matching private key
● Encryption and Decryption are inverse operations (math details later)
○ Work of Euclid, Euler, and Fermat provide the mathematical foundation of RSA
● Eavesdropper Eve cannot easily derive the secret (math details later)
○ Unless she solves “hard” number theory problems that are computationally intractable
7
8. 8
....
Note: There is a change in the
notation of symbols in other
publications since 1977.
We will not use symbols w, r, s.
e will be used but with a different
meaning (explained in next slides).
9. 9
Notations and Facts for RSA
GCD(x, y): The greatest common divisor that divides integers x and y
Co-prime: If gcd(x, y) = 1, then x and y are co-primes
Zn
= { 0, 1, 2, …, n-1 }, n > 0; we may imagine Zn
as a circular wall clock
Z*
n
= { x ∈ Zn
| gcd(x, n) = 1 }; (Z*
n
is a multiplicative group)
φ(n): Euler’s Totient function denotes the number of elements in Z*
n
φ(nm) = φ(n).φ(m) (This property is called multiplicative)
φ(p) = p-1, if p is a prime number
x ≡ y (mod n) denotes that n divides x-y; x is congruent to y mod n
10. RSA - Key Generation Algo. (Fits on one page)
1. Select an appropriate bitlength of the RSA modulus n (e.g., 2048 bits)
○ Value of the parameter n is not chosen until step 3; small n is dangerous (details later)
2. Pick two independent, large random primes, p and q, of half of n’s bitlength
○ In practice, p and q are not close to each other to avoid attacks (e.g., Fermat’s factorization)
3. Compute n = p.q (n is also called the RSA modulus)
4. Compute Euler’s Totient (phi) Function φ(n) = φ(p.q) = φ(p)φ(q) = (p-1)(q-1)
5. Select numbers e and d from Zn
such that e.d ≡ 1(mod φ(n))
○ Many implementations set e to be 65537 (Note: gcd(e, φ(n)) = 1)
○ e must be relatively prime to φ(n) otherwise d cannot exist (i.e., we cannot decrypt)
○ d is the multiplicative inverse of e in Zn
6. Public key is the pair <n, e> and private key is 4-tuple <φ(n), d, p, q>
Note: If p, q, d, or φ(n) is leaked, RSA is broken immediately
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11. Formal definition of the RSA function
● RSA: Zn
→ Zn
● Let m and c ∈ Zn
● c = RSA(m) = me
mod n
● m = RSA-1
(c) = cd
mod n
● e and d are called encryption and decryption exponents, respectively
● Usually a padding scheme is applied before encryption
○ Not relevant for this presentation
● Note: Attackers know c, e, and n but not d
11
12. RSA Source Code (sample)
● RSACoreEngine.java implements the RSA and RSA-1
functions
○ See the method processBlock
● RSA(input) = inpute
mod n
{
return input.modPow(
key.getExponent(), key.getModulus());
}
● The implementation for RSA-1
is a bit involved (Chinese-Remainder Theorem)
○ To speed-up the computation of inputd
mod n, it computes inputd
mod p and mod q
○ Not relevant for this presentation
12
13. Security assumption of RSA: Factorization Problem
● 15 = 5 . 3
● 21 = 7 . 3
● 143 = 11 . 13
● Given a number n find two prime numbers p and q such that n = p . q
● If this problem is solved for a large n, security of RSA is compromised
13
14. Problem statement: Break the RSA-768 bits
● Recall that c = RSA(m) = me
mod n
○ c is the ciphertext and m is the plaintext
● Breaking of the RSA function means finding m from c
○ Equivalently, reverse the plaintext from the ciphertext
● Formally, given <c, e, n> find m such that RSA(m) = c
14
15. RSA-768 bits Example
15
RSA-768 has 232 decimal digits (768 bits), and was factored on
December 12, 2009 over the span of two years, by Thorsten Kleinjung, Kazumaro
Aoki, Jens Franke, Arjen K. Lenstra, Emmanuel Thomé, Pierrick Gaudry, Alexander Kruppa, Peter
Montgomery, Joppe W. Bos, Dag Arne Osvik, Herman te Riele, Andrey Timofeev, and Paul Zimmermann
16. RSA-768 bits - Encryption
public class RSAWeakKeyDemo {
public static void main(String [] args) throws Exception {
BigInteger n = new
BigInteger("1230186684530117755130494958384962720772853569595334792197322452
1517264005072636575187452021997864693899564749427740638459251925573263034
5373154826850791702612214291346167042921431160222124047927473779408066535
1419597459856902143413");
BigInteger e = new BigInteger("65537");
PublicKey pubKey = RSAService.getPublicKey(n, e);
byte[] ciphertext = RSAService.encrypt(pubKey, args[0]); // error check for args[0]?
System.out.println(javax.xml.bind.DatatypeConverter.printHexBinary(ciphertext));
}
}
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17. RSA-768 - Encrypt a plaintext
$ java RSAWeakKeyDemo "Glenn was here"
7C93E5C41364520991CF359321991B7AC2118E02ADC81913B8D6A70B08B38
4BB8EBF96C5C31C92F48DEDA3080D427E622A365682B64890BCB1F5C3571
9D06A03F0BCF3F5C55A160A5A9C754700348A6A9B11B8F5E06AA428BA1A5
F54B471C64D
❖ I will demonstrate how to take the ciphertext and reconstruct the plaintext
➢ This was possible because the prime factors of the public modules n are available
17
18. Let’s decrypt the ciphertext
● Recall that the public modulus n has 768 bits and was already factored
● p = 33478071698956898786044169848212690817704794983713768568912431388982883793
878002287614711652531743087737814467999489
● q = 36746043666799590428244633799627952632279158164343087642676032283815739666
511279233373417143396810270092798736308917
● We know that φ(n) = φ(p.q) = φ(p)φ(q) = (p-1)(q-1)
● To decrypt we need to find the decryption exponent d such that e.d ≡ 1(mod φ(n))
● Thus, we can find d by taking the inverse of e in φ(n)
18
20. Demo: Let’s reverse the plaintext from ciphertext
$ time java RSABreakWeakKey
7C93E5C41364520991CF359321991B7AC2118E02ADC81913B8D6A70B08B38
4BB8EBF96C5C31C92F48DEDA3080D427E622A365682B64890BCB1F5C3571
9D06A03F0BCF3F5C55A160A5A9C754700348A6A9B11B8F5E06AA428BA1A5
F54B471C64D
Plaintext: Glenn was here
real 0m0.973s
user0m0.540s
sys 0m0.068s
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21. Discussion/Conclusion
21
● Security of RSA depends on computation hardness of integer factorization
● A list of known RSA numbers is: https://en.wikipedia.org/wiki/RSA_numbers
● This list shows that so far 768-bits RSA modulus is factored
○ Given a public key, the private prime numbers factors p and q are known to the world
● The demo showed how to decrypt ciphertext when prime factors are known!
● Implementations have to keep ahead by choosing a large public key size
○ At the time of writing, 1024-bits are not factored; 2048-bits is recommended
22. References
● W. Diffie and M. E. Hellman, “New Directions in Cryptography,” IEEE
Transactions on Information Theory, vol. IT-22, no. 6, November, 1976.
● R. L. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital
signatures and public-key cryptosystems,” CACM 21, 2, February, 1978.
● A. Menezes, P. van Oorschot, and S. Vanstone, “Handbook of Applied
Cryptography,” CRC Press, 1996.
● C. Paar and J. Pelzl. “Understanding Cryptography: A Textbook for Students
and Practitioners,” Springer, 2011.
● https://en.wikipedia.org/wiki/RSA_numbers
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