The RSA cryptosystem document discusses:
1) The RSA cryptosystem uses a public and private key to encrypt and decrypt messages based on large prime number factorization.
2) An example is provided where a message is encrypted with a public key and decrypted with a private key.
3) The security of RSA relies on the difficulty of factoring large numbers, as factorization algorithms take exponential time relative to the number of bits.
Ruby Supercomputing - Using The GPU For Massive Performance Speedup v1.1Preston Lee
For MountainWest RubyConf 2011 in Salt Lake City, Utah. By Preston Lee.
Twitter: @prestonism
Blog: http://prestonlee.com
Code: https://github.com/preston/ruby-gpu-examples
Slides: http://www.slideshare.net/preston.lee/
Ruby Supercomputing - Using The GPU For Massive Performance Speedup v1.1Preston Lee
For MountainWest RubyConf 2011 in Salt Lake City, Utah. By Preston Lee.
Twitter: @prestonism
Blog: http://prestonlee.com
Code: https://github.com/preston/ruby-gpu-examples
Slides: http://www.slideshare.net/preston.lee/
If you've tried Apache Solr 1.4, you've probably had a chance to take it for a spin indexing and searching your data, and getting acquainted with its powerful, versatile new features and functions. Now, it's time to roll up your sleeves and really master what Solr 1.4 has to offer.
WEBINAR ON FUNDAMENTALS OF DIGITAL IMAGE PROCESSING DURING COVID LOCK DOWN by by K.Vijay Anand , Associate Professor, Department of Electronics and Instrumentation Engineering , R.M.K Engineering College, Tamil Nadu , India
Bioastronautics: Space Exploration and its Effects on the Human Body Course S...Jim Jenkins
This three-day course is intended for technical and managerial personnel who wish to be introduced to the effects of the space environment on humans. This course introduces bioastronautics from a fundamental perspective, assuming no prior knowledge of biology, physiology, or chemistry. The objective of the course is to provide the student with basic knowledge that will allow him or her to contribute more effectively to the human space exploration program. The human body, that through evolution is uniquely designed to function on the Earth, adapts to the space environment characterized by weightlessness and enhanced radiation. These alterations can impact the health and performance of astronauts, especially on return to the Earth.
Fundamentals Of Space Systems & Space Subsystems course samplerJim Jenkins
This course in space systems and space subsystems is for technical and management personnel who wish to gain an understanding of the important technical concepts in the development of space instrumentation, subsystems, and systems. The goal is to assist students to achieve their professional potential by endowing them with an understanding of the subsystems and supporting disciplines important to developing space instrumentation, space subsystems, and space systems. It designed for participants who expect to plan, design, build, integrate, test, launch, operate or manage subsystems, space systems, launch vehicles, spacecraft, payloads, or ground systems. The objective is to expose each participant to the fundamentals of each subsystem and their inter-relations, to not necessarily make each student a systems engineer, but to give aerospace engineers and managers a technically based space systems perspective. The fundamental concepts are introduced and illustrated by state-of-the-art examples. This course differs from the typical space systems course in that the technical aspects of each important subsystem are addressed.
ATI's Quantitative Methods course: Bridging Project Management and System Eng...Jim Jenkins
This 3-day course is de¬signed for the professional program manager, system engineer, or project manager engaged in technically challenging projects where close technical collaboration between engineering and management is a must. To that end, this course addresses major topics that bridge the disciplines of project management and system engineering. Each of the selected topics is presented from the perspective of quantitative methods. Students first learn a theory or narrative, and then related methods or practices. Ideas are demonstrated that are immediately applicable to programs and projects. Attendees receive a copy of the instructor’s text, Quantitative Methods in Project Management.
Fundamentals of Engineering Probability Visualization Techniques & MatLab Cas...Jim Jenkins
This four-day course gives a solid practical and intuitive understanding of the fundamental concepts of discrete and continuous probability. It emphasizes visual aspects by using many graphical tools such as Venn diagrams, descriptive tables, trees, and a unique 3-dimensional plot to illustrate the behavior of probability densities under coordinate transformations. Many relevant engineering applications are used to crystallize crucial probability concepts that commonly arise in aerospace CONOPS and tradeoffs
If you've tried Apache Solr 1.4, you've probably had a chance to take it for a spin indexing and searching your data, and getting acquainted with its powerful, versatile new features and functions. Now, it's time to roll up your sleeves and really master what Solr 1.4 has to offer.
WEBINAR ON FUNDAMENTALS OF DIGITAL IMAGE PROCESSING DURING COVID LOCK DOWN by by K.Vijay Anand , Associate Professor, Department of Electronics and Instrumentation Engineering , R.M.K Engineering College, Tamil Nadu , India
Bioastronautics: Space Exploration and its Effects on the Human Body Course S...Jim Jenkins
This three-day course is intended for technical and managerial personnel who wish to be introduced to the effects of the space environment on humans. This course introduces bioastronautics from a fundamental perspective, assuming no prior knowledge of biology, physiology, or chemistry. The objective of the course is to provide the student with basic knowledge that will allow him or her to contribute more effectively to the human space exploration program. The human body, that through evolution is uniquely designed to function on the Earth, adapts to the space environment characterized by weightlessness and enhanced radiation. These alterations can impact the health and performance of astronauts, especially on return to the Earth.
Fundamentals Of Space Systems & Space Subsystems course samplerJim Jenkins
This course in space systems and space subsystems is for technical and management personnel who wish to gain an understanding of the important technical concepts in the development of space instrumentation, subsystems, and systems. The goal is to assist students to achieve their professional potential by endowing them with an understanding of the subsystems and supporting disciplines important to developing space instrumentation, space subsystems, and space systems. It designed for participants who expect to plan, design, build, integrate, test, launch, operate or manage subsystems, space systems, launch vehicles, spacecraft, payloads, or ground systems. The objective is to expose each participant to the fundamentals of each subsystem and their inter-relations, to not necessarily make each student a systems engineer, but to give aerospace engineers and managers a technically based space systems perspective. The fundamental concepts are introduced and illustrated by state-of-the-art examples. This course differs from the typical space systems course in that the technical aspects of each important subsystem are addressed.
ATI's Quantitative Methods course: Bridging Project Management and System Eng...Jim Jenkins
This 3-day course is de¬signed for the professional program manager, system engineer, or project manager engaged in technically challenging projects where close technical collaboration between engineering and management is a must. To that end, this course addresses major topics that bridge the disciplines of project management and system engineering. Each of the selected topics is presented from the perspective of quantitative methods. Students first learn a theory or narrative, and then related methods or practices. Ideas are demonstrated that are immediately applicable to programs and projects. Attendees receive a copy of the instructor’s text, Quantitative Methods in Project Management.
Fundamentals of Engineering Probability Visualization Techniques & MatLab Cas...Jim Jenkins
This four-day course gives a solid practical and intuitive understanding of the fundamental concepts of discrete and continuous probability. It emphasizes visual aspects by using many graphical tools such as Venn diagrams, descriptive tables, trees, and a unique 3-dimensional plot to illustrate the behavior of probability densities under coordinate transformations. Many relevant engineering applications are used to crystallize crucial probability concepts that commonly arise in aerospace CONOPS and tradeoffs
Mobile data traffic is growing year to year. Mobile operators are facing a different situation from voice legacy business. The growth of data traffic is not as high as one of revenue. They need to lower cost of Mbps to survive otherwise they will collapse.
ATI Systems Engineering - The People Dimension Professional Development Techn...Jim Jenkins
This course provides perspective and insight into a part of the system engineering process that is critical to the success of any project: the people, and the leadership and management of people. It includes a short review of system engineering and it's associated processes, especially the people related aspects. It discusses the subjects of leadership and management, and their differences, and how they relate to system engineering.
The course is valuable to program and Line Management, as well as to technical and administrative personnel who are a part of the system engineering process.
ATI's Total Systems Engineering Development & Management technical training c...Jim Jenkins
This three-day ATI professional development course, Total Systems Engineering Development & Management, course, covers four system
development fundamentals: (1) a sound
engineering management infrastructure within
which work may be efficiently accomplished, (2)
define the problem to be solved (requirements and
specifications), (3) solve the problem (design,
integration, and optimization), and (4) prove that
the design solves the defined problem
(verification).
ATI's Systems Engineering - Requirements technical training course samplerJim Jenkins
This ATI professional
development course, Systems Engineering - Requirements, provides system engineers, team leaders, and managers with a clear understanding about how to develop good specifications affordably using modeling methods that encourage identification of the essential characteristics that must be respected in the subsequent design process.
This three day course is intended for practicing systems engineers who want to learn how to apply model-driven systems Successful systems engineering requires a broad understanding of the important principles of modern spacecraft communications. This three-day course covers both theory and practice, with emphasis on the important system engineering principles, tradeoffs, and rules of thumb. The latest technologies are covered. <p>
Applied Physical Oceanography And ModelingJim Jenkins
This three-day course is designed for engineers, physicists, acousticians, climate scientists, and managers who wish to enhance their understanding of this discipline or become familiar with how the ocean environment can affect their individual applications. Examples of remote sensing of the ocean, in situ ocean observing systems and actual examples from recent oceanographic cruises are given.
The students will be able to access educational Java applets to visualize waves and key acoustic phenomena: Click here to view
Other web-based resources include acoustic demonstration podcasts and iPod apps to conduct acoustic measurements. The student will also be armed with Internet resources for up-to-date information on sonar systems, undersea sound propagation models, and environmental databases. The student will leave with a clear understanding of how the ocean influences undersea sound propagation and scattering.
This is the DeepStochLog presentation, published at AAAI22 (Association for the Advancement of Artificial Intelligence 2022).
Authors: Thomas Winters*, Giuseppe Marra*, Robin Manhaeve, Luc De Raedt
*equal contribution
Code: https://github.com/ml-kuleuven/deepstochlog
Abstract: Recent advances in neural symbolic learning, such as DeepProbLog, extend probabilistic logic programs with neural predicates. Like graphical models, these probabilistic logic programs define a probability distribution over possible worlds, for which inference is computationally hard. We propose DeepStochLog, an alternative neural symbolic framework based on stochastic definite clause grammars, a type of stochastic logic program, which defines a probability distribution over possible derivations. More specifically, we introduce neural grammar rules into stochastic definite clause grammars to create a framework that can be trained end-to-end. We show that inference and learning in neural stochastic logic programming scale much better than for neural probabilistic logic programs. Furthermore, the experimental evaluation shows that DeepStochLog achieves state-of-the-art results on challenging neural symbolic learning tasks.
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.
Everything I always wanted to know about crypto, but never thought I'd unders...Codemotion
For many years, I had entirely given up on ever understanding the anything about cryptography. However, I’ve since learned it’s not nearly as hard as I thought to understand many of the important concepts. In this talk, I’ll take you through some of the underlying principles of modern applications of cryptography. We’ll talk about our goals, the parts are involved, and how to prevent and understand common vulnerabilities. This’ll help you to make better choices when you implement crypto in your products, and will improve your understanding of how crypto is applied to things you already use.
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!
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.
RSA and OAEP
Diffe-Hellman Key Exchange and its Security Aspects
Model of Asymmetric Key Cryptography
Factorization and other methods for Public Key Cryptography
1. RSA Cryptosystem 6/8/2002 2:20 PM
Outline
Euler’s theorem (§10.1.3)
RSA cryptosystem (§10.2.3)
RSA Cryptosystem Definition
Example
Bits PCs Memory Security
430 1 128MB Correctness
760 215,000 4GB
Algorithms for RSA
1,020 342×106 170GB
Modular power (§10.1.4)
1,620 1.6×1015 120TB
Modular inverse (§10.1.5)
Randomized primality testing (§10.1.6)
6/8/2002 2:20 PM RSA Cryptosystem 1 6/8/2002 2:20 PM RSA Cryptosystem 2
Euler’s Theorem RSA Cryptosystem
The multiplicative group for Zn, denoted with Z*n, is the subset of Setup: Example
elements of Zn relatively prime with n n = pq, with p and q Setup:
The totient function of n, denoted with φ(n), is the size of Z*n primes p = 7, q = 17
Example e relatively prime to n = 7⋅17 = 119
φ(n) = (p − 1) (q − 1) φ(n) = 6⋅16 = 96
Z*10 = { 1, 3, 7, 9 } φ(10) = 4
d inverse of e in Zφ(n) e=5
If p is prime, we have
Keys: d = 77
Z*p = {1, 2, …, (p − 1)} φ(p) = p − 1
Public key: KE = (n, e) Keys:
Euler’s Theorem public key: (119, 5)
Private key: KD = d
For each element x of Z*n, we have xφ(n) mod n = 1 private key: 77
Example (n = 10) Encryption: Encryption:
3φ(10) mod 10 = 34 mod 10 = 81 mod 10 = 1 Plaintext M in Zn M = 19
7φ(10) mod 10 = 74 mod 10 = 2401 mod 10 = 1 C = Me mod n C = 195 mod 119 = 66
9φ(10) mod 10 = 94 mod 10 = 6561 mod 10 = 1 Decryption: Decryption:
M = Cd mod n C = 6677 mod 119 = 19
6/8/2002 2:20 PM RSA Cryptosystem 3 6/8/2002 2:20 PM RSA Cryptosystem 4
Complete RSA Example Security
Setup: Encryption The security of the RSA In 1999, a 512-bit number was
cryptosystem is based on the factored in 4 months using the
p = 5, q = 11 C = M3 mod 55 widely believed difficulty of following computers:
n = 5⋅11 = 55 Decryption factoring large numbers
160 175-400 MHz SGI and Sun
φ(n) = 4⋅10 = 40 M = C27 mod 55 The best known factoring
algorithm (general number 8 250 MHz SGI Origin
e=3
field sieve) takes time 120 300-450 MHz Pentium II
d = 27 (3⋅27 = 81 = 2⋅40 + 1) exponential in the number of 4 500 MHz Digital/Compaq
bits of the number to be
factored Estimated resources needed to
M 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 factor a number within one year
The RSA challenge, sponsored
C 1 8 27 9 15 51 13 17 14 10 11 23 52 49 20 26 18 2 by RSA Security, offers cash Bits PCs Memory
M 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 prizes for the factorization of
430 1 128MB
C 39 25 21 33 12 19 5 31 48 7 24 50 36 43 22 34 30 16 given large numbers
In April 2002, prizes ranged 760 215,000 4GB
M 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
C 53 37 29 35 6 3 32 44 45 41 38 42 4 40 46 28 47 54 from $10,000 (576 bits) to 1,020 342×106 170GB
$200,000 (2048 bits) 1,620 1.6×1015 120TB
6/8/2002 2:20 PM RSA Cryptosystem 5 6/8/2002 2:20 PM RSA Cryptosystem 6
1
2. RSA Cryptosystem 6/8/2002 2:20 PM
Correctness Algorithmic Issues
We show the correctness of Thus, we obtain The implementation of Setup
the RSA cryptosystem for the (Me)d mod n = the RSA cryptosystem Generation of random
case when the plaintext M Med mod n = requires various numbers with a given
does not divide n Mkφ(n) + 1 mod n = number of bits (to generate
algorithms
Namely, we show that MMkφ(n) mod n = candidates p and q)
M (Mφ(n))k mod n = Overall Primality testing (to check
(Me)d mod n = M
M (Mφ(n) mod n)k mod n = Representation of integers that candidates p and q are
Since ed mod φ(n) = 1, there is of arbitrarily large size and
M (1)k mod n = prime)
an integer k such that arithmetic operations on
M mod n = Computation of the GCD (to
ed = kφ(n) + 1 them verify that e and φ(n) are
Since M does not divide n, by M Encryption relatively prime)
Euler’s theorem we have See the book for the proof of Modular power Computation of the
correctness in the case when multiplicative inverse (to
Mφ(n) mod n = 1 the plaintext M divides n Decryption compute d from e)
Modular power
6/8/2002 2:20 PM RSA Cryptosystem 7 6/8/2002 2:20 PM RSA Cryptosystem 8
Modular Power Modular Inverse
The repeated squaring Example Theorem Given positive integers a and b,
algorithm speeds up the 318 mod 19 (18 = 10010) Given positive integers a the extended Euclid’s algorithm
computation of a modular and b, let d be the smallest computes a triplet (d,i,j) such that
Q1 = 31 mod 19 = 3
power ap mod n d = gcd(a,b)
Q2 = (32 mod 19)30 mod 19 = 9 positive integer such that
Write the exponent p in binary d = ia + jb
Q3 = (92 mod 19)30 mod 19 = d = ia + jb
p = pb − 1 pb − 2 … p1 p0 To test the existence of and
81 mod 19 = 5 for some integers i and j.
Start with Q4 = (52 mod 19)31 mod 19 = We have compute the inverse of x ∈ Zn, we
Q1 = apb − 1 mod n (25 mod 19)3 mod 19 =
execute the extended Euclid’s
d = gcd(a,b) algorithm on the input pair (x,n)
Repeatedly compute 18 mod 19 = 18
Example Let (d,i,j) be the triplet returned
Qi = ((Qi − 1)2 mod n)apb − i mod n Q5 = (182 mod 19)30 mod 19 = a = 21
(324 mod 19) mod 19 = d = ix + jn
We obtain b = 15
17⋅19 + 1 mod 19 = 1 Case 1: d = 1
Qb = ap mod n d=3
i is the inverse of x in Zn
The repeated squaring p5 − 1 1 0 0 1 0 i = 3, j = −4
Case 2: d > 1
algorithm performs O (log p) 2 p5 − i 3 1 1 3 1 3 = 3⋅21 + (−4)⋅15 =
arithmetic operations 63 − 60 = 3 x has no inverse in Zn
Qi 3 9 5 18 1
6/8/2002 2:20 PM RSA Cryptosystem 9 6/8/2002 2:20 PM RSA Cryptosystem 10
Pseudoprimality Testing Randomized Primality Testing
The number of primes less than or equal to n is about n / ln n Compositeness witness function
witness(x, n) with error probability Algorithm RandPrimeTest(n, k)
Thus, we expect to find a prime among, O(b) randomly generated
q for a random variable x Input integer n,confidence
numbers with b bits each parameter k and composite
Case 1: n is prime
Testing whether a number is prime (primality testing) is believed witness function witness(x,n)
witness w(x, n) = false with error probability q
to be a hard problem Case 2: n is composite Output an indication of
An integer n ≥ 2 is said to be a base-x pseudoprime if witness w(x, n) = false with whether n is composite or prime
xn − 1 mod n = 1 (Fermat’s little theorem) probability q < 1 with probability 2−k
Composite base-x pseudoprimes are rare: Algorithm RandPrimeTest tests
whether n is prime by repeatedly t ← k/log2(1/q)
A random 100-bit integer is a composite base-2 pseudoprime with for i ← 1 to t
evaluating witness(x, n)
probability less than 10-13
A variation of base- x x ← random()
The smallest composite base-2 pseudoprime is 341
pseudoprimality provides a if witness(x,n)= true
Base-x pseudoprimality testing for an integer n: suitable compositeness witness return “n is composite”
Check whether xn − 1 mod n = 1 function for randomized primality return “n is prime”
Can be performed efficiently with the repeated squaring algorithm testing (Rabin-Miller algorithm)
6/8/2002 2:20 PM RSA Cryptosystem 11 6/8/2002 2:20 PM RSA Cryptosystem 12
2