This document provides an overview of elliptic curve cryptography (ECC). It begins with background on ECC, describing how it was independently proposed in 1985 as an approach to asymmetric cryptography. It then covers the basics of asymmetric cryptosystems and how ECC compares to RSA and Diffie-Hellman. The document goes on to explain elliptic curves over real and finite numbers, how points are added and doubled on elliptic curves, and how this relates to discrete logarithm problems. It discusses implementations of ECC for cryptography and comparisons to RSA in terms of key size and performance. Finally, it covers efficient implementations of ECC for smart cards.
The following slides explains about elliptic curves, their interpretation over Gallois finite fields, algorithms that reduces arithmetic computational requirements and primarly applications of the ECC.
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 Elliptic Curve Cryptography and the basis of elliptic curve arithmetics.
The following slides explains about elliptic curves, their interpretation over Gallois finite fields, algorithms that reduces arithmetic computational requirements and primarly applications of the ECC.
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 Elliptic Curve Cryptography and the basis of elliptic curve arithmetics.
Substitution cipher and Its CryptanalysisSunil Meena
Substitution Cipher
classical cipher and monoalphabetic and polyalphabetic cipher and its cryptanalysis . Correctness and security and learning analysis
Elliptic Curve Cryptography for those who are afraid of mathsMartijn Grooten
A low level introduction into elliptic curve cryptography, as presented at BSides San Francisco 2016.
NB don't be put off by the 100 slides; every transition is on its own slide.
Discrete Logarithmic Problem- Basis of Elliptic Curve CryptosystemsNIT Sikkim
ECC was developed in 1985 independently by Neal Koblitz and Victor Miller. Both men saw the application of the elliptic curve discrete log problem (ECDLP) as a replacement for the conventional discrete log problem (DLP) which is used in DSA, and the integer factorization problem found in RSA. For both problems, sub-exponential solutions have been generated; the
same which cannot be said for ECDLP . In addition to offering increased security for a smaller key size, operations of adding and doubling can be optimized successfully on a mobile
platform . ECC offers a viable replacement to the most common public-key cryptography algorithms on mobile devices.
Digital Signature Recognition using RSA AlgorithmVinayak Raja
• OBJECTIVE: Basically, the idea behind digital signatures is the same as your handwritten signature. You use it to authenticate the fact that you promised something that you can't take back later. A digital signature doesn't involve signing something with a pen and paper then sending it over the Internet. But like a paper signature, it attaches the identity of the signer to a transaction.
• PROBLEM SOLVED: Signer authentication , Message authentication, Non-repudiation , Message integrity
Introduction to Public key Cryptosystems with block diagrams
Reference : Cryptography and Network Security Principles and Practice , Sixth Edition , William Stalling
Gives a basic idea of Finite field theory and its uses in Elliptic cure cryptography. ECDLP and Diffie Helman key exchange and Elgamal Encryption with ECC.
Talk given at Devoxx UK 2014
Caveat - without the video these slides can be taken out of context, see Parleys for the full video.
RSA is the oldest kid in the public-key cryptography playground, and its position of toughest and fastest is under sharp competition from ECC (Elliptic Curve Cryptography). We look at the mathematical difference between the two cryptosystems, showing why ECC is faster and “harder” than RSA, but also very energy efficient hence its unique advantage in the mobile space. We show how to use ECC in your Java and Android applications. Before finally summarising the “state of the union” for RSA and ECC in the light of the Snowden leaks, and the likely near-future for public-key cryptography.
Substitution cipher and Its CryptanalysisSunil Meena
Substitution Cipher
classical cipher and monoalphabetic and polyalphabetic cipher and its cryptanalysis . Correctness and security and learning analysis
Elliptic Curve Cryptography for those who are afraid of mathsMartijn Grooten
A low level introduction into elliptic curve cryptography, as presented at BSides San Francisco 2016.
NB don't be put off by the 100 slides; every transition is on its own slide.
Discrete Logarithmic Problem- Basis of Elliptic Curve CryptosystemsNIT Sikkim
ECC was developed in 1985 independently by Neal Koblitz and Victor Miller. Both men saw the application of the elliptic curve discrete log problem (ECDLP) as a replacement for the conventional discrete log problem (DLP) which is used in DSA, and the integer factorization problem found in RSA. For both problems, sub-exponential solutions have been generated; the
same which cannot be said for ECDLP . In addition to offering increased security for a smaller key size, operations of adding and doubling can be optimized successfully on a mobile
platform . ECC offers a viable replacement to the most common public-key cryptography algorithms on mobile devices.
Digital Signature Recognition using RSA AlgorithmVinayak Raja
• OBJECTIVE: Basically, the idea behind digital signatures is the same as your handwritten signature. You use it to authenticate the fact that you promised something that you can't take back later. A digital signature doesn't involve signing something with a pen and paper then sending it over the Internet. But like a paper signature, it attaches the identity of the signer to a transaction.
• PROBLEM SOLVED: Signer authentication , Message authentication, Non-repudiation , Message integrity
Introduction to Public key Cryptosystems with block diagrams
Reference : Cryptography and Network Security Principles and Practice , Sixth Edition , William Stalling
Gives a basic idea of Finite field theory and its uses in Elliptic cure cryptography. ECDLP and Diffie Helman key exchange and Elgamal Encryption with ECC.
Talk given at Devoxx UK 2014
Caveat - without the video these slides can be taken out of context, see Parleys for the full video.
RSA is the oldest kid in the public-key cryptography playground, and its position of toughest and fastest is under sharp competition from ECC (Elliptic Curve Cryptography). We look at the mathematical difference between the two cryptosystems, showing why ECC is faster and “harder” than RSA, but also very energy efficient hence its unique advantage in the mobile space. We show how to use ECC in your Java and Android applications. Before finally summarising the “state of the union” for RSA and ECC in the light of the Snowden leaks, and the likely near-future for public-key cryptography.
Elliptic Curve Cryptography and Zero Knowledge ProofArunanand Ta
Elliptic Curve Cryptography and Zero Knowledge Proof
Presentation by Nimish Joseph, at College of Engineering Cherthala, Kerala, India, during Faculty Development Program, on 06-Nov-2013
Zero to ECC in 30 Minutes: A primer on Elliptic Curve Cryptography (ECC)Entrust Datacard
As both standalone and networked computing capabilities continue to grow in-line with Moore’s law, key sizes for the most widely used public-key cryptographic systems have to grow disproportionately fast. This trend makes a switch to elliptic-curve cryptography (ECC) more and more attractive.
Unfortunately, ECC has a reputation for being difficult to understand. And this reputation, deserved or not, deters many from exploring the principles on which it is based.
The basic principles, on the other hand, are easily understood by anyone who studied mathematics through high school. And a wider understanding of the basics will result in a wider circle of informed discussion. This white paper dispels the myth that knowledge of ECC is out of reach to all but the mathematical elite.
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 ...
Low Power FPGA Based Elliptical Curve CryptographyIOSR Journals
Abstract: Cryptography is the study of techniques for ensuring the secrecy and authentication of the information. The development of public-key cryptography is the greatest and perhaps the only true revolution in the entire history of cryptography. Elliptic Curve Cryptography is one of the public-key cryptosystem showing up in standardization efforts, including the IEEE P1363 Standard. The principal attraction of elliptic curve cryptography compared to RSA is that it offers equal security for a smaller key-size, thereby reducing the processing overhead. As a Public-Key Cryptosystem, ECC has many advantages such as fast speed, high security and short key. It is suitable for the hardware of implementation, so ECC has been more and more focused in recent years. The hardware implementation of ECC on FPGA uses the arithmetic unit that has small area, small storage unit and fast speed, and it is an extremely suitable system which has limited computation ability and storage space.[1][2] The modular arithmetic division operations are carried out using conditional successive subtractions, thereby reducing the area. The system is implemented on Vertex-Pro XCV1000 FPGA. Index Terms – VHDL, FSM, FPGA, Elliptic Curve Cryptography.
Low Power FPGA Based Elliptical Curve CryptographyIOSR Journals
Cryptography is the study of techniques for ensuring the secrecy and authentication of the
information. The development of public-key cryptography is the greatest and perhaps the only true revolution in
the entire history of cryptography. Elliptic Curve Cryptography is one of the public-key cryptosystem showing
up in standardization efforts, including the IEEE P1363 Standard. The principal attraction of elliptic curve
cryptography compared to RSA is that it offers equal security for a smaller key-size, thereby reducing the
processing overhead. As a Public-Key Cryptosystem, ECC has many advantages such as fast speed, high
security and short key. It is suitable for the hardware of implementation, so ECC has been more and more
focused in recent years. The hardware implementation of ECC on FPGA uses the arithmetic unit that has small
area, small storage unit and fast speed, and it is an extremely suitable system which has limited computation
ability and storage space.[1][2] The modular arithmetic division operations are carried out using conditional
successive subtractions, thereby reducing the area. The system is implemented on Vertex-Pro XCV1000 FPGA
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.
RSA and OAEP
Diffe-Hellman Key Exchange and its Security Aspects
Model of Asymmetric Key Cryptography
Factorization and other methods for Public Key Cryptography
Ijcatr03051008Implementation of Matrix based Mapping Method Using Elliptic Cu...Editor IJCATR
Elliptic Curve Cryptography (ECC) gained a lot of attention in industry. The key attraction of ECC over RSA is that it
offers equal security even for smaller bit size, thus reducing the processing complexity. ECC Encryption and Decryption methods can
only perform encrypt and decrypt operations on the curve but not on the message. This paper presents a fast mapping method based on
matrix approach for ECC, which offers high security for the encrypted message. First, the alphabetic message is mapped on to the
points on an elliptic curve. Later encode those points using Elgamal encryption method with the use of a non-singular matrix. And the
encoded message can be decrypted by Elgamal decryption technique and to get back the original message, the matrix obtained from
decoding is multiplied with the inverse of non-singular matrix. The coding is done using Verilog. The design is simulated and
synthesized using FPGA.
Design and Implementation of Variable Radius Sphere Decoding Algorithmcsandit
Sphere Decoding (SD) algorithm is an implement deco
ding algorithm based on Zero Forcing
(ZF) algorithm in the real number field. The classi
cal SD algorithm is famous for its
outstanding Bit Error Rate (BER) performance and de
coding strategy. The algorithm gets its
maximum likelihood solution by recursive shrinking
the searching radius gradually. However, it
is too complicated to use the method of shrinking t
he searching radius in ground
communication system. This paper proposed a Variabl
e Radius Sphere Decoding (VR-SD)
algorithm based on ZF algorithm in order to simplif
y the complex searching steps. We prove the
advantages of VR-SD algorithm by analyzing from the
derivation of mathematical formulas and
the simulation of the BER performance between SD an
d VR-SD algorithm.
Implementation of Elliptic Curve Digital Signature Algorithm Using Variable T...ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Implementation of Elliptic Curve Digital Signature Algorithm Using Variable T...ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Public-Key Cryptography.pdfWrite the result of the following operation with t...FahmiOlayah
Write the result of the following operation with the correct number of significant figure of 0.248?Write the result of the following operation with the correct number of signi
Securing the Data in Big Data Security Analytics by Kevin Bowers, Nikos Triandopoulos of RSA Laboratories and catherine Hart and Ari Juels of Bell Canada
IBM Security Strategy Intelligence, Integration and Expertise
by Marc van Zadelhoff, VP, WW Strategy and Product Management and Joe Ruthven IBM MEA Security Leader
Search and Society: Reimagining Information Access for Radical FuturesBhaskar Mitra
The field of Information retrieval (IR) is currently undergoing a transformative shift, at least partly due to the emerging applications of generative AI to information access. In this talk, we will deliberate on the sociotechnical implications of generative AI for information access. We will argue that there is both a critical necessity and an exciting opportunity for the IR community to re-center our research agendas on societal needs while dismantling the artificial separation between the work on fairness, accountability, transparency, and ethics in IR and the rest of IR research. Instead of adopting a reactionary strategy of trying to mitigate potential social harms from emerging technologies, the community should aim to proactively set the research agenda for the kinds of systems we should build inspired by diverse explicitly stated sociotechnical imaginaries. The sociotechnical imaginaries that underpin the design and development of information access technologies needs to be explicitly articulated, and we need to develop theories of change in context of these diverse perspectives. Our guiding future imaginaries must be informed by other academic fields, such as democratic theory and critical theory, and should be co-developed with social science scholars, legal scholars, civil rights and social justice activists, and artists, among others.
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
"Impact of front-end architecture on development cost", Viktor TurskyiFwdays
I have heard many times that architecture is not important for the front-end. Also, many times I have seen how developers implement features on the front-end just following the standard rules for a framework and think that this is enough to successfully launch the project, and then the project fails. How to prevent this and what approach to choose? I have launched dozens of complex projects and during the talk we will analyze which approaches have worked for me and which have not.
4. Asymmetric Cryptosystems
Two mathematically related keys
Public key for encryption
Private key for decryption
Private key can not be easily deduced
from the public key
Security depends on a mathematical function
whose inverse is difficult to calculate
6. Elliptic Curves
Elliptic curves are not
ellipses (the name
comes from elliptic
integrals)
Circle
x2 + y2 = r2
Ellipsis
a·x2 + b·y2 = c
Elliptic curve
y2 = x3 + a·x + b
7. Elliptic Curves Over Real Numbers
An elliptic curve over reals is the set of points
(x,y) which satisfy the equation y2 = x3 + a·x + b,
where x, y, a, and b are real numbers
If 4·a3 + 27·b2 is not 0 (i.e. x3 + a·x + b contains no
repeated factors), then the elliptic curve can be
used to form a group
An elliptic curve group consists of the points on
the curve and a special point O
Elliptic curves are additive groups
Addition can be defined geometrically or algebraically
8. Adding Points P and Q
Draw a line that intersects
distinct points P and Q
The line will intersect a
third point -R
Draw a vertical line
through point -R
The line will intersect a
fourth point R
Point R is defined as the
summation of points P
and Q
R=P+Q
9. Adding Points P and -P
Draw a line that
intersects points P
and -P
The line will not
intersect a third point
For this reason,
elliptic curves include
O, a point at infinity
P + (-P) = O
O is the additive
identity
10. Doubling the Point P
Draw a line tangent to
point P
The line will intersect a
second point -R
Draw a vertical line
through point -R
The line will intersect a
third point R
Point R is defined as the
summation of point P with
itself
R = 2·P
11. Doubling the Point P if yP = 0
Draw a line tangent to
point P
If yP = 0, the line will
not intersect a second
point
2·P = O when yP = 0
3·P = P (2·P + P)
4·P = O (2·P + 2·P)
5·P = P (2·P + 2·P + P)
12. Algebraic Approach
Point Addition
R=P+Q
s = (yP – yQ) / (xP – xQ)
xR = s2 – xP – xQ
yR = -yP + s(xP – xR)
Point Doubling
R = 2·P
s = (3·xP2 + a) / (2·yP)
xR = s2 – 2·xP
yR = -yP + s(xP – xR)
13. Cryptography with Elliptic Curves
Calculations with real numbers are slow
and rounding causes inaccuracy
Speed and accuracy are important for
cryptography
Use elliptic curve groups over the finite
field Fp *
Elliptic curves are formed by choosing a
and b within the field Fp
y2 mod p = x3 + a·x + b mod p
* can also use F2m, but I’m skipping it
14. Cryptography with Elliptic Curves
Because it’s a finite field, a finite number
of points make up the curve
This means there is no true curve anymore
But also no more rounding
Geometric definitions of addition and
doubling don’t work on these curves
Algebraic definitions still hold
15. The Discrete Logarithm
Problem
The discrete logarithm problem for ECC is
the inverse of point multiplication
Point multiplication is simply calculating
Q=kP, where k is an integer and P is a
point on the curve
16. Elliptic Curve Discrete Logarithm
Given points P and Q, find a number k
such that k·P = Q
P is the base point on a specific, published
curve
Q is the public key
k is the private key (very large prime number)
With doubling, we can go from P to 2·P
With addition, we can go from 2·P to 3·P
17. The Discrete Logarithm
Problem
Determining the point k·P in this way is
referred to as the scalar multiplication of a
point
Scalar multiplication is intractable
Elliptic Curve Discrete Logarithm Problem
k is the discrete logarithm of Q to the base P
Brute force attacks range up to 3x10 57
operations by a stepping process
Applies to NIST-defined P192 curve
18. Attacking ECC
ECC is not susceptible to index-calculus attacks
Index-calculus relies on group properties that ECC
groups do not have
Brute force does not fair well either as shown
Best possible way is a ‘collision attack’ known as
Pollard’s rho attack
As field size increases, the attack becomes harder at
an exponential rate
19. Security Performance
Implementation allows for a significant
reduction in key size
ECC key of 163 bits is equivalent to RSA key
of 1024 bits
ECC key of 256 bits is equivalent to RSA key
of 3072 bits
ECC’s main advantage: as key length
increases, so does the difficulty of the
inversion process
20. Performance Analysis - Speed
ECC performance is dependent on field
operations
Arithmetic involved in ECC
Algorithmic Level (addition and subtraction
chains)
Curve Arithmetic Level (selection of
coordinate representation)
Field Arithmetic Level (basis selection,
multiplier and inverter structures)
21. Performance Analysis - Speed
How can ECC performance increase?
Increase efficiency of finite field mathematics
The performance of ECC relies heavily on the
speed of the computations in the finite field
Use particular finite fields and elliptic curves
where applicable
Implementing the right field representation
22. Representations
Types of representations for elements in a
finite field
Normal Basis
Takes the form {1, α, α2,…, αn-1}
Type I and Type II representations optimized for N
Polynomial Basis
Takes the form {α, α2, α2^2,…, α2^(n-1)}
α is a root of an irreducible polynomial f(x)
that has a degree N in a field
23. Which is better?
PB does inversion 10% faster
NB does scalar multiplication 12% faster
Both perform basic addition and subtraction
efficiently
Performance depends on implementation
Ex. ElGamel protocol - encryption using EC runs 22%
faster when combined with NB rather than PB
Using other protocols may show different results as
well
Performance is also related to hardware design
24. Performance Comparison
Key sizes for EC using PB are 155 and 183
respectively
Key sizes for EC using NB are 155 and 173
respectively
26. Elliptic Curve Cryptosystems (ECC)
Merits:
A 160 bit ECC has roughly the same security
as 1024 bit RSA.
Limited memory and computational power.
Purpose:
Algorithms to achieve optimized
implementation of the ECDSA over the field
GF(p) on smart cards.
Algorithms for modular reduction, modular
inversion and scalar multiplication.
27. Discrete Logarithm Problem
Based on the difficulty of elliptic curve discrete
logarithm problem (DLP).
DLP applies to mathematical structures called
groups.
For higher security the rate of increase key size
is much slower for RSA key sizes.
Faster implementation using less bandwidth and
power- crucial for smart cards.
IEEE Std 1363-2000, WAP (Wireless
Application Protocol), ANSI X9.62, ANSI X9.63
and ISO CD 14888-3) employs ECC.
28. Elliptic curve over a Galois field
with p elements
E : y2 = x3 + ax + b (mod p)
Addition and doubling of points are the group
operations along with the identity element.
Definition ECDLP:
Given the prime modulus p, the curve constants a
and b and two points P and Q, find a scalar k such
that Q = kP
Efficient Field Arithmetic in crypto coprocessor.
Effect of coordinate systems on speed of the
scalar multiplication operations.
29. Smart Card Hardware
Motorola M-Smart JupiterTM smart card based on Java
CardTM 2.1 technology and an ARM processor with a
word size of 32 bits, 64KB of ROM,32KB of EEPROM,
3KB RAM and a modular arithmetic coprocessor (crypto
coprocessor).
32. Modular arithmetic of GF(p)
Modular Addition and Subtraction.
Modular Reduction (multiplication) algorithms:
Barrett reduction.
Montgomery reduction.
NIST primes by Brown et al., very fast (6% and 33%)
but specialized reduction algorithm.
Pseudo-Mersenne prime.
Modular Inversion (Division)
Binary extended GCD (BEGCD) algorithm
Extended Euclidean algorithm (EEA)
Exponentiation method (Fermat’s little theorem)
33. Scalar multiplication
Basic crypto operation of an ECC.
Series of point addition and doubling.
Binary method due to no pre-computation
phase .
Faster processing when using signed
representation of the scalar value.
34. Point coordinates and Scalar
Multiplication
Addition and Doubling
Affine - a point is represented as (xA, yA).
Projective - (X, Y,Z) where xA = XZ−1 and yA = Y
Z−1.
Jacobian, Modified Jacobian and Chudnovsky
Jacobian.
Issue of Temporary variables required by
each algorithm.
Mixed coordinate multiplication.
35. Background References
Elliptic Curve Cryptography at the Wikipedia
http://en.wikipedia.org/wiki/Elliptic_curve_cryptography
http://en.wikipedia.org/wiki/Elliptic_curves
Elliptic curve cryptography FAQ by George Barwood
http://www.cryptoman.com/elliptic.htm
Elliptic Curve Cryptography according to Steven
Galbraith
http://www.isg.rhul.ac.uk/~sdg/ecc.html
An Elliptic Curve Cryptography (ECC) Primer by certicom
http://www.deviceforge.com/articles/AT4234154468.html
Online Elliptic Curve Cryptography Tutorial by certicom
http://www.certicom.com/index.php?action=ecc_tutorial,home
36. Performance References
Bednara, M. et. al. “Tradeoff Analysis of
FPGA Based Elliptic Curve Cryptography.”
Circuits and Systems, 29 May 2002.
Qizhi, Qui “Research on Elliptic Curve
Cryptography.” Computer Supported
Cooperative Work in Design. 26 May 2004
37. Application References
Implementing an efficient elliptic curve cryptosystem over GF(p) on
a smart card, Yvonne Hitchcock, Edward Dawson, Andrew Clark,
Paul Montague, October 2002.
THE ELLIPTIC CURVE CRYPTOSYSTEM FOR SMART CARDS, A
Certicom White Paper, Published: May 1998
Editor's Notes
Graph from An intro to Elliptical Curve Cryptography at http://www.deviceforge.com/articles/AT4234154468.html, which is a reproduction of An Elliptic Curve Cryptography (ECC) Primer by Certicom.