rsSolutionHow RSA works (sorry if I use differ.pdf

•

0 j'aime•6 vues

rs Solution How RSA works (sorry if I use different variables from your class; the writing on the slide you included is too small to see): 1) Choose two distinct primes p and q. Let pq = n. 2) Calculate (n) = (pq) = (p - 1)(q - 1) (where is Euler\'s totient function, i.e., (n) = the number of positive integers less than or equal to n that are coprime to n) 3) Find an e such that gcd((n), e) = 1 4) Compute d such that de = 1 (mod (n)) (i.e., d the modular multiplicative inverse of e mod (pq)) e and n will be made public for people to encrypt messages with. d should remain private and will decrypt messages. (Obviously, (n) should also remain private as it would allow people to find d. For strong cryptography, n=pq should be sufficiently large that it is hard to factor and thus it is hard to guess (n).) To encrypt a message m (where m < n): 1) Compute c = me (mod n). This is the encrypted message. To decrypt c: 1) Compute cd (mod n) = (me)d (mod n) = (med) (mod n) =* m1 (mod n) = m. *this equality works because of Euler\'s Theorem, a^((p)) = 1 mod p (and here we have de = 1 (mod (n))) Your problem: Encrypt \"HOW ARE YOU\". First convert to an integer string as described. A = 00, B = 01, C = 02, D = 03, E = 04, F = 05, G = 06, H = 07, I = 08, J = 09, K = 10, L = 11, M = 12, N = 13, O = 14, P = 15, Q = 16, R = 17, S = 18, T = 19, U = 20, V = 21, W = 22, X = 23, Y = 24, Z = 25, space = 26. HOW ARE YOU = 07 14 22 26 00 17 04 26 24 14 20 = 0714222600170426241420 To encrypt, we need the message integer to be less than our chosen n, and n = 100. So we\'ll keep the message broken up into chunks of two. (Interestingly, n should be a product of two primes, and they\'ve given you an n that isn\'t. But we\'ll apply the RSA encryption algorithm anyway.) To encrypt each message chunk: c = me (mod n) where c is the encrypted message chunk, m is the original message chunk, e is the given value of 13, and n is the given value of 100. Encrypt each chunk: m = 07 c = 713 (mod 100) c = 7 m = 14 c = 1413 (mod 100) c = 44 m = 22 c = 2213 (mod 100) c = 52 m = 26 c = 2613 (mod 100) c = 76 m = 00 c = 013 (mod 100) c = 0 m = 17 c = 1713 (mod 100) c = 37 m = 04 c = 413 (mod 100) c = 64 m = 26 c = 2613 (mod 100) c = 76 m = 24 c = 2413 (mod 100) c = 24 m = 14 c = 1413 (mod 100) c = 44 m = 20 c = 2013 (mod 100) c = 0 Combine all the c\'s together into an encrypted message. For each m -> c chunk we have: 07 -> 07 14 -> 44 22 -> 52 26 -> 76 00 -> 00 17 -> 37 04 -> 64 26 -> 76 24 -> 24 14 -> 44 20 -> 00 So the whole encrypted message c is: 07 44 52 76 00 37 64 76 24 44 00 or 0744527600376476244400. c = 2613 (mod 100) c = 76.

Formation

*this equality works because of Euler's Theorem, a^((p)) = 1 mod p (and here we have de = 1
(mod (n)))
Your problem:
Encrypt "HOW ARE YOU". First convert to an integer string as described.
A = 00, B = 01, C = 02, D = 03, E = 04, F = 05, G = 06, H = 07, I = 08, J = 09, K = 10, L = 11,
M = 12, N = 13, O = 14, P = 15, Q = 16, R = 17, S = 18, T = 19, U = 20, V = 21, W = 22, X =
23, Y = 24, Z = 25, space = 26.
HOW ARE YOU = 07 14 22 26 00 17 04 26 24 14 20
= 0714222600170426241420
To encrypt, we need the message integer to be less than our chosen n, and n = 100. So we'll keep
the message broken up into chunks of two. (Interestingly, n should be a product of two primes,
and they've given you an n that isn't. But we'll apply the RSA encryption algorithm anyway.)
To encrypt each message chunk:
c = me (mod n)
where c is the encrypted message chunk, m is the original message chunk, e is the given value of
13, and n is the given value of 100. Encrypt each chunk:
m = 07
c = 713 (mod 100)
c = 7
m = 14
c = 1413 (mod 100)
c = 44
m = 22
c = 2213 (mod 100)
c = 52
m = 26
c = 2613 (mod 100)
c = 76
m = 00
c = 013 (mod 100)
c = 0
m = 17
c = 1713 (mod 100)
c = 37
m = 04

c = 413 (mod 100)
c = 64
m = 26
c = 2613 (mod 100)
c = 76
m = 24
c = 2413 (mod 100)
c = 24
m = 14
c = 1413 (mod 100)
c = 44
m = 20
c = 2013 (mod 100)
c = 0
Combine all the c's together into an encrypted message. For each m -> c chunk we have:
07 -> 07
14 -> 44
22 -> 52
26 -> 76
00 -> 00
17 -> 37
04 -> 64
26 -> 76
24 -> 24
14 -> 44
20 -> 00
So the whole encrypted message c is:
07 44 52 76 00 37 64 76 24 44 00
or
0744527600376476244400.
c = 2613 (mod 100)
c = 76

Contenu connexe

Similaire à rsSolutionHow RSA works (sorry if I use differ.pdf

Rsa

naniix21_3

The explosive growth of content-on-the-move, such as video streaming to mobile devices, has propelled research on multimedia broadcast and multicast schemes. Multi-rate transmission strategies have been proposed as a means of delivering layered services to users experiencing different downlink channel conditions. In this presentation, we consider random linear network coding for its inherent reliability features and study two encoding approaches, which are appropriate for layered services. We derive packet error probability expressions and use them as performance metrics in the formulation of resource allocation frameworks. The aim of these frameworks is both the optimization of the transmission scheme and the minimization of the number of broadcast packets on each downlink channel, while offering service guarantees to a predetermined fraction of users. Our proposed frameworks are adapted to the LTE stack and the integrated eMBMS technology. We focus on the delivery of a video service based on the H.264/SVC standard and demonstrate the advantages of layered network coding over multi-rate transmission. Furthermore, we establish that the choice of both the network coding technique and the resource allocation method play a critical role in the footprint of a service, as determined by the quality of each received video layer.

Talk on Resource Allocation Strategies for Layered Multimedia Multicast Services

Andrea Tassi

The Mathematics of RSA Encryption

Nathan F. Dunn

a digital code (generated and authenticated by public key encryption) which is attached to an electronically transmitted document to verify its contents and the sender's identity. This definition explains how digital signatures work and what they are used for. Learn about the mathematical underpinnings of digital signature technology A digital signature is basically a way to ensure that an electronic document (e-mail, spreadsheet, text file, etc.) is authentic. Authentic means that you know who created the document and you know that it has not been altered in any way since that person created it. https://signyourdoc.com/

Signyourd digital signature certificate provider

Kishankant Yadav

Divide-and-Conquer & Dynamic Programming Divide-and-Conquer: Divide a problem to independent subproblems, find the solutions of the subproblems, and then merge the solutions of the subproblems to the solution of the original problem. Dynamic Programming: Solve the subproblems (they may overlap with each other) and save their solutions in a table, and then use the table to solve the original problem. Example 1:　Compute Fibonacci number f(0)=0, f(1）=1， f(n)=f(n-1)+f(n-2) Using Divide-and-Conquer: F(n) = F(n-1) + F(n-2) F(n-2) + F(n-3) + F(n-3) + F(n-4) 　F(n-3)+F(n-4) + F(n-4)+F(n-5) + F(n-4)+F(n-5) + F(n-5) + F(n-6) 　　　　　　　……………………. Computing time:　T(n) = T(n-1) + T(n-2), T(1) = 0　 T(n)＝O(2　） Using Dynamic Programmin: Computing time=O(n) n Chapter 8 Dynamic Programming (Planning) F(0) F(1) F(2) F(3) F(4) …… F(n) Example 2 The matrix-train mutiplication problem * Structure of an optimal parenthesization Matrix Size A1 30×35 A2　　　　35×15 A3　　　　15×5 A4　　　　5×10 A5　　　　10×20 A6　　　　20×25 Input 6 5 3 2 4 1 3 3 3 3 3 3 3 3 5 1 2 3 4 5 i j 1 2 3 4 5 s[i,j] 6 5 3 2 4 i j m[i,j] 15125 10500 5375 3500 5000 0 11875 7125 2500 1000 0 9375 4375 750 0 7875 2625 0 15750 0 0 1 2 　3 4 5 　6 １ i-1 k j Matrix-Chain-Order(p) １　n := length[p] -1; 2 for i　= 1　to n 3 do m[i,i] := 0; 4 for l =2 to n 5 do {for i=1 to n-l +1 6 do { j := i+ l-1; 7 m[i,j] := ; 8 for k = i to j-1 9 do {q := m[i,k]+m[k+1,j] +p p p ; 10 if q < m[i,j] 11 then {m[i,j] :=q; s[i,j] := k}; }; }; 13 return m, s; Input of algorithm:　p , p , … , p 　　　　（The size of A = ｐ *ｐ） Computing time O(n ) 8 0 1 n i i+1 i 3 Example 3 Longest common subsequence (LCS) A problem from Bioinformatics: the DNA of one organism may be S1 = ACCGGTCGAGTGCGCGGAAGCCGGCCGAAA, while the DNA of another organism may be S2 = GTCGTTCTTAATGCCGTTGCTCTGTAAA. One goal of comparing two strands of DNA is to determine how “similar” the two strands are, as some measure of how closely related the two organisms are. Problem Formulization Given a sequence X = ( ), another sequence Z = ( ) is a subsequence of X if there exists a strictly increasing sequence ( ) of indices of X such that for all j = 1, 2, …k, we have . Theorem Let X = ( ) and Y = ( ...

Divide-and-Conquer & Dynamic ProgrammingDivide-and-Conqu.docx

jacksnathalie

Digital Signal Processing

aj ahmed

Rsa cryptosystem

Abhishek Gautam

Cryptanalysis Project Report

shahparin

DAA - UNIT 4 - Engineering.pptx

vaishnavi339314

machinelearning project

Lianli Liu

AJMS_476_23.pdf

BRNSS Publication Hub

Proof of Kraft Mc-Millan theorem - nguyen vu hung

Vu Hung Nguyen

We examine the effectiveness of randomized quasi Monte Carlo (RQMC) to improve the convergence rate of the mean integrated square error, compared with crude Monte Carlo (MC), when estimating the density of a random variable X defined as a function over the s-dimensional unit cube (0,1)^s. We consider histograms and kernel density estimators. We show both theoretically and empirically that RQMC estimators can achieve faster convergence rates in some situations. This is joint work with Amal Ben Abdellah, Art B. Owen, and Florian Puchhammer.

QMC: Transition Workshop - Density Estimation by Randomized Quasi-Monte Carlo...

The Statistical and Applied Mathematical Sciences Institute

Number theory

manikanta361

In this work, we describe advanced numerical tools for working with multivariate functions and for the analysis of large data sets. These tools will drastically reduce the required computing time and the storage cost, and, therefore, will allow us to consider much larger data sets or ner meshes. Covariance matrices are crucial in spatio-temporal statistical tasks, but are often very expensive to compute and store, especially in 3D. Therefore, we approximate covariance functions by cheap surrogates in a low-rank tensor format. We apply the Tucker and canonical tensor decompositions to a family of Matern- and Slater-type functions with varying parameters and demonstrate numerically that their approximations exhibit exponentially fast convergence. We prove the exponential convergence of the Tucker and canonical approximations in tensor rank parameters. Several statistical operations are performed in this low-rank tensor format, including evaluating the conditional covariance matrix, spatially averaged estimation variance, computing a quadratic form, determinant, trace, loglikelihood, inverse, and Cholesky decomposition of a large covariance matrix. Low-rank tensor approximations reduce the computing and storage costs essentially. For example, the storage cost is reduced from an exponential O(nd) to a linear scaling O(drn), where d is the spatial dimension, n is the number of mesh points in one direction, and r is the tensor rank. Prerequisites for applicability of the proposed techniques are the assumptions that the data, locations, and measurements lie on a tensor (axesparallel) grid and that the covariance function depends on a distance,...

$New data structures and algorithms for \\post-processing large data sets and ...$ $New data structures and algorithms for \\post-processing large data sets and ...$

New data structures and algorithms for \\post-processing large data sets and ...

Alexander Litvinenko

Lecture03 p1

aa11bb11

RSA final notation change2

Coleman Gorham

5. Summing Series.pptx

HassanShah396906

Hierarchical matrices for approximating large covariance matries and computin...

Alexander Litvinenko

In this paper, we consider an RSA modulus N=pq, where the prime factors p, q are of the same size. We present an attack on RSA when the decryption exponent d is in the form d=Md1+d0 where M is a given positive integer and d1 and d0 are two suitably small unknown integers. In 1999, Boneh and Durfee presented an attack on RSA when 0.292 d < N . When d=Md1+d0, our attack enables one to overcome Boneh and Durfee's bound and to factor the RSA modulus

A NEW ATTACK ON RSA WITH A COMPOSED DECRYPTION EXPONENT

ijcisjournal

Similaire à rsSolutionHow RSA works (sorry if I use differ.pdf (20)

Rsa

Talk on Resource Allocation Strategies for Layered Multimedia Multicast Services

The Mathematics of RSA Encryption

Signyourd digital signature certificate provider

Divide-and-Conquer & Dynamic ProgrammingDivide-and-Conqu.docx

Digital Signal Processing

Rsa cryptosystem

Cryptanalysis Project Report

DAA - UNIT 4 - Engineering.pptx

machinelearning project

AJMS_476_23.pdf

Proof of Kraft Mc-Millan theorem - nguyen vu hung

QMC: Transition Workshop - Density Estimation by Randomized Quasi-Monte Carlo...

Number theory

$New data structures and algorithms for \\post-processing large data sets and ...$ $New data structures and algorithms for \\post-processing large data sets and ...$

New data structures and algorithms for \\post-processing large data sets and ...

Lecture03 p1

RSA final notation change2

5. Summing Series.pptx

Hierarchical matrices for approximating large covariance matries and computin...

A NEW ATTACK ON RSA WITH A COMPOSED DECRYPTION EXPONENT

Plus de ALASEEMKOTA

Salad Toppings: Donald Wheeler is ordering a salad at a Ruby Tuesday restaurant. He can purchase a salad consisting of just lettuce, or he can add any of the following items: cucumber, onion, tomato, carrot, green pepper, olive, mushroom. How many different variations of a salad are possible? Solution There are 7 different topping choices (ignore lettuce, since he has to get that), so we need the number of subsets of a set with 7 elements: 2^n=2^7= 128 choices.

Salad Toppings Donald Wheeler is ordering a salad at a Ruby Tuesday.pdf

rsSolutionHow RSA works (sorry if I use differ.pdf

Recommandé

Recommandé

Contenu connexe

Similaire à rsSolutionHow RSA works (sorry if I use differ.pdf

Similaire à rsSolutionHow RSA works (sorry if I use differ.pdf (20)

Plus de ALASEEMKOTA

Plus de ALASEEMKOTA (20)

Dernier

Dernier (20)

rsSolutionHow RSA works (sorry if I use differ.pdf