Report project fundamentals of information security digital signature generation algorithm using rsa

MINISTRY OF INFORMATION AND COMMUNICATIONS POSTS AND TELECOMMUNICATIONS INSTITUTE OF TECHNOLOGY REPORT PROJECT Fundamentals of Information Security DIGITAL SIGNATURE GENERATION ALGORITHM

MINISTRY OF INFORMATION AND COMMUNICATIONS POSTS AND TELECOMMUNICATIONS INSTITUTE OF

TECHNOLOGY

REPORT PROJECT Fundamentals of Information Security

DIGITAL SIGNATURE GENERATION ALGORITHM USING

RSA

Class: E21CQCN03-B Group: 03

Group of student: Mã sinh viên

Lê Minh Tiến B21DCCN704 Dương Kim Tùng B21DCDT235 Phan Hoàng Trung B21DCCN735 Nguyễn Văn Thắng B21DCDT206

Hà Nội – 2024

I Introduction 3

II Description 5

1 Architecture 5

1.1 Idea 5

1.2 Architecture 6

2 Algorithm 7

2.1 Key generation, digital signature creation, and signature verification 7

2.2 Use a hash function to map data into fixed-size values 9

3 Weakness 12

3.1 Execution speed: 12

3.2 Cost 12

3.3 Performance 12

4 Attacker 12

4.1 Find the private key 12

4.2 Forgery of signatures (without directly computing the private key 13

5 Applications 14

III Conclusion 15

IV References 16

I Introduction

With so many articles being published that highlight how important encryption is nowadays, we must stay aware of every possible route to enforce such standards The RSA algorithm has been a reliable source of security since the early days of computing, and it keeps solidifying itself as a definitive weapon in the line of cybersecurity

Before moving forward with the algorithm, let’s get a refresher on asymmetric encryption since it verifies digital signatures according to

asymmetric cryptography architecture, also known as public-key cryptography architecture

The first question is: “ What does a digital signature mean and how does the RSA algorithm relate to it? ”

A digital signature is a cryptographic technique used to validate the authenticity and integrity of digital messages, documents, or transactions It functions similarly to a handwritten signature or a stamped seal on a physical document but in the digital realm

Also, the RSA algorithm is a widely used asymmetric cryptographic algorithm, named after its inventors Ron Rivest, Adi Shamir, and Leonard Adleman It is used for secure data transmission, digital signatures, and encryption of sensitive information over networks RSA is based on the mathematical properties of large prime numbers and the difficulty of factoring the product of two large prime numbers

For many of those questions, we should continue with: “How does Asymmetric Encryption work in digital signature with RSA algorithm ?”

In Asymmetric Encryption algorithms, you use two different keys, one for encryption and the other for decryption The key used for encryption is the

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public key, and the key used for decryption is the private key But, of course, both the keys must belong to the receiver

RSA encryption relies on a few basic assets and quite a bit of math These elements are required:

 A public key (e)

 A private key (d)

 Two prime numbers (p and q), multiplied (n)

The RSA algorithm for creating digital signatures also involves using a hash function

A hash function is a mathematical function that converts a message of any length into a fixed-length sequence of bits This sequence of bits is called a message digest or hash value, which represents the original message

In some cases, the information being sent may be encrypted or vulnerable to unsafe values In practical RSA implementations, there's often a random transformation involved, such as:

 Utilizing standards like PKCS (Public Key Cryptography Standards)

 PKCS standards also incorporate additional features to ensure the security

of RSA signatures, such as the Probabilistic Signature Scheme for RSA (RSA-PSS)

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II Description

1 Architecture

1.1 Idea

Example: Alice has two keys, one public key, and one private key Alice provides her public key to Bob but keeps her private key for herself When Alice wants to transfer documents to Bob, she can sign these documents using her private key and send them to Bob Bob can then use Alice's public key to verify that the documents she received were indeed sent by Alice

1.2 Architecture

Signing Phase:

 Compute the message digest (hash value) of the message using a SHA-1 hashing algorithm

 Sign the message digest using the sender's private key and the RSA signature generation algorithm The result obtained is the digital signature

S of the message

 The original message is concatenated with the digital signature to M S

form the signed message S

 The signed message + is sent to the recipient.S M

Verification Phase:

 Separate the RSA digital signature and the original message from the signed message for individual processing

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 Compute the message digest of the original message using the hashing D

algorithm SHA-1 used during signing

 Use the sender's public key to decrypt the RSA digital signature, we obtained the message digest

 Compare and D :

o If , the signature verification is successful The message ensures integrity and genuinely originates from the sender (since the public key is authenticated)

o If , the signature is invalid The message may have been tampered with or does not genuinely originate from the sender

2 Algorithm

2.1 Key generation, digital signature creation, and signature verification Schema

Key Generation:

 Consider two prime numbers and p q

 Compute ,

 Choose e such that

 Calculate d such that mod

 Public Key ; Private Key

Digital signature creation:

 Convert the message into the integer such that P M

 Compute

 Send and to BobS M

Verification:

 Correctly authenticate Alice's public key as and , S M

 Check mod n

 Confirm Alice’s signature

Example for a key generation:

Generating public key:

 Select two prime no's Suppose = 53 and = 59.p q

 Now the first part of the Public key: = n p q = 3127

 We also need a small exponent say e

 But Must be e

o An integer

o Not be a factor of n

o

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 Let us now consider to be equal to 3e

 The public key has been made of and n e

Generating private key:

 We need to calculate

o Such that so, Φ(n) = 3016

 Now calculate : d

o for some integer x

o For = 2, the value of is 2011x d

 The private key has been made of d

2.2 Use a hash function to map data into fixed-size values

In practice, to create a digital signature using RSA, it's common to use a hash function of the data instead of directly using the data itself

For example: Alice wants to send Bob a document with her signature To do this, Alice generates a hash value of the document to be signed and computes its value mod The final value is the electronic signature of the documentn

under consideration When Bob receives the document along with the electronic signature, he calculates mod of the signature while also computing n

the hash value of the document If these two values are equal, Bob knows that the signer knows Alice's private key and the document has not been altered since signing This approach offers many benefits such as:

 Hash functions are one-way functions, so even if you have the hash, you cannot determine the original message

 The hash length is fixed and usually very small, so it won't take up much space

 The hash value can also be used to verify whether the received message is intact or not

2.3 Install algorithm in Java

Using BigInteger in the java.math.* package provides most constructors and arithmetic functions to conveniently manipulate large integers

Some functions include:

Constructor BigInteger(int bitLength, int certainty, Random rnd): Generate a random prime integer with the specified bit length

Method BigInteger add(BigInteger val): Add two large integers

Method BigInteger subtract(BigInteger val): Subtract two large integers Method BigInteger multiply(BigInteger val): Multiply two large integers Method gcd(BigInteger val): Find the greatest common divisor (GCD) of two large integers

Method mod(BigInteger m): Calculate the modulo (remainder) of integer division

Method BigInteger modInverse(BigInteger m): Calculate the modular

multiplicative inverse (this^-1 mod m)

Method BigInteger modPow(BigInteger exponent, BigInteger m): Calculate (this^exponent mod m)

For example, RSA encryption can be implemented using these functionalities

Generating keys:

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Encryption: (This code is used in RSA, while in digital signatures, d is privately encrypted)

Decryption: (This code is used in RSA, while in digital signatures, e is publicly encrypted)

Demo’s interface:

3 Weakness

3.1 Execution speed:

The execution speed of the RSA algorithm is one of its weaknesses compared to symmetric-key cryptography systems

According to estimates, RSA key generation is slower by more than 100 times compared to the symmetric-key DES system when performed using software Additionally, it is slower by more than 1000 times compared to DES when performed using hardware

→ This method also introduces new security challenges One example is the necessity of generating truly random symmetric keys Otherwise, attackers may bypass RSA and focus on guessing the symmetric keys

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3.2 Cost

To perform the RSA algorithm, the majority of the computational cost lies

in basic operations such as key generation, encryption, and decryption

The process of signing and verifying signatures is equivalent to the cost of performing exponentiation modulo of n

To ensure the security of the private key, it is common to choose a public exponent much smaller than the secret exponent e d

3.3 Performance

Due to the relatively slow speed of RSA, it is not commonly used for signing large volumes of data Instead, RSA is often used in conjunction with hashing algorithms to enhance flexibility in signing and verifying signatures

4 Attacker

4.1 Find the private key

One of the values , or is leaked:p q, n

 During the key generation process, an attacker can easily calculate the private key using the formula: mod

 Knowing the private key, an attacker can forge the user's signature

 Solution: Ensure the secrecy of p q,, and during the key generation n

process

Attack based on the public key and of the signer:n e

 The attacker will attempt to factorize the value of into two prime n

factors, and From there, they can calculate p q and ultimately compute the private key d

 Solution: To prevent this attack, it's essential to choose prime numbers p

and large enough to make factoring into the product of two prime q n

factors computationally infeasible in real-time In practice, large numbers (at least 100 digits) are typically generated and then checked for primality

Using small prime factors in or :

 If we are careless in selecting parameters and such that p q or

has small prime factors, the security of the signature scheme becomes compromised When or has small prime factors,

we can easily use Pollard's algorithm to factorize the modulo of into its n

prime factors

 Solution: Choose parameters p and q so that and have large prime factors

4.2 Forgery of signatures (without directly computing the private key

We have: With being a valid signature of , it's possible to S m

construct a pair of valid messages/signatures without knowing the private key

With two signatures, we can create a third signature without knowing the private key

Solution: Encrypt first, sign later

 First, encrypt using an encryption function m

 Then sign

 Where the encryption function mentioned has two types:

o Encodings for this type of attack (Ad-hoc encodings)

 PKCS#1 v1.5, ISO 9796-1, ISO 9796-2

 Designed to prevent specific attacks but may reveal some weaknesses

o Encodings that ensure security (Provably secure encodings)

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 RSA-FDH, RSA-PSS

 Proven to be secure under defined assumptions

5 Applications

Used to ensure data integrity: Signatures, official documents, files, sent over the Internet environment of individuals, organizations, etc Specifically, in: Communication (Email, SMS): When writing an email, we do not need to trust the email provider regarding privacy and forgery We can encrypt the email with the recipient's public key and sign it alongside This way, the sender is assured that there is no forgery, and the message is from the right sender The only issue that may occur is if the provider doesn't deliver the message at all Open-source contributions: Many codes are written in the form of open-source by various individuals They may be users of that open-source code or may be paid for those contributions Maintainers of projects need to ensure that all contributions are useful, but they may lack time to check each contribution They need to be able to trust some people Usually, the first few people are thoroughly checked, but over time, even anyone can become a contributor Everyone trusts each other But they need to be sure it's the same person They need to ensure that each person's contribution doesn't change For this reason, every contribution needs to be signed

Software updates: Think about Smart TVs / Alexa / FritzBox All these devices need updates Suppose we can plug in a USB with an update file into the device As a manufacturer, they want to ensure that the update file is not tampered with They want to ensure that the device will continue to operate So, they share the company's public key in the device When the device finds the update file, it verifies that the company is the source by checking the signature

of the update

Digital certificates: When applying for a job, potential new employers of the candidate may want to see the candidate's reference letters and diplomas

Especially when the coronavirus is raging worldwide, those documents are being delivered digitally How do labor users verify if a diploma is valid? Digital signatures can help The candidate will need a digitally signed version of the diploma and may share the public key in a trustworthy manner with the candidate's employer

Cryptocurrency: To prove that someone owns Bitcoin, the system uses asymmetric cryptography From the beginning, someone is assured to be the legitimate owner of the coin Then, the legitimate owner is identified as the owner of the private key, corresponding to a certain public key Please note that digital signatures only prove ownership at a specific point in time They do not solve the problem where the owner can spend the coin twice - the double-spending problem

III Conclusion

The report has introduced the architecture, implementation of the algorithm, weaknesses, and various types of attacks on digital signatures using the RSA algorithm

 Creating and verifying digital signatures

 Implementation of experiments and verification of digital signatures to ensure data integrity

IV References

https://www.cs.cornell.edu/courses/cs5430/2015sp/notes/rsa_sign_vs_dec php https://geeklaunch.net/blog/what-does-my-rsa-public-key-actually-mean/ http://www2.lawrence.edu/fast/GREGGJ/CMSC510/Ch31/RSA.html

https://cp-algorithms.com/algebra/module-inverse.html

https://www.geeksforgeeks.org/rsa-and-digital-signatures/

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