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Table of contents
1.
Introduction
2.
Digital Signature Schemes
3.
SECURITY
4.
APPLICATIONS
5.
CLASSIFICATION
5.1.
Integer Factorization (IF)
5.2.
Discrete Logarithm (DL)
5.3.
Elliptic Curve (EC)
6.
Elliptic Curve Digital Signature Algorithm(ECDSA)
7.
Concepts Related to ECDSA
7.1.
Private Key
7.2.
Public Key
7.3.
Signature
8.
The Advantages and Disadvantages
9.
Implementation of ECDSA
10.
Frequently Asked Questions
10.1.
How Secure is Elliptic Curve Cryptography?
10.2.
What Is an Elliptic Curve Digital Signature?
10.3.
What is Elliptic Curve Cryptography Used For?
11.
Conclusion
Last Updated: Mar 27, 2024

# The Elliptic Curve DSA

Nilesh Kumar
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## Introduction

The Digital Signature Algorithm's (DSA) elliptic curve equivalent is the Elliptic Curve Digital Signature Algorithm (ECDSA) (DSA). It was approved as an ANSI standard in 1999 and as an IEEE and NIST standard in 2000. Additionally, it was approved as an ISO standard in 1998, and its inclusion in additional ISO standards is currently being discussed. There is no subexponential-time algorithm for the elliptic curve discrete logarithm problem, in contrast to the common discrete logarithm problem and the integer factorization problem. For this reason, an elliptic curve-based algorithm has a significantly higher strength-per-key-bit. The ANSI X9.62 ECDSA is described in this document, along with related security, implementation, and compatibility concerns.

## Digital Signature Schemes

Digital counterparts of handwritten signatures are provided via digital signature schemes (and more). A digital signature is information that relies on a secret (the signer's private key) and the information in the message being signed. Signatures must be independently verified, and if there is a disagreement over whether a particular entity signed a document, a neutral third party should be able to settle the issue amicably without needing access to the signer's private key. Disputes may occur when a signer tries to retract a signature it produced or when a forger makes a false claim.

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## SECURITY

In theory, a digital signature system should be impossible to fake using a chosen-message attack. Goldwasser, Micali, and Rivest introduced this idea of security. It says that an adversary who can get an entity's signatures for any messages of their choosing cannot successfully forge an entity's signature on even one additional note.

## APPLICATIONS

Three essential cryptographic services that digital signature schemes can offer are data integrity, data origin authentication (the proof that the source of data is as claimed), and non-repudiation (the assurance that an entity cannot deny prior actions or obligations). Commonly used as primitives in cryptographic protocols that also offer other services like entity authentication are digital signature techniques.

## CLASSIFICATION

The complex underlying mathematical issue that forms the basis of the current digital signature techniques can be used to categorize them:

### Integer Factorization (IF)

The security of integer factorization (IF) techniques is based on the intractability of the issue. The RSA and Rabin signature schemes are some prominent examples of this.

### Discrete Logarithm (DL)

Discrete Logarithm (DL) methods, which rely on intractability for security

Consider the finite field discrete logarithm problem (typical). Illustrative of these

include the Nyberg-Rueppel, DSA, Schnorr, and ElGamal signature systems.

### Elliptic Curve (EC)

Elliptic Curve (EC) techniques are based on the intractability of the discrete logarithm problem for elliptic curves.

This article will discuss Elliptic Curve Digital Signature Algorithm in Detail.

## Elliptic Curve Digital Signature Algorithm(ECDSA)

The ECDSA (Elliptic Curve Digital Signature Algorithm) is a cryptographically secure digital signature system (ECC) based on elliptic-curve cryptography. The cyclic groups of elliptic curves over finite fields and the difficulty of the ECDLP issue are the mathematical foundations of ECDSA (elliptic-curve discrete logarithm problem). The following describes how the EC point multiplication-based ECDSA sign/verify algorithm operates. For the same level of security, ECDSA keys and signatures are shorter than those of RSA. The security level of a 256-bit ECDSA signature is equivalent to that of a 3072-bit RSA signature.

Cryptographic elliptic curves (EC) over finite fields in the standard Weierstrass form are used by ECDSA. These curves' EC domain parameters are described by different cryptographic standards, including SECG: SEC 2 and Brainpool (RFC 5639).

For ECDSA to function, an elliptic curve must first be evaluated, then a point on the curve must be chosen. A new topic on the curve is produced by multiplying that point by another number. Even with the original issue, locating the ice on the curve is challenging. ECDSA is more resistant to current techniques for decryption breaking because of its complexity. ECDSA has several additional advantages besides being more secure against current attack techniques.

## Concepts Related to ECDSA

### Private Key

A private number that is only known to the person who created it. In essence, a private key is a number that was chosen at random. In Bitcoin, the funds can only be spent by someone with a private key. A private key in Bitcoin is a single unsigned 256-bit integer (32 bytes).

### Public Key

A number that can be revealed but still corresponds to a private key. A private key can be used to calculate a public key, but not the other way around. Without disclosing the private key, a public key can be used to check whether a signature is authentic (that is, made with the correct key). Public keys in Bitcoin can either be compressed or not. A 256-bit integer named x and a prefix of either 0x02 or 0x03 make up the 33 bytes that make up compressed public keys. The earlier uncompressed keys include a constant prefix (0x04), two 256-bit integers called x and y (2 * 32 bytes), and are 65 bytes long.

### Signature

A number that marks the completion of a signature operation. A hash of the object to be signed and a private key is mathematically combined to create a signature. The signature is made up of the two integers r and s. Without having access to the private key, a mathematical procedure using the public key on the signature can be used to establish that it was initially created from the hash and the private key. The resulting signatures are 73, 72, or 71 bytes long, with probabilities of approximately 25%, 50%, and 25%, respectively. However, sizes considerably smaller than that are achievable with exponentially diminishing possibilities.

## The Advantages and Disadvantages

The fact that ECDSA is so new compared to other public key cryptography is advantageous. Compared to the most widely used public key cryptography algorithm, RSA, standardized in 1995, ECDSA was just released in 2005. Hackers have had less time to study how to break ECDSA because it has only been available for a short while. This makes upgrading to ECDSA seem like a more appealing alternative every year, along with ECDSA's complexity. Because of these advantages, more recent protocols favor ECDSA over RSA.

## Implementation of ECDSA

ECDSA can be used any place RSA has been with the same result. Therefore it is wider than certificate signing. Everything from TLS/SSL to code signing uses public key cryptography techniques. The government uses ECDSA to safeguard internal communications and Tor to guarantee user anonymity. These are just a few applications that ECDSA can be used for. However, the rise of quantum computing poses a problem for all cryptosystems. All conventional cryptosystems, such as RSA, ECDSA, and AES, risk becoming outdated due to quantum computing. Due to the usage of quantum cryptography, formerly practical approaches like ECDSA will need to be updated or risk becoming obsolete.

## Frequently Asked Questions

### How Secure is Elliptic Curve Cryptography?

Elliptic curve cryptography may be susceptible to side-channel and twist-security attacks, among other flaws. Both types attempt to undermine the private key security provided by the ECC.

Side-channel attacks, such as differential power assaults, fault analysis, simple power attacks, and simple timing attacks, frequently cause information leaks. There are straightforward defenses against all varieties of side-channel attacks.

### What Is an Elliptic Curve Digital Signature?

ECC keys are used by an elliptic curve digital signature algorithm (ECDSA) to guarantee that each user is distinct and that each transaction is secure. Although this type of digital signature algorithm (DSA) provides a result that is functionally identical to that of other DSAs, it is more effective because it uses the more minor keys associated with ECC.

### What is Elliptic Curve Cryptography Used For?

One of the most popular methods for implementing digital signatures in cryptocurrencies is ECC. The Elliptic Curve Digital Signature Algorithm (ECDSA) is used to sign transactions in both Bitcoin and Ethereum. ECC is utilized outside of cryptocurrency, though. In the future, most web apps will utilize this encryption standard due to its effectiveness and reduced key length.

## Conclusion

In this article, we have extensively discussed the details of The Elliptic Curve DSA along with the details of Digital Signature Schemes, Classification, Concept Related to ECDSA, Advantages, Disadvantages, and Implementation of ECDSA.

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