> For the complete documentation index, see [llms.txt](https://satoshichain.gitbook.io/satoshi-chain/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://satoshichain.gitbook.io/satoshi-chain/background/3.2-digital-signatures/3.2.2-ecdsa-signature-algorithm.md).
# 3.2.2 ECDSA Signature Algorithm
### ECDSA Signature Algorithm
Elliptic Curve Digital Signature Algorithm (ECDSA) is a cryptographic algorithm for creating digital signatures. More concretely,\
\
**Setup**
```
● Public Parameters: Let 𝐹 be a finite field, two parameters 𝑎 and 𝑏 define an elliptic
𝑞
curve 𝐶 over 𝐹 , a seed which validates 𝐶, a prime integer
𝑞
𝐺∈ 𝐶 of order 𝑛 where 𝑞 is either prime or a power of 2.
● Private Key: An integer 𝑑 in [1, 𝑛 − 1].
● Public Key: 𝑄 = 𝑑𝐺.
```
**Signature generation for a given message 𝑀:**
```
● Generate 𝑘∈ [1, 𝑛 − 1]
● Compute
(𝑥1, 𝑦1) = 𝑘𝐺
𝑟 = 𝑥1 𝑚𝑜𝑑 𝑛
𝑠 = 𝐻(𝑀) + 𝑑𝑟 𝑚𝑜𝑑 𝑛
● If 𝑟 = 0 or 𝑠 = 0, try again. The signature is (𝑟, 𝑠).
● Signature: (𝑀, 𝑟 , 𝑠 ).
```
**Verification:**
```
● Given (𝑀, 𝑟 ', 𝑠' ).
● Verify if 𝑟' and 𝑠' are in [1, 𝑛 − 1] and that 𝑟' = 𝑥 𝑚𝑜𝑑 𝑛 for
(𝑥 , 𝑦 ) = 𝑢 𝐺 + 𝑢 𝑄 = 𝐻(𝑀) 𝑚𝑜𝑑 𝑛, and 𝑢 = 𝑟' 𝑚𝑜𝑑 𝑛.s')
𝑢
1 1 1 2 1 𝑠' 2 𝑠'
```
---
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