7.2. Cryptographic Security Methods

Binance Smart Chain (BSC) employs advanced cryptographic methods to ensure the security of transactions, signatures, and data storage. The primary algorithms include the Elliptic Curve Digital Signature Algorithm (ECDSA) based on the secp256k1 curve and the Keccak256 hash function to guarantee data immutability.

The ECDSA algorithm provides authentication for transactions and protection against forgery. The main steps of the algorithm involve key generation, signature creation, and verification. Key generation includes the formation of a private and a public key.

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The private key kkk is randomly selected as follows:

$$k \in \{1, 2, \ldots, n - 1\}$$

• n — is the order of the point group G (the generator of the elliptic curve).

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The public key is calculated as:

$$P = k \cdot G$$

• G — is the base point of the elliptic curve.

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Steps for creating a signature:

1. Calculate the hash mark of the message M:

$$z = \operatorname{keccak256}(M)$$

2. Select a random number r:

$$r \in \{1, 2, \ldots, n - 1\}$$

3. Compute a point on the curve:

$$R = r \cdot G$$

and take the $x_{R}$ coordinate:

$$r = x_{R} \bmod n$$

If r=0, select another r.

4. Calculate the second part of the signature:

$$s = r^{-1} \cdot (z + k \cdot r) \bmod n$$

If s=0, select another r.

The resulting signature is the pair r,s.

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To Verify a Signature r,s Using the Public Key P:

1. Calculate the hash mark of the message:

$$z = \operatorname{keccak256}(M)$$

2. Compute the parameters:

$$u_1 = s^{-1} \cdot z \bmod n, \quad u_2 = s^{-1} \cdot r \bmod n$$

3. Compute the point:

$$Q = u_1 \cdot G + u_2 \cdot P$$

4. If $x_{Q} \bmod n = r$, the signature is considered valid.

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The secp256k1 Elliptic Curve is Defined by the Equation:

$$y^2 = x^3 + 7 \bmod p$$

where:

• $p = 2^{256} - 2^{32} - 977$ — a prime number (modulus),

• G — the base point of the curve with fixed coordinates,

• n — the order of point G.

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Features of secp256k1:

• High computational speed due to optimizations for modern processors.

• Compactness: keys are 32 bytes (256 bits) in size.

• Resistance to attacks based on the discrete logarithm problem.

The Keccak256 function (a variant of SHA-3) is used for hashing data and transaction identifiers in BSC. The hash function takes input data of any size and returns a fixed-size hash (256 bits).

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Hashing Formula:

$$H(M) = \operatorname{Keccak256}(M)$$

where:

• M — the original message,

• H(M) — its hash.

Hashing is irreversible, preventing recovery of the original message.

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Every transaction is signed with the sender's private key. The signature is verified by network nodes, ensuring the authenticity of the transaction. Merkle trees are used to verify the integrity of block data. The root hash combines all transactions in the block, and any change in the data results in a change in the root hash. Validators in the Proof-of-Stake Authority algorithm use cryptographic methods to validate blocks and participate in voting.

The cryptographic algorithms used in Binance Smart Chain provide high speed, data compactness, and resistance to attacks. These methods make BSC a secure and efficient blockchain platform.