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Elliptic curves provide the mathematical foundation for cryptographic operations, including zk-SNARKs and digital signatures. These operations are secure, efficient, and compact.
Equation:
y^2 = x^3 + ax + b (mod p)
Explanation:
- p: A prime number defining the finite field GF(p).
- a, b: Parameters of the curve that define its shape.
- 4a^3 + 27b^2 ≠ 0 (mod p): Ensures no singularities such as cusps or self-intersections.
Operations:
1. Point Addition (P ≠ Q):
λ = (y2 - y1) / (x2 - x1) (mod p)
x3 = λ^2 - x1 - x2 (mod p)
y3 = λ(x1 - x3) - y1 (mod p)
2. Point Doubling (P = Q):
λ = (3x1^2 + a) / (2y1) (mod p)
x3 = λ^2 - 2x1 (mod p)
y3 = λ(x1 - x3) - y1 (mod p)
3. Key Generation:
Q = d * G
where d: Private key, Q: Public key, G: Generator point.
Expanded Explanation: Elliptic curves are used to create compact public and private keys. These keys allow secure transactions and efficient verification in Abstract Chain. The discrete logarithm problem, which underpins the security, is computationally infeasible to solve with current technology.
Equation:
y^2 = x^3 + ax + b (mod p)
Explanation:
- p: A prime number defining the finite field GF(p).
- a, b: Parameters of the curve that define its shape.
- 4a^3 + 27b^2 ≠ 0 (mod p): Ensures no singularities such as cusps or self-intersections.
Operations:
1. Point Addition (P ≠ Q):
λ = (y2 - y1) / (x2 - x1) (mod p)
x3 = λ^2 - x1 - x2 (mod p)
y3 = λ(x1 - x3) - y1 (mod p)
2. Point Doubling (P = Q):
λ = (3x1^2 + a) / (2y1) (mod p)
x3 = λ^2 - 2x1 (mod p)
y3 = λ(x1 - x3) - y1 (mod p)
3. Key Generation:
Q = d * G
where d: Private key, Q: Public key, G: Generator point.
Expanded Explanation: Elliptic curves are used to create compact public and private keys. These keys allow secure transactions and efficient verification in Abstract Chain. The discrete logarithm problem, which underpins the security, is computationally infeasible to solve with current technology.
Polynomial commitments allow zk-SNARKs to prove polynomial relationships without revealing the data itself.
Equation:
C_P = g^(P(s)) (mod p)
Explanation:
- P(x): Polynomial defined as P(x) = a0 + a1x + a2x^2 + ... + adx^d.
- g: Generator of the cryptographic group.
- s: Secret random value known as "toxic waste."
Usage:
1. To prove P(z) = v:
Q(x) = (P(x) - v) / (x - z)
Verification:
C_P = C_Q * g^v * (g^z)^(-1) (mod p)
Expanded Explanation: Polynomial commitments ensure both privacy and efficiency. Abstract Chain uses this mechanism to validate computations in zk-SNARKs. The binding and hiding properties make it impossible to alter committed values without detection.
Equation:
C_P = g^(P(s)) (mod p)
Explanation:
- P(x): Polynomial defined as P(x) = a0 + a1x + a2x^2 + ... + adx^d.
- g: Generator of the cryptographic group.
- s: Secret random value known as "toxic waste."
Usage:
1. To prove P(z) = v:
Q(x) = (P(x) - v) / (x - z)
Verification:
C_P = C_Q * g^v * (g^z)^(-1) (mod p)
Expanded Explanation: Polynomial commitments ensure both privacy and efficiency. Abstract Chain uses this mechanism to validate computations in zk-SNARKs. The binding and hiding properties make it impossible to alter committed values without detection.
Batching optimizes throughput by aggregating multiple transactions into a single proof.
Equation:
Z = (T1 * T2 * ... * Tn) (mod p)
Explanation:
- Ti: Individual transaction data.
- n: Number of transactions in the batch.
- Z: Aggregated result of all transactions.
Usage:
Batching reduces the volume of data sent to the Ethereum mainnet, lowering gas costs and improving scalability.
Expanded Explanation: By grouping thousands of transactions into a single proof, Abstract Chain achieves significant cost and performance benefits. This batching mechanism ensures that users experience faster confirmation times and reduced fees.
Equation:
Z = (T1 * T2 * ... * Tn) (mod p)
Explanation:
- Ti: Individual transaction data.
- n: Number of transactions in the batch.
- Z: Aggregated result of all transactions.
Usage:
Batching reduces the volume of data sent to the Ethereum mainnet, lowering gas costs and improving scalability.
Expanded Explanation: By grouping thousands of transactions into a single proof, Abstract Chain achieves significant cost and performance benefits. This batching mechanism ensures that users experience faster confirmation times and reduced fees.
FFT is used to efficiently evaluate polynomials at multiple points
Equation:
P(ω^k) = Σ (aj * (ω^k)^j), j = 0...n-1
Explanation:
- ω: Primitive nth root of unity in the field.
- aj: Coefficients of the polynomial.
- k: Index for evaluation.
Expanded Explanation: FFT transforms computationally expensive operations on polynomials into faster matrix multiplications. This is crucial in zk-SNARKs for reducing proof generation times, allowing Abstract Chain to scale transaction throughput.
Equation:
P(ω^k) = Σ (aj * (ω^k)^j), j = 0...n-1
Explanation:
- ω: Primitive nth root of unity in the field.
- aj: Coefficients of the polynomial.
- k: Index for evaluation.
Expanded Explanation: FFT transforms computationally expensive operations on polynomials into faster matrix multiplications. This is crucial in zk-SNARKs for reducing proof generation times, allowing Abstract Chain to scale transaction throughput.
Bilinear pairings allow efficient verification of zk-SNARK proofs.
Equation:
e(g1^a, g2^b) = e(g1, g2)^(a*b)
Explanation:
- g1, g2: Elements of cryptographic groups G1 and G2.
- a, b: Scalars used in the pairing.
- e: Bilinear pairing function.
Expanded Explanation: Pairings ensure that zk-SNARKs can verify correctness with minimal overhead. This operation is integral to combining multiple proof systems into a single validated output.
Equation:
e(g1^a, g2^b) = e(g1, g2)^(a*b)
Explanation:
- g1, g2: Elements of cryptographic groups G1 and G2.
- a, b: Scalars used in the pairing.
- e: Bilinear pairing function.
Expanded Explanation: Pairings ensure that zk-SNARKs can verify correctness with minimal overhead. This operation is integral to combining multiple proof systems into a single validated output.
zk-SNARKs enable proofs of correctness without revealing private data.
Equation:
A * B = C
QAP Equation:
P(x) * Q(x) = H(x) * Z(x) (mod p)
Explanation:
- A, B, C: Matrices representing the computation.
- P(x), Q(x), H(x), Z(x): Polynomials encoding the computation circuit.
Usage:
Proves the correctness of computations without disclosing the inputs or outputs.
Expanded Explanation: zk-SNARKs form the cryptographic backbone of Abstract Chain. They allow the network to process transactions and validate complex computations without revealing sensitive user data, ensuring both privacy and security.
Equation:
A * B = C
QAP Equation:
P(x) * Q(x) = H(x) * Z(x) (mod p)
Explanation:
- A, B, C: Matrices representing the computation.
- P(x), Q(x), H(x), Z(x): Polynomials encoding the computation circuit.
Usage:
Proves the correctness of computations without disclosing the inputs or outputs.
Expanded Explanation: zk-SNARKs form the cryptographic backbone of Abstract Chain. They allow the network to process transactions and validate complex computations without revealing sensitive user data, ensuring both privacy and security.
Verification combines polynomial commitments, bilinear pairings, and batching.
Equation:
e(C_P, g2) = e(C_Q, g2) * e(g^v, g2) * e(g^(z), g2)^(-1)
Explanation:
- CP: Commitment to the original polynomial.
- CQ: Commitment to the quotient polynomial.
- g, g2: Generators of cryptographic groups.
- v, z: Known values used in verification.
Usage:
Simplifies large-scale verification of batched transactions.
Expanded Explanation: Proof verification in Abstract Chain ensures the integrity of transactions and computations. The use of batching and zk-SNARKs minimizes computational resources while maintaining a high level of trust.
Equation:
e(C_P, g2) = e(C_Q, g2) * e(g^v, g2) * e(g^(z), g2)^(-1)
Explanation:
- CP: Commitment to the original polynomial.
- CQ: Commitment to the quotient polynomial.
- g, g2: Generators of cryptographic groups.
- v, z: Known values used in verification.
Usage:
Simplifies large-scale verification of batched transactions.
Expanded Explanation: Proof verification in Abstract Chain ensures the integrity of transactions and computations. The use of batching and zk-SNARKs minimizes computational resources while maintaining a high level of trust.
Abstract Chain integrates these mathematical technologies to create a highly scalable, private, and efficient blockchain system. By combining elliptic curves, polynomial commitments, batching, and zk-SNARKs, it achieves a balance between performance, cost, and security.
