Understanding R1CS: Rank-One Constraint Systems

Rank-One Constraint Systems, the properties, and its advantages.

What is R1CS?

R1CS or Rank-One Constraint System, is a mathematical construct used to encode polynomial equations in matrices 𝐴, 𝐵, and 𝐶, where each row of the matrices corresponds to a system of equations of the form:

$$𝐴𝑧 × 𝐵𝑧 = 𝐶𝑧.$$

, where 𝐴, 𝐵, 𝐶 are matrices with few nonzero entries and the product symbol (×) means that the matrices are multiplied entry-wise. The variables in each equation have a maximum degree of two. Moreover, any bounded computation can be expressed as an R1CS, and the same arithmetic circuits can be represented by different R1CSs.

Application of R1CS

R1CS has numerous applications, including in zero-knowledge proof systems such as zkSNARK constructions. It allows for efficient verification of computations without revealing any sensitive information. Additionally, R1CS can be used in cryptographic protocols for secure communication.

Limitations of R1CS

While R1CS is a powerful tool, it has some limitations. One of the main limitations is that it does not work well with folding schemes. A folding scheme is a cryptographic protocol that allows for the efficient verification of multiple computations with a single proof. However, R1CS is not well-suited for this purpose.

Overcoming these limitations

Folding protocols, which use vector and polynomial commitments, can overcome these limitations and provide efficient verification of multiple computations with a single proof. In the next section, we describe how Folding protocols work and present their advantages.