Notations & Definitions
This page is a glossary for notation and concepts present in the documentation.
Sets and Groups #
is the set of integers, .
is the set of integers greater than or equal to 0, .
is the finite set of integers .
are the integers modulo , a set associated with the equivalence classes of integers .
is the multiplicative group of integers modulo : an element from is in iff , that is . When is prime, then .
is the finite field of order ; when is a prime number, these are the integers modulo , ; when is a prime power , these are Galois fields.
is the order of a set , i.e., its number of elements. For example, , and for a prime , .
: An evaluation domain, typically used in FFT-based polynomial commitment schemes (e.g., domains of size a power of two).
: A relation or constraint system, such as an R1CS (Rank-1 Constraint System).
: Reed-Solomon codes.
Vectors #
is a vector with for all .
denotes the scalar product .
denotes the inner product
Sampling #
In protocol specifications, we will often need to uniformly sample elements from sets. We will use the following notation:
, where is uniformly sampled from the set .
Assertions #
We will use assertions in protocol descriptions. When the assertions do not hold, the protocol must abort to avoid leaking secret information.
, requires , and aborts otherwise
, requires , and aborts otherwise
, requires that is in the set , and aborts otherwise.
Hash Functions #
is a cryptographically secure domain-separated hash function.
is a cryptographically secure domain-separated hash function with specific output-size of kk-bits.
Special Functions
: The degree of a polynomial , i.e., the highest exponent with non-zero coefficient in .
: The logarithm of (base context-dependent, often base 2 in crypto).
: The maximum of a set of values.
: The minimum of a set of values.
is the positive greatest common divisor of integers and ; when
, and are said to be coprime.
is Euler’s totient function; for , it is the number of integers in coprime with .
Others
A Montgomery form of is represented by , given a modulus and a Montgomery radix such that .
References #
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