cardinal

cardinal

Package cardinal provides representations for cardinal numbers (cardinalities) used to express the size of algebraic structures such as groups, rings, and fields.

Overview

In cryptographic contexts, the cardinality of a structure may be known, unknown (hidden for security a la groups of unknown order), or infinite. This package provides three implementations of the Cardinal interface to handle these cases uniformly. We stress that the intention of this package is to provide clarity when dealing with cardinalities in a cryptographic context, NOT to implement a mathematically faithful version of cardinal arithmetics.

Key Types

• Known: A cardinal with a concrete numeric value, stored as big-endian bytes.
• Unknown: Represents a cardinal whose value is intentionally hidden (e.g., the order of an RSA group without knowledge of the factorization).
• Infinite: Represents an infinite cardinality (e.g., the integers Z or rationals Q).

Comparison Semantics

The comparison methods follow specific rules based on the nature of each cardinal type:

Equal

• Known.Equal(Known): True if and only if the numeric values are identical.
• Unknown.Equal(any): Always false. Unknown values cannot be compared for equality since their values are hidden.
• Infinite.Equal(Infinite): True. All infinite cardinals are considered equal.

IsLessThanOrEqual

• Known.IsLessThanOrEqual(Known): Standard numeric comparison.
• Known.IsLessThanOrEqual(Unknown/Infinite): False. Known values cannot be compared to unknown or infinite values.
• Unknown.IsLessThanOrEqual(any): Always false. No ordering can be established for unknown values.
• Infinite.IsLessThanOrEqual(Infinite): True. Infinity is considered equal to itself.
• Infinite.IsLessThanOrEqual(Known/Unknown): False. Infinity is not less than any finite or unknown value.

Type Predicates

• IsFinite(): True for Known and Unknown, false for Infinite.
• IsUnknown(): True only for Unknown.
• IsZero(): True only for Known with value zero.