Identity element
An element that, under a specified binary operation, leaves every element in the operation’s domain unchanged when combined with it.
Real-World Cryptography
An identity element is an element that leaves every other element unchanged when combined with it under the group's operation.
- How it is used
- Supports group-based key exchanges by providing the unchanged result required by the group operation.
- How deep it goes
- Defines the property, gives multiplicative and elliptic-curve examples, and applies it to subgroup validation without formal proof.
- Examples
- Uses 1 under multiplication, O under elliptic-curve addition, and subgroup-order checks that return the identity.
- What you can do after
- Evaluate/understand group operations by identifying the element that leaves every operand unchanged.
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Geometry for Programmers
An identity element is an element that leaves every compatible element unchanged when combined under an operation.
- How it is used
- Uses the identity element to characterize algebraic systems and relate addition, inverses, and matrix multiplication.
- How deep it goes
- Defines the property, demonstrates it with integer addition, and places it among group properties without developing formal proofs.
- Examples
- Uses integer addition, zero, additive inverses, and matrix multiplication as concrete examples.
- What you can do after
- Evaluate/understand whether an element leaves others unchanged and relate that test to inverses and algebraic structure.
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Functional Programming in Scala, Second Edition
Identity is the property that an operation has an element whose combination with another element leaves that element unchanged.
- How it is used
- Uses a neutral value to define monoids and support generic combination, folding, and composition.
- How deep it goes
- Defines two-sided identity laws and connects them to monoids and monads without developing their implementations in depth.
- Examples
- Uses empty strings, zero, one, Boolean constants, Nil, monoid traits, and monadic unit as examples.
- What you can do after
- Evaluate/understand whether an operation has a neutral value and recognize its role in generic algebraic interfaces.
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100 Java Mistakes and How to Avoid Them
An identity element is a value that leaves another operand unchanged when combined with it by an operation.
- How it is used
- Explains the identity requirement for arguments supplied to Stream.reduce operations.
- How deep it goes
- Defines identity algebraically, demonstrates a violating accumulator, and stops short of general reduction implementation details.
- Examples
- Uses List.hashCode(), an initial value of 1, and a multiplication-and-addition accumulator as the central counterexample.
- What you can do after
- Diagnose reduction bugs by checking identity and associativity laws, then testing behavior with parallel streams.
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Elliptic Curve Cryptography for Developers
An identity element is an element that leaves another element unchanged under a specified operation, such as zero for addition or one for multiplication.
- How it is used
- Supports arithmetic in finite fields, elliptic-curve point operations, and pairing constructions.
- How deep it goes
- Defines additive and multiplicative identities, then treats the elliptic-curve point at infinity as a special computational case.
- Examples
- Uses zero, one, modular clock arithmetic, the point at infinity, and roots of unity as concrete examples.
- What you can do after
- Evaluate/understand why arithmetic systems require neutral elements and how elliptic-curve code represents its special identity.
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Related concepts
- Monoidpart of · A monoid is an algebraic structure consisting of a set with an associative binary operation and an identity element.
- point at infinityincludes · A point at infinity is a formal point added to a geometric space to represent behavior at infinity, such as a direction, limiting location, or elliptic-curve identity.
- mathematical calculationimplemented by · A mathematical calculation is an operation or expression that derives a result from one or more input values.
- type-lifting functionincludes · A type-lifting function embeds a value into a corresponding computational context, container, or type, often a monadic type.
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