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GuidePublished 7 Aug 2026Updated 13 Aug 20268 min readBy Kevin Jogin
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Engineering  /  Mathematics  — Computation and Sources

Parameter Sizes, Records and Live References

Where to obtain current key size recommendations, factoring records and post-quantum guidance, and why they are not reproduced here.

Page KV-MATH-0474Reading time 4 minReviewed 2026-08-07Author Kevin Jogin

Executive summary

Key size recommendations, computational records and migration timelines all change, and a static page carrying them becomes misleading rather than merely stale.

This page records the durable method for deriving a recommendation and routes to maintained sources for the figures.

Learning objectives

  1. Understand how key size recommendations are derived.
  2. Identify the categories of authoritative source.
  3. Recognise which facts are durable and which are not.

01The sourcing policy

Caution
This collection deliberately does not reproduce current record sizes, effort figures, recommended key lengths or migration deadlines. Records fall, guidance is revised, and a page carrying stale numbers is worse than one carrying none.

The same policy applies across the KEVOS mathematics collections. The durable layer — definitions, algorithms, complexity results and the reasoning by which parameters are chosen — is authored here. Numeric data with a shelf life is routed to maintained sources.

What this collection carries and what it does not
DurableNot durable
Why NFS has L(1/3) complexityThe current factoring record
How a key size follows from a recordThe recommended key size
Why quantum computers break RSAThe migration deadline
What a security level meansWhich algorithms meet it today

02How a recommendation is derived

  1. Take a completed record

    A published factorisation or discrete logarithm computation with its effort accounting in core-years.

  2. Extrapolate by the complexity formula

    The L(1/3) expression predicts effort at other parameter sizes.

  3. Project hardware improvement

    Allow for cost per operation falling over the intended protection period.

  4. Add a margin for algorithmic risk

    Allow for improvements to the algorithms themselves, which cannot be predicted.

  5. Publish a minimum size

    Standards bodies convert the result into recommendations by protection period.

Caution
Every step after the first is a judgement. Different bodies make different assumptions and publish different numbers for the same protection period, which is why comparison sites exist and why a single figure quoted without its source is not useful.

The extrapolation also assumes no algorithmic breakthrough. A materially better factoring algorithm would invalidate every recommendation simultaneously.

03Categories of source

  • Standards bodies

    National and international organisations publish key length recommendations by protection period, revised periodically.

  • Comparison services

    Maintained sites aggregating the recommendations of different bodies side by side, which makes the spread of expert opinion visible.

  • Record announcements

    Published by the teams performing the computations, with full effort accounting — the primary data for extrapolation.

  • Post-quantum guidance

    Migration timelines and algorithm selections, published separately and currently moving faster than classical guidance.

For any specific decision, consult a current source at the time of the decision. The reasoning on this page tells you what to look for and how to interpret it; it deliberately does not tell you the answer, because the answer changes.

Note
The post-quantum dimension is the one moving fastest. A sufficiently large fault-tolerant quantum computer would break RSA and discrete-log cryptography outright via Shor's algorithm, rather than merely requiring larger parameters. Migration guidance is being revised on a much shorter cycle than classical key size guidance, and anything written here would date quickly.

04Frequently asked questions

Why not include figures with a review date?

Because pages are read long after they are written and the review date is easily missed. Routing to a maintained source is more reliable than dating a static figure, and it fails safe.

Does the Riemann hypothesis affect key sizes?

No. It concerns the distribution of primes, not the difficulty of factoring. A proof would sharpen various estimates and settle conditional theorems, and would supply no factoring algorithm.

How urgent is post-quantum migration?

Urgent for data requiring long-term confidentiality, because encrypted traffic captured now could be decrypted once a sufficiently large quantum computer exists. Current guidance from standards bodies is the appropriate source for timelines.

Related pages

  • The Number Field Sieve and Factoring Records
  • The RSA Cryptosystem
  • The Diffie-Hellman Key Establishment Protocol
  • Arbitrary Precision Arithmetic in Practice

Sources and method

Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — orientation page, no single source section.

This page carries the durable method layer only: definitions, constructions, algorithms, complexity results and selection criteria, authored originally for KEVOS. No text is transcribed or paraphrased from the source, and no numeric tables or benchmark data are reproduced — these are routed to live authoritative sources instead.

Forward reference: this page extends beyond the source text and is flagged as post-source.

Author: Kevin Jogin. Last reviewed 2026-08-07.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Parameter Sizes, Records and Live References. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat Parameter Sizes, Records and Live References as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—records, post-quantum, guidance, parameter, sizes—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

  1. Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
  2. Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
  3. Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
  4. Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
  5. Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

StageRecordQuality check
InputObjects, domain, notation, assumptionsEvery symbol is defined
MethodPermitted operation or cited result at each stepAll hypotheses hold
OutputExact result and representationCorrect type, domain and form
VerificationSubstitution, invariant or alternative derivationIndependent agreement
Boundary testZero, identity, degenerate or failed hypothesisScope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

  • Can every symbol be traced to a definition or prior result?
  • Which hypothesis does each major step use?
  • Does the method cover zero, identity, degenerate and boundary cases?
  • Can the conclusion be checked by a second representation or calculation?
  • Are mandatory requirements distinguished from recommendations and illustrative values?
  • Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
  • Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
  • Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying Parameter Sizes, Records and Live References?

Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.

Which assumption about records would change the result most if it proved false?

Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.

What evidence would allow an independent reviewer to reproduce or challenge the conclusion?

Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.

Which boundary, exception or failure case has not yet been tested?

Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.

What must be handed over, monitored or reviewed after the immediate work is complete?

Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.

Authoritative references and use notes

The sources below were selected as institutional or primary guidance for the broader practice. They support the handbook method; they do not imply that every statement or clause in a source applies to every project. Confirm the current edition, jurisdiction, contract and application before treating any requirement as mandatory.

  • MIT OpenCourseWare — Number Theory I — Massachusetts Institute of Technology. Used for algebraic and analytic number theory. Accessed 2026-08-13.
  • MIT OpenCourseWare — Algebra I — Massachusetts Institute of Technology. Used for groups, vector spaces, linear transformations and linear groups. Accessed 2026-08-13.

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