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GuidePublished 7 Aug 2026Updated 13 Aug 20268 min readBy Kevin Jogin
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KEVOS AIGenerating a Random Number from a Given Interval

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Engineering  /  Mathematics  — Probabilistic Algorithms

Generating a Random Number from a Given Interval

Sampling uniformly from an arbitrary range using a source of random bits, and controlling the resulting bias.

Page KV-MATH-0360Reading time 3 minReviewed 2026-08-07Author Kevin Jogin

Executive summary

A source of uniform bits generates uniform values on powers of two. Producing a uniform value on an arbitrary range requires either rejection sampling, which is exact, or reduction, which is biased.

The bias from reduction is quantifiable and can be driven below any threshold by drawing extra bits.

Learning objectives

  1. Implement rejection sampling and bound its expected cost.
  2. Quantify the bias of the modular reduction method.
  3. Choose between the two approaches.

01Rejection sampling

Algorithm

Uniform sampling by rejection

Inputbound n, source of uniform bits
Outputuniform value in [0, n)
  1. Let k be the number of bits such that 2^k ≥ n.
  2. Draw k uniform random bits, forming a value v in [0, 2^k).
  3. If v < n, return v.
  4. Otherwise discard v and repeat.
Cost  expected fewer than 2 iterations

The output is exactly uniform, with no approximation. The acceptance probability is n/2^k, which exceeds one half by the choice of k, so the expected iteration count is below two.

Caution
The running time is unbounded, though its tail decays geometrically. For constant-time cryptographic code this is a genuine problem, since the number of iterations is data-dependent and observable.

02Reduction and its bias

Drawing a k-bit value and reducing modulo n is bounded-time but not uniform: residues below 2^k mod n occur once more often than the rest.

Theorem

Bias bound

The statistical distance between (uniform k-bit value) mod n and the uniform distribution on [0, n) is at most n / 2^k.

Bias from modular reduction
Excess bits k − len(n)Bias boundSuitable for
0≈ 1Nothing
82^{−8}Non-critical sampling
642^{−64}Most cryptographic uses
1282^{−128}High-security parameters

Drawing 64 or 128 bits beyond the length of n makes the bias negligible while keeping the operation bounded-time, which is why this is the standard approach in cryptographic libraries.

03Choosing between them

  • Rejection

    Exactly uniform, simple to reason about, unbounded time. Right where exactness matters and timing is not observable.

  • Reduction with excess bits

    Bounded time, negligible bias, slightly more expensive per draw. Right for constant-time cryptographic code.

Caution
Reducing a value of the same length as n — the naive rand() % n — produces bias close to one and has caused real vulnerabilities. Nonce generation with even a few bits of consistent bias permits full private key recovery from a modest number of signatures via lattice techniques.

04Frequently asked questions

Why is the acceptance probability above one half?

Because k is chosen minimally with 2^k ≥ n, so n > 2^{k−1}, giving n/2^k > 1/2. A larger k would lower acceptance and raise the expected iteration count.

Can rejection sampling be made constant-time?

Not while remaining exact. Fixing the iteration count reintroduces a failure probability, converting it into a bounded-time method with residual bias — essentially the reduction approach in another form.

Does the bias matter outside cryptography?

Rarely. For simulation and randomised algorithms a bias of 2^{−8} is immaterial. It matters where an adversary can accumulate many samples and exploit a systematic skew.

Related pages

  • Statistical Distance
  • Flipping a Coin Until a Head Appears
  • Generating a Random Prime

Sources and method

Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 159-162.

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.

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 Generating a Random Number from a Given Interval. 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 Generating a Random Number from a Given Interval as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—sampling, rejection, random, bits, bias—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 Generating a Random Number from a Given Interval?

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 sampling 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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