mathematics

Richard Schroeppel

Richard Schroeppel is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Richard Schroeppel rather than just read about it. In short: Richard C. Schroeppel (born 1948) is an American mathematician born in Illinois.

Key takeaways

Richard Schroeppel belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Richard Schroeppel to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Richard Schroeppel from memory before moving on to harder problems.

Reference excerpt

Richard C. Schroeppel (born 1948) is an American mathematician born in Illinois. His research has included magic squares, elliptic curves, and cryptography. In 1964, Schroeppel won first place in the United States among over 225,000 high school students in the Annual High School Mathematics Examination, a contest sponsored by the Mathematical Association of America and the Society of Actuaries. In both 1966 and 1967, Schroeppel scored among the top 5 in the U.S. in the William Lowell Putnam Mathematical Competition. In 1973 he discovered that there are 275,305,224 normal magic squares of order 5. In 1998–1999 he designed the Hasty Pudding Cipher, which was a candidate for the Advanced Encryption Standard, and he is one of the designers of the SANDstorm hash, a submission to the NIST SHA-3 competition. Among other contributions, Schroeppel was the first to recognize the sub-exponential running time of certain integer factoring algorithms. While not entirely rigorous, his proof that Morrison and Brillhart's continued fraction factoring algorithm ran in roughly e 2 ln ⁡ n ln ⁡ ln ⁡ n {\displaystyle e^{\sqrt {2\ln {n}\ln {\ln {n}}}}} steps was an important milestone in factoring and laid a foundation for much later work, including the current "champion" factoring algorithm, the number field sieve. Schroeppel analyzed Morrison and Brillhart's algorithm, and saw how to cut the run time to roughly e ln ⁡ n ln ⁡ ln ⁡ n {\displaystyle e^{\sqrt {\ln {n}\ln {\ln {n}}}}} by modifications that allowed sieving. This improvement doubled the size of numbers that could be factored in a given amount of time. Coming around the time of the RSA algorithm, which depends on the difficulty of factoring for its security, this was a critically important result. Due to Schroeppel's apparent prejudice against publishing (though he freely circulated his ideas within the research community), and in spite of Carl Pomerance noting that his quadratic sieve factoring algorithm owed a debt to Schroeppel's earlier work, Schroeppel's contribution is often overlooked. Schroeppel's Erdős number is 2.

Illustrations

Worked examples

Example 1 — a first encounter with Richard Schroeppel

Start with the simplest possible case. Write down what Richard Schroeppel claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Richard Schroeppel before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Richard Schroeppel ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Richard Schroeppel

In research: Richard Schroeppel appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Richard Schroeppel in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Richard Schroeppel is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1948 births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Richard Schroeppel outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Richard Schroeppel” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Richard Schroeppel in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Richard Schroeppel means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Richard Schroeppel out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Richard Schroeppel in simple terms?

Richard C. Schroeppel (born 1948) is an American mathematician born in Illinois.

Why does Richard Schroeppel matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Richard Schroeppel?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Richard Schroeppel.