** Chapter 8: Patent Protection of Software:
Statutory Subject Matter in the
Supreme Court and the Federal Circuit
D. Statutory Subject Matter in Computerized
Methods of Doing Business (Continued)**

**Case-Law Developments After State Street and AT&T (Cont'd):
Bilski and Alice in the Supreme Court and Beyond That**

Alice Corp. Pty. Ltd. v. CLS Bank Int’l

United States Supreme Court

573 U.S. 208 (2014)Justice Thomas delivered the opinion of the Court.

The patents at issue in this case disclose a computer-implemented scheme for mitigating “settlement risk” (i.e., the risk that only one party to a financial transaction will pay what it owes) by using a third-party intermediary. The question presented is whether these claims are patent-eligible under 35 U. S. C. § 101, or are instead drawn to a patent-ineligible abstract idea. We hold that the claims at issue are drawn to the abstract idea of intermediated settlement, and that merely requiring generic computer implementation fails to transform that abstract idea into a patent-eligible invention. We therefore affirm the judgment of the United States Court of Appeals for the Federal Circuit.I

APetitioner Alice Corporation is the assignee of several patents that disclose schemes to manage certain forms of financial risk. According to the specification, the invention “enabl[es] the management of risk relating to specified, yet unknown, future events.” The specification further explains that the “invention relates to methods and apparatus, including electrical computers and data processing systems applied to financial matters and risk management.”

The claims at issue are designed to facilitate the exchange of financial obligations between two parties by using a computer system as a third-party intermediary. The intermediary creates “shadow” credit and debit records (i.e., account ledgers) that mirror the balances in the parties’ real-world accounts at “exchange institutions” (e.g., banks). The intermediary updates the shadow records in real time as transactions are entered, allowing “only those transactions for which the parties’ updated shadow records indicate sufficient resources to satisfy their mutual obligations.” At the end of the day, the intermediary instructs the relevant financial institutions to carry out the “permitted” transactions in accordance with the updated shadow records, thus mitigating the risk that only one party will perform the agreed-upon exchange.

In sum, the patents in suit claim (1) the foregoing method for exchanging obligations (the method claims), (2) a computer system configured to carry out the method for exchanging obligations (the system claims), and (3) a computer-readable medium containing program code for performing the method of exchanging obligations (the media claims). All of the claims are implemented using a computer; the system and media claims expressly recite a computer, and the parties have stipulated that the method claims require a computer as well.B

Respondents CLS Bank International and CLS Services Ltd. (together, CLS Bank) operate a global network that facilitates currency transactions. In 2007, CLS Bank filed suit against petitioner, seeking a declaratory judgment that the claims at issue are invalid, unenforceable, or not infringed. Petitioner counterclaimed, alleging infringement. Following this Court’s decision in

Bilski v. Kappos, 561 U. S. 593 (2010), the parties filed cross-motions for summary judgment on whether the asserted claims are eligible for patent protection under 35 U. S. C. § 101. The District Court held that all of the claims are patent-ineligible because they are directed to the abstract idea of “employing a neutral intermediary to facilitate simultaneous exchange of obligations in order to minimize risk.”

A divided panel of the United States Court of Appeals for the Federal Circuit reversed, holding that it was not “manifestly evident” that petitioner’s claims are directed to an abstract idea. The Federal Circuit granted rehearingen banc, vacated the panel opinion, and affirmed the judgment of the District Court in a one-paragraphper curiamopinion. Seven of the ten participating judges agreed that petitioner’s method and media claims are patent ineligible. With respect to petitioner’s system claims, theen bancFederal Circuit affirmed the District Court’s judgment by an equally divided vote.

Writing for a five-member plurality, Judge Lourie concluded that all of the claims at issue are patent ineligible. In the plurality’s view, under this Court’s decision inMayo Collaborative Services v. Prometheus Laboratories, Inc., 566 U. S. 66 (2012), a court must first “identif[y] the abstract idea represented in the claim,” and then determine “whether the balance of the claim adds ‘significantly more.’ ” The plurality concluded that petitioner’s claims “draw on the abstract idea of reducing settlement risk by effecting trades through a third-party intermediary,” and that the use of a computer to maintain, adjust, and reconcile shadow accounts added nothing of substance to that abstract idea.

Chief Judge Rader concurred in part and dissented in part. In a part of the opinion joined only by Judge Moore, Chief Judge Rader agreed with the plurality that petitioner’s method and media claims are drawn to an abstract idea. In a part of the opinion joined by Judges Linn, Moore, and O’Malley, Chief Judge Rader would have held that the system claims are patent-eligible because they involve computer “hardware” that is “specifically programmed to solve a complex problem.” Judge Moore wrote a separate opinion dissenting in part, arguing that the system claims are patent eligible. Judge Newman filed an opinion concurring in part and dissenting in part, arguing that all of petitioner’s claims are patent eligible. Judges Linn and O’Malley filed a separate dissenting opinion reaching that same conclusion.

We granted certiorari and now affirm.II

“We have long held that this provision contains an important implicit exception: Laws of nature, natural phenomena, and abstract ideas are not patentable.”

Myriad. We have interpreted § 101 and its predecessors in light of this exception for more than 150 years.Bilski; see alsoO’Reilly v. Morse, 56 U.S. (15 How.) 62, 112–120 (1853);Le Roy v. Tatham, 14 How. 156, 174–175 (1853).

We have described the concern that drives this exclusionary principle as one of pre-emption. See, e.g.,Bilski(upholding the patent “would pre-empt use of this approach in all fields, and would effectively grant a monopoly over an abstract idea”). Laws of nature, natural phenomena, and abstract ideas are “the basic tools of scientific and technological work.”Myriad. “[M]onopolization of those tools through the grant of a patent might tend to impede innovation more than it would tend to promote it,” thereby thwarting the primary object of the patent laws.Mayo; see U. S. Const., Art. I, § 8, cl. 8 (Congress “shall have Power . . . To promote the Progress of Science and useful Arts”). We have “repeatedly emphasized this . . . concern that patent law not inhibit further discovery by improperly tying up the future use of ” these building blocks of human ingenuity.Mayo(citingMorse).

At the same time, we tread carefully in construing this exclusionary principle lest it swallow all of patent law.Mayo. At some level, “all inventions . . . embody, use, reflect, rest upon, or apply laws of nature, natural phenomena, or abstract ideas.”Id. Thus, an invention is not rendered ineligible for patent simply because it involves an abstract concept. SeeDiamond v. Diehr, 450 U.S. 175, 187 (1981). “[A]pplication[s]” of such concepts “to a new and useful end,” we have said, remain eligible for patent protection.Gottschalk v. Benson, 409 U. S. 63, 67 (1972). Accordingly, in applying the § 101 exception, we must distinguish between patents that claim the “buildin[g] block[s]” of human ingenuity and those that integrate the building blocks into something more,Mayo, thereby “transform[ing]” them into a patent-eligible invention,id. The former “would risk disproportionately tying up the use oft he underlying” ideas,id., and are therefore ineligible for patent protection. The latter pose no comparable risk of pre-emption, and therefore remain eligible for the monopoly granted under our patent laws.III

In

Mayo, we set forth a framework for distinguishing patents that claim laws of nature, natural phenomena, and abstract ideas from those that claim patent-eligible applications of those concepts. First, we determine whether the claims at issue are directed to one of those patent-ineligible concepts. If so, we then ask, “[w]hat else is there in the claims before us?”Id. To answer that question, we consider the elements of each claim both individually and “as an ordered combination” to determine whether the additional elements “transform the nature of the claim” into a patent-eligible application.Id. We have described step two of this analysis as a search for an “ ‘inventive concept’”—i.e., an element or combination of elements that is “sufficient to ensure that the patent in practice amounts to significantly more than a patent upon the [ineligible concept] itself.”Id.A

We must first determine whether the claims at issue [here] are directed to a patent-ineligible concept. We conclude that they are: These claims are drawn to the abstract idea of intermediated settlement. The “abstract ideas” category embodies “the long-standing rule that ‘[a]n idea of itself is not patentable.’ ”

Benson(quotingRubber-Tip Pencil Co. v. Howard, 20 Wall. 498, 507 (1874)); see alsoLe Roy(“A principle, in the abstract, is a fundamental truth; an original cause; a motive; these cannot be patented, as no one can claim in either of them an exclusive right ”). InBenson, for example, this Court rejected as ineligible patent claims involving an algorithm for converting binary-coded decimal numerals into pure binary form, holding that the claimed patent was “in practical effect . . . a patent on the algorithm itself.” And inParker v. Flook, we held that a mathematical formula for computing “alarm limits” in a catalytic conversion process was also a patent-ineligible abstract idea.

We most recently addressed the category of abstract ideas inBilski. The claims at issue inBilskidescribed a method for hedging against the financial risk of price fluctuations. Claim 1 recited a series of steps for hedging risk, including: (1) initiating a series of financial transactions between providers and consumers of a commodity; (2) identifying market participants that have a counter risk for the same commodity; and (3) initiating a series of transactions between those market participants and the commodity provider to balance the risk position of the first series of consumer transactions. Claim 4 “pu[t] the concept articulated in claim 1 into a simple mathematical formula.” The remaining claims were drawn to examples of hedging in commodities and energy markets.

“[A]ll members of the Court agree[d]” that the patent at issue inBilskiclaimed an “abstract idea.” Specifically, the claims described “the basic concept of hedging, or protecting against risk.” The Court explained that “‘[h]edging is a fundamental economic practice long prevalent in our system of commerce and taught in any introductory finance class.’” “The concept of hedging” as recited by the claims in suit was therefore a patent-ineligible “abstract idea, just like the algorithms at issue inBensonandFlook.”

It follows from our prior cases, andBilskiin particular, that the claims at issue here are directed to an abstract idea. Petitioner’s claims involve a method of exchanging financial obligations between two parties using a third party intermediary to mitigate settlement risk. The intermediary creates and updates “shadow” records to reflect the value of each party’s actual accounts held at “exchange institutions,” thereby permitting only those transactions for which the parties have sufficient resources. At the end of each day, the intermediary issues irrevocable instructions to the exchange institutions to carry out the permitted transactions.

On their face, the claims before us are drawn to the concept of intermediated settlement, i.e., the use of a third party to mitigate settlement risk. Like the risk hedging inBilski, the concept of intermediated settlement is “a fundamental economic practice long prevalent in our system of commerce.” See, e.g., Emery,Speculation on the Stock and Produce Exchanges of the United States, in 7Studies in History, Economics and Public Law283, 346-356 (1896) (discussing the use of a "clearing-house" as an intermediary to reduce settlement risk). The use of a third-party intermediary (or “clearing house”) is also a building block of the modern economy. Thus, intermediated settlement, like hedging, is an “abstract idea” beyond the scope of § 101.

Petitioner acknowledges that its claims describe intermediated settlement, but rejects the conclusion that its claims recite an “abstract idea.” Drawing on the presence of mathematical formulas in some of our abstract-ideas precedents, petitioner contends that the abstract-ideas category is confined to “preexisting, fundamental truth[s]” that “‘exis[t] in principle apart from any human action.”

Bilskibelies petitioner’s assertion. The concept of risk hedging we identified as an abstract idea in that case cannot be described as a “preexisting, fundamental truth.” The patent inBilskisimply involved a “series of steps instructing how to hedge risk.” Although hedging is a longstanding commercial practice, it is a method of organizing human activity, not a “truth” about the natural world that has always existed, seeFlook. One of the claims inBilskireduced hedging to a mathematical formula, but the Court did not assign any special significance to that fact, much less the sort of talismanic significance petitioner claims. Instead, the Court grounded its conclusion that all of the claims at issue were abstract ideas in the understanding that risk-hedging was a “fundamental economic practice.”

In any event, we need not labor to delimit the precise contours of the “abstract ideas” category in this case. It is enough to recognize that there is no meaningful distinction between the concept of risk-hedging inBilskiand the concept of intermediated settlement at issue here. Both are squarely within the realm of “abstract ideas” as we have used that term.B

Because the claims at issue are directed to the abstract idea of intermediated settlement, we turn to the second step in Mayo’s framework. We conclude that the method claims, which merely require generic computer implementation, fail to transform that abstract idea into a patent-eligible invention.

1

At

Mayostep two, we must examine the elements of the claim to determine whether it contains an “‘inventive concept’” sufficient to “transform” the claimed abstract idea into a patent-eligible application. A claim that recites an abstract idea must include “additional features” to ensure “that the [claim] is more than a drafting effort designed to monopolize the [abstract idea].”Mayomade clear that transformation into a patent-eligible application requires “more than simply stat[ing] the [abstract idea] while adding the words ‘apply it.’ ”

Mayoitself is instructive. The patents at issue inMayoclaimed a method for measuring metabolites in the bloodstream in order to calibrate the appropriate dosage of thiopurine drugs in the treatment of autoimmune diseases. The respondent in that case contended that the claimed method was a patent-eligible application of natural laws that describe the relationship between the concentration of certain metabolites and the likelihood that the drug dosage will be harmful or ineffective. But methods for determining metabolite levels were already “well known in the art,” and the process at issue amounted to “nothing significantly more than an instruction to doctors to apply the applicable laws when treating their patients.” “Simply appending conventional steps, specified at a high level of generality,” was not “enough” to supply an “‘inventive concept.’ ”Id.

The introduction of a computer into the claims does not alter the analysis atMayostep two. InBenson, for example, we considered a patent that claimed an algorithm implemented on “a general-purpose digital computer.” Because the algorithm was an abstract idea, the claim had to supply a “ ‘new and useful’ ” application of the idea in order to be patent eligible. But the computer implementation did not supply the necessary inventive concept; the process could be “carried out in existing computers long in use.”Ibid. We accordingly “held that simply implementing a mathematical principle on a physical machine, namely a computer, [i]s not a patentable application of that principle.”Mayo(citingBenson)).

Flookis to the same effect. There, we examined a computerized method for using a mathematical formula to adjust alarm limits for certain operating conditions (e.g., temperature and pressure) that could signal danger in a catalytic conversion process. Once again, the formula itself was an abstract idea, and the computer implementation was purely conventional. In holding that the process was patent ineligible, we rejected the argument that “implement[ing] a principle in some specific fashion” will “automatically fal[l] within the patentable subject matter of § 101.” Thus, “Flookstands for the proposition that the prohibition against patenting abstract ideas cannot be circumvented by attempting to limit the use of [the idea] to a particular technological environment.”Bilski.

InDiehr, by contrast, we held that a computer-implemented process for curing rubber was patent eligible, but not because it involved a computer. The claim employed a “well-known” mathematical equation, but it used that equation in a process designed to solve a technological problem in “conventional industry practice.” The invention inDiehrused a “thermocouple” to record constant temperature measurements inside the rubber mold—something “the industry ha[d] not been able to obtain.” The temperature measurements were then fed into a computer, which repeatedly recalculated the remaining cure time by using the mathematical equation. These additional steps, we recently explained, “transformed the process into an inventive application of the formula.”Mayo. In other words, the claims inDiehrwere patent eligible because they improved an existing technological process, not because they were implemented on a computer.

These cases demonstrate that the mere recitation of a generic computer cannot transform a patent-ineligible abstract idea into a patent-eligible invention. Stating an abstract idea “while adding the words ‘apply it’” is not enough for patent eligibility.Mayo. Nor is limiting the use of an abstract idea “to a particular technological environment.”Bilski. Stating an abstract idea while adding the words “apply it with a computer” simply combines those two steps, with the same deficient result. Thus, if a patent’s recitation of a computer amounts to a mere instruction to “implemen[t]” an abstract idea “on . . . a computer,”Mayo, that addition cannot impart patent eligibility. This conclusion accords with the preemption concern that undergirds our § 101 jurisprudence. Given the ubiquity of computers, wholly generic computer implementation is not generally the sort of “additional featur[e]” that provides any “practical assurance that the process is more than a drafting effort designed to monopolize the [abstract idea] itself.”Mayo.

The fact that a computer “necessarily exist[s] in the physical, rather than purely conceptual, realm” is beside the point. There is no dispute that a computer is a tangible system (in § 101 terms, a “machine”), or that many computer-implemented claims are formally addressed to patent-eligible subject matter. But if that were the end of the § 101 inquiry, an applicant could claim any principle of the physical or social sciences by reciting a computer system configured to implement the relevant concept. Such a result would make the determination of patent eligibility “depend simply on the draftsman’s art,”Flook,supra, at 593, thereby eviscerating the rule that “‘[l]aws of nature, natural phenomena, and abstract ideas are not patentable,’”Myriad.2

The representative method claim in this case recites the following steps: (1) “creating” shadow records for each counterparty to a transaction; (2) “obtaining” start-of-day balances based on the parties’ real-world accounts at exchange institutions; (3) “adjusting” the shadow records as transactions are entered, allowing only those transactions for which the parties have sufficient resources; and (4) issuing irrevocable end-of-day instructions to the exchange institutions to carry out the permitted transactions. Petitioner principally contends that the claims are patent eligible because these steps “require a substantial and meaningful role for the computer.” As stipulated, the claimed method requires the use of a computer to create electronic records, track multiple transactions, and issue simultaneous instructions; in other words, “[t]he computer is itself the intermediary.”

The relevant question is whether the claims here do more than simply instruct the practitioner to implement the abstract idea of intermediated settlement on a generic computer. They do not.

Taking the claim elements separately, the function performed by the computer at each step of the process is “[p]urely conventional.”Mayo. Using a computer to create and maintain “shadow” accounts amounts to electronic record keeping—one of the most basic functions of a computer. The same is true with respect to the use of a computer to obtain data, adjust account balances, and issue automated instructions; all of these computer functions are “well-understood, routine, conventional activit[ies]” previously known to the industry. In short, each step does no more than require a generic computer to perform generic computer functions.

Considered “as an ordered combination,” the computer components of petitioner’s method “ad[d] nothing . . . that is not already present when the steps are considered separately.”Id. Viewed as a whole, petitioner’s method claims simply recite the concept of intermediated settlement as performed by a generic computer. The method claims do not, for example, purport to improve the functioning of the computer itself. Nor do they effect an improvement in any other technology or technical field. See, e.g.,Diehr. Instead, the claims at issue amount to “nothing significantly more” than an instruction to apply the abstract idea of intermediated settlement using some unspecified, generic computer. Under our precedents, that is not “enough” to transform an abstract idea into a patent-eligible invention.C

Petitioner’s claims to a computer system and a computer-readable medium fail for substantially the same reasons. Petitioner conceded below that its media claims rise or fall with its method claims. As to its system claims, petitioner emphasizes that those claims recite “specific hardware” configured to perform “specific computerized functions.” But what petitioner characterizes as specific hardware—a “data processing system” with a “communications controller” and “data storage unit,” for example—is purely functional and generic. Nearly every computer will include a communications controller” and “data storage unit” capable of performing the basic calculation, storage, and transmission functions required by the method claims. As a result, none of the hardware recited by the system claims “offers a meaningful limitation beyond generally linking ‘the use of the [method] to a particular technological environment,’ that is, implementation via computers.” See

Bilski.

Put another way, the system claims are no different from the method claims in substance. The method claims recite the abstract idea implemented on a generic computer; the system claims recite a handful of generic computer components configured to implement the same idea. This Court has long “warn[ed] . . . against” interpreting § 101 “in ways that make patent eligibility ‘depend simply on the draftsman’s art.’ ”Mayo(quotingFlook, 437 U.S. at 593); seeid., at 590 (“The concept of patentable subject matter under § 101 is not ‘like a nose of wax which may be turned and twisted in any direction . . .’”). Holding that the system claims are patent eligible would have exactly that result.

Because petitioner’s system and media claims add nothing of substance to the underlying abstract idea, we hold that they too are patent ineligible under §101.Justice Sotomayor, with whom Justices Ginsburg and Breyer join, concurring.

I adhere to the view that any “claim that merely describes a method of doing business does not qualify as a ‘process’ under § 101.”Bilski v. Kappos, 561 U.S. 593, 614 (2010) (Stevens, J., concurring in judgment); see alsoIn re Bilski, 545 F.3d 943, 972 (Fed. Cir. 2008) (Dyk, J., concurring) (“There is no suggestion in any of th[e] early [English] consideration of process patents that processes for organizing human activity were or ever had been patent- able”). As inBilski, however, I further believe that the method claims at issue are drawn to an abstract idea. I therefore join the opinion of the Court.

Notes1. Link to oral argument in

Alicecase.

2. Here (at right) is a jaundiced take on theAliceopinion by lawcomics. What counter-arguments might you make in support of what this commentator condemns?

3. One of the dissenting opinions in the Federal Circuit in theAlicecase had predicted that the decision, if affirmed, would be “the death of hundreds of thousands of business method, financial system, and software patents.” During oral argument before the Supreme Court, Alice’s counsel insisted that if the Court held software to be patent-ineligible “unless the software somehow actually improves the computer, . . . this would inherently declare and in one fell swoop hundreds of thousands of patents invalid, and the consequences of that it seems to me are utterly unknowable.” Many stakeholders in such patents were alarmed.

During oral argument, Justice Sotomayor repeatedly asked counsel whetherAlicewas a software case:Why do we need to reach . . . software patents at all in this case?

What's the necessity for us to announce a general rule with respect to software? There is no software being patented in this case.

Do you think we have to reach the patentability of software to answer this case?Does the opinion ever discuss the patent eligibility of software, as such? Does it mention software at all? (It mentions computer implementation of methods of organizing human activity and other ideas, but that is a different question.) The opinion leaves open, at least in principle, the possibility of patents on computer-related inventions. Given the Court’s analysis of the patent eligibility of abstract ideas, however, is this seeming possibility illusory? Was the alarm justified? How does the ruling in this case affect the viability of business method, financial system, and software patents?

Gene Quinn, a vociferous advocate of patenting software had this to say about theAlicedecision:The Supreme Court decision in

Alicewill render many hundreds of thousands of software patents completely useless. … On first read I don’t see how any software patent claims written as method or systems claims can survive challenge. … What this means is that companies like Apple, IBM, Microsoft, Google and others have had the value of their patent portfolios nearly completely erased today. … getting a software patent will be much more difficult than it ever has been. … Most already issued software patents will not be able to be saved.Do you agree with this assessment? What should be done? What should a lawyer who has advised her cleint to invest in patenting software say to the client when the resulting investment seems to be heading south? Or, as Quinn puts it, the patent portfolio is erased?

4. Consider the requirement that the implementation of an abstract idea must itself embody an inventive concept, or be inventive—that the computerization (if that is the implementation) must not be routine or conventional. Does thisFlook-Mayo-Alicerule threaten the patent eligibility of software patents? By implication at least, theAliceopinion leaves room for patents on software that improves technological and industrial processes. For example, the Court said that the claims in theDiehrcase using the Arrhenius Equation were patent eligible “because they improved an existing technological process.” (By the way, is this take onDiehrdifferent from that ofMayo?)

Software on the internal functioning of computers would also appear patent eligible. In criticizing Alice’s claims, the Court said, “The method claims do not, for example, purport to improve the functioning of the computer itself.” While this is no express endorsement of such software as patent eligible, counsel for the owner of a patent on software for improving internal operations of a computer can take some comfort from, and surely will try to make the most of, this language. What is your take? Should you rely on the quoteddicta? Is it at least an implication that Justice Thomas or some justices will look favorably on the patent eligibility of such software?

What about data-compression and encryption patents? Are they technological enough? At least, they are not just “methods of organizing human activity”. But do they do more than routinely implement mathematical ideas, such as the idea that it is comparatively easy to multiply two large prime numbers but it is hard to factor the resulting product? (RSA encryption involves multiplying two large primes and then using so-called modular arithmetic. More on this later in this chapter—a Note on Encryption.)

On data compression, journalist Timothy Lee said this in Vox, “The Supreme Court doesn't understand software, and that's a problem”:If a patent claims a mathematical formula simple enough for a judge to understand how it works, she is likely to recognize that the patent claims a mathematical formula and invalidate it. But if the formula is too complex for her to understand, then she concludes that it’s something more than a mathematical algorithm and upholds it.

But this makes the law highly unpredictable, since it effectively depends on the mathematical sophistication of the judge who happens to take the case. And it's also logically incoherent. The courts originally excluded algorithms from patent protection because they are basic building blocks for innovation—that's as true of complex algorithms like data compression as of simple ones.

Do you agree? Did a majority of the Court think it understood the Flook algorithm but another majority didn’t understand the Arrhenius equation? Does this put a premium on counsel’s devising a simple, intuitive characterizartion of the claimed invention? Thus:

Diehr’s claimed invention was a fancy way of saying put a slab of beef in the oven with a meat thermometer stuck into it that is set to “medium rare” and then watch the dial until the black needle moves around until it coincides with the red marker.

Benson’s claimed invention was that “1492” means (1000 times 1) plus (100 times 4) plus (10 times 9) plus (1 times 2). Or, if you prefer, it was that (10 × n)

_{base 10}= n(2^{1}+ 2^{3})_{base 10}≡ [(sum of a shift left and two more shifts left) × n]_{base 2}.What about a new BIOS? A new reduced instruction set? A new microprocessor architecture? New nanocode?

5. During the several-month pendency ofAlice, the Court held threecertioraripetitions pending—Ultramercial, which is covered at length earlier in this section of the course materials;Accenture Global Services, GmbH v. Guidewire Software, Inc., 728 F.3d 1336 (Fed. Cir. 2013); andBancorp Services, LLC v. Sun Life Assurance Co., 687 F.3d 1266 (Fed. Cir. 2012). On June 30, 2014, the last day of the October 2013 Term, about a week after theAlicedecision, the Court disposed of the petitions. It vacated theUltramercialjudgment and remanded for the Federal Circuit to reconsider in light ofAlice.

InAccenture, the Federal Circuit had held patent ineligible a scheme for “generating tasks based on rules to be completed upon the occurrence of an event,” which required a computer system to update a database upon occurrence of an insurance-related event. In so ruling the court used a methodology similar to that ofAlice—one based onMayo, saying that “simply implementing an abstract concept on a computer, without meaningful limitations to that concept, does not transform a patent-ineligible claim into a patent-eligible one.”

InBancorp, the patent claimed a scheme for “managing a stable value protected life insurance policy,” again involving storage and data manipulation by computer. In this case the Federal Circuit held that the use of a computer was not sufficiently “integral to the claimed invention” to avoid patent ineligibility under the abstract-idea exception.

The Supreme Court deniedcertiorariinAccentureandBancorp, leaving the judgments of patent ineligibility undisturbed. What do you infer from that and the vacation of the eligibility judgment inUltramercial?

6. That theAliceopinion provides no detailed guidance on when computer implementation is effective to confer patent-eligibility is consistent with the recent course of Supreme Court decision in this area and should therefore be no surprise, at least to careful observers. As Professor John Duffy (now a refugee from GW Law School, but still well remembered here) observed: “[T]he Supreme Court has been remarkably resistant to providing clear guidance in this area, and this case continues that trend.” John Duffy,The uncertain expansion of judge-made exceptions to patentability, SCOTUSblog. But that reticence has disappointed many who had hoped, first inBilskiand then inAlice, for clear, categorical statements. Indeed,Mayo’s relatively more expansive discussion (in comparison with the other opinions) whetted their appetites and stirred their aspirations. But now, some commentators deplore the Court's series of narrow rulings and fear that they will never see a clear ruling on software or business methods from the Court. They fear that the Court will just keep on issuing narrow rulings on case-specific bases, out of anxiety lest a broad, informative statement from the Court have unintended, negative economic consequences or dash settled investor expectations or even stifle innovation.

They were particularly exasperated by Justice Thomas’s characteristic refusal to engage in development of theoretical infrastructure for the judgment or Socratic exploration of hypothetical cases to test the limits or soundness of the legal ruling. To them the highpoint (or perhaps low point) of all that inAlicewas his “we not labor” to spell out what is or isn't too abstract for patent eligibility:In any event, we need not labor to delimit the precise contours of the “abstract ideas” category in this case. It is enough to recognize that there is no meaningful distinction between the concept of risk hedging in

Bilskiand the concept of intermediated settlement at issue here. Both are squarely within the realm of "abstract ideas" as we have used that term.Professor Merges, for example, in

Go ask Alice—what can you patent after Alice v. CLS Bank?, protested, “To say we did not get an answer is to miss the depth of the non-answer we did get.” To similar effect, see note 2 and the accompanying cartoon.

But are the relatively doctrinally sparse, narrow, and fact-specific opinions in these cases—particularlyBilskiandAlice. butMyriadtoo—the expectable price of unanimity in a nine-member tribunal? Are the costs of uncertainty over specific case patterns outweighed by the greater sensed legitimacy and precedential stability that these opinions provide—which are among the benefits ofstare decisis? These unanimous opinions are unlikely to be overturned soon, if at all. Contrast the zig-zags fromFlooktoDiehr.

7. Even more fundamentally, are those who mourn the lack of “clear guidance in this area” and “the depth of the non-answer” that they got miguided—engaged in a quixotic search in pursuit of an unreachable goal? Consider a parallel pursuit in copyright law—the line of demarcation between idea and expression:

In copyright law, the “abstractions” or “patterns” or “concentric circles” analysis forexpression(say, the center of the bull’s-eye or the circles near it) andidea(say, the outer circles of the target) measures the difference by the magnitude of the body of limitations, in terms (for example) of plot and characterization. Thus, a play about a riotous knight and a foppish steward will not infringeTwelfth Night, for the shared pattern is too abstract, but as further limitations are added—such as the knight likes practical jokes, the steward is amorous of his mistress, the knight disturbs the household by carousing late at night, the steward is persuaded to write love notes and wear crossed garters, the knight is a kinsman of the mistress, the steward is thought mad and is locked up in a dark room, and so on—the characters more closely approach Sir Toby Belch and Malvolio, and the second play more closely approachesTwelfth Night, and at a certain radius the second play becomes infringing. SeeNichols v. Universal Pictures Corp., 45 F.2d 119, (2d Cir.1930) (L. Hand, J.); see alsoSheldon v. Metro-Goldwyn Pictures Corp., 81 F.2d 49 (2d Cir. 1936) (L. Hand, J.). In theNicholscase Judge Hand explained theabstractionsprinciple in these terms:Upon any work … a great number of patterns of increasing generality will fit equally well, as more and more of the incident is left out. The last may perhaps be no more than the most general statement of what the play is about, and at times might consist only of its title; but there is a point in this series of abstractions where they are no longer protected, since otherwise the playwright could prevent the use of his “ideas,” to which, apart from their expression, his property is never extended.

But moving from theory to application is problematic. There is no general theory of how to locate the correct radius. In

Peter Pan Fabrics v. Martin Weiner Corp., 274 F.2d 467 (2d Cir. 1960), Judge Hand's last copyright opinion written shortly before his death, he confessed:Obviously, no principle can be stated as to when an imitator has gone beyond copying the “idea,” and has borrowed its “expression.” . . . [O]ne cannot say how far an imitator must depart from an undeviating reproduction to escape infringement.

Is the result that one is thrown back on a case-by-case, fact-specific analysis—almost the equivalent of “I know it when I see it”?

Is the copyright mode of concentric-circles analysis helpful for purposes of § 101 analysis in this case? Does it suggest a structural relationship between abstractness and the specifics of implementation (such as the particular apparatus)? Does copyright law suggest that providing clear, precise, categorical demarcation for the concept of abstract ideas may be a hopeless task? To paraphrase Judge Hand in thePeter Pancase, is it so in patent law that “no principle can be stated” as to where the frontier of abstract idea is? In any case, has a satisfactory such principle yet been articulated? And is the result that one is thrown back on a case-by-case, fact-specific analysis—almost the equivalent of “I know it when I see it”?

8. On August 4, 2014, the PTO issued this statement addressing “our ongoing implementation of the June 19, 2014, unanimous Supreme Court decision inAlice”:[T]he USPTO has applications that were indicated as allowable prior to

Alice Corp., but that have not yet issued as patents. Given our duty to issue patents in compliance with existing case law, we have taken steps to avoid granting patents on those applications containing patent ineligible claims in view ofAlice Corp. To this end, our primary examiners and supervisory patent examiners (SPEs) promptly reviewed the small group of such applications that were most likely to be affected by theAlice Corp. ruling.

We withdrew notice of allowances for some of these applications due to the presence of at least one claim having an abstract idea and no more than a generic computer to perform generic computer functions. After withdrawal, the applications were returned to the originally assigned examiner for further prosecution. Over the past several days, our examiners have proactively notified those applicants whose applications were withdrawn.USPTO Commissioner for Patents Peggy Focarino, Update on USPTO's Implementation of

Alice.... BeforeAlice, it was apparently unclear to the PTO that theMayo“framework” applied to abstract ideas as well as to laws of nature and natural phenomena, involved inFunkandMayo. See Memo to Patent Examining Corps from Andrew H. Hirschfeld, Deputy Commissioner for Patent Examination Policy, Preliminary Examination Instructions in view of the Supreme Court Decision inAlice Corporation Pty. Ltd. v. CLS Bank International, et al. (June 25, 2014). After the Supreme Court’sAlicedecision came down, the PTO understood the message that it did.

9. Shortly after this, commentator Michael Borella pointed out that “the USPTO [in the guidelines] does not address whether prior art should be used to establish that an abstract idea is ‘fundamental’ or to determine that an additional limitation is more than routine or conventional.” Should the PTO use and cite references to support an assertion that the expedient recited in a limitation is routine or conventional? In theAliceopinion, in concluding that intermediated settlement was an old concept, the Court said, citing references:Like the risk hedging in

Bilski, the concept of intermediated settlement is “a fundamental economic practice long prevalent in our system of commerce.” See, e.g., Emery,Speculation on the Stock and Produce Exchanges of the United States, in 7Studies in History, Economics and Public Law283, 346-356 (1896) (discussing the use of a "clearing-house" as an intermediary to reduce settlement risk).But in finding the implementation routine and conventional, the Court essentially took judicial notice that the recited elements were routine and conventional:

Nearly every computer will include a “communications controller” and “data storage unit” capable of performing the basic calculation, storage, and transmission functions required by the method claims. See 717 F.3d at 1290 (Lourie, J., concurring). As a result, none of the hardware recited by the system claims “offers a meaningful limitation beyond generally linking ‘the use of the [method] to a particular technological environment,’ that is, implementation via computers.”

Is there a good reason to expect to see a reference for the conclusion that the alleged abstract idea is old and well known, but not to demand that for the conclusion that the implementation is old?

10. Another commentator on the guidelines complained that the PTO was “running wild” in its use ofAlice:[I]n what surprised many in this field, in response to the

Alicedecision the USPTO is now running wild with the “abstract idea” exception and applying it to all sorts of claims (in various art units) that are not directed as a whole to “abstract ideas” as identified by the precedent. In these new Section 101 rejections, examiners currently appear to be creatively identifying abstract ideas in claims and discounting any structure as “conventional” and concluding claims are directed to “abstract ideas.” Any claim (whether directed to software or not) can be rejected under Section 101 using this approach.Can the Board be relied on to curb this? If not, will the Federal Circuit do so?

12. Here is a commentator’s complaint that an examiner is fishing a § 101 abstract-idea rejection out of thin air, instead of citing “a college textbook describing the idea, or an industry reference publication, or a technical or journal article providing a survey of basic industry practices”:

“

First, we find the bedbugs, Ma’am. And then, we drive these tiny stakes through their wee, black hearts.”Consider the following recent rejection: “The claims are directed toward the abstract idea of viewing formula dependencies in a spreadsheet.”

This idea is obviously not a fundamental building block of science or commerce, or even computer science. You will not find it discussed in textbooks in computer science or engineering. First, whatever the invention is, it is specific to spreadsheets. While spreadsheets are common, that does not make them fundamental building blocks of human ingenuity. An invention that is specific to spreadsheets, or word processors, or other particular tools (whether computer-implemented or not) should never qualify as fundamental. This rejection demonstrates the further the risk of conflating Section 101 with Section 102; here, the examiner’s underlying concern is more likely that the claims are anticipated, but if that is the case, she must cite a reference.Is this a fair comment? Should the ground of abstract-idea rejection instead be mental steps? Would a reference be expected for that? (Courts ordinarily just say that a person could do this in their head or with a pencil and paper. See, e.g.,

Dietgoal Innovations, LLC v. Bravo Media LLC, (S.D.N.Y July 8, 2014) (patent on use of computer to plan meals and achieve diet goals held to be invalid because claimed process was directed to an abstract idea that could have been performed by human beings);Comcast IP Holdings, LLC v. Sprint Communications Co. (D. Del. July 16, 2014) (patent on computerized telecommunications system that checked with user before deciding whether or not to establish a new connection invalidated because claimed process was directed to abstract idea that could have been performed by human beings).)

Note on Encryption SoftwareIn the oral argument of the

Alicecase, the issue was raised what kind of software would be patent-eligible under theMayoanalysis. Counsel for CLS Bank suggested that data compression and data encryption were software technologies that are likely to be patent-eligible because they address “a business problem, a social problem, or a technological problem.” The Solicitor General, asamicus curiae, said that it would be difficult to identify a patent-eligible business method unless it involved an improved technology, such as “a process for additional security point-of-sale credit card transactions using particular encryption technology” – “that might well be patent eligible.” This note examines those issues further.

Wikipedia says: “Encryption software executes an algorithm that is designed to encrypt computer data in such a way that it cannot be recovered without access to the key.”

That means we can speak of two functions, F and its inverse F^{-1}. F(x) = y, and F^{-1}(y) = x. F is the function that encrypts x into F(x), which is y. F(x) = y. The inverse function F^{-1}is the inverse of F and decrypts the encrypted x, i.e., F(x) = y, back into the original x. F^{-1}(y) = x. Or F^{-1}{F(x)] = x. Those equations provide the paradigms for encryption and decryption.

In cryptographic parlance, what has just been represented as the independent variable x is the “plaintext,” the original message text; what has just been represented as the dependent variable y is the “ciphertext,” the encrypted message. F and F^{-1}represent the processes of encryption and decryption.

The task of the would-be encryptor Alice is to devise a system that permits her to easily convert a plaintext P into a ciphertext C that her colleague Bob can easily decrypt, using a key known only to him. But for others, such as Charlie, who do not know the key, it should be very hard to decrypt the ciphertext C and thus ascertain the plaintext P. The encryption function is referred to as a “one way” function. It is easy to encrypt the message, but hard for the uninitiated to decrypt it. (Think of going the wrong way on a one-way street in rush hour or think of diodes.)

This diagram illustrates the idea of the encryption and decryption processes:

Restating this, the idea of an encryption - decryption system is to devise a set of functions F and F

^{-1}such that a first party, Alice, starts out with a plaintext P or x. She encrypts it (in effect, locks it up) with encrypting function F in order to provide an encoded message F(x) = y, the ciphertext. Then Alice sends ciphertext C over insecure communication links (the Internet, a radio broadcast, etc.) to a second party, Bob. Nobody who intercepts it (for example, such a third party as Charlie) is supposed to be able to understand C. (They do not have access to the key.) The recipient Bob then decrypts ciphertext C by using decryption function F^{-1}to produce the original plaintext P. (Bob has a secret key to unlock the locked-up text.)

This requires that F(P) = C, and F^{-1}(C) = P. That is, F^{-1}(F[P]) = P. The decrypted (encrypted-P) is P, for any message sent using the system. Devising and “breaking” such F and F^{-1}systems has occupied Alices, Bobs, and Charlies for several millennia.

Historical examples

For example, Julius Caesar is said to have used the “shift-2” cipher, meaning substitute for each letter the letter two later in the alphabet – so A → C, B → D, C → E, etc. Here F(x) is shifting letter ordinal place twice ahead to produce y, the ciphertext, from x, the plaintext. To decipher y, F^{-1}is the operation of shifting the letter ordinal place twice back to produce x, the plaintext. That is, A ← C, B ← D, C ← E, etc. In this system, the plaintext message “SECURE” becomes the ciphertext “UGEWTG.” When we shift each letter back two places, to decrypt, we get “SECURE” again.

Another simple cipher is XOR (exclusive OR, also written as ⊕). Two XORs on a datastream p, using the same arbitrary keystream q, give you back the original datastream. (p ⊕ q) ⊕ q = p. Thus:

Plaintext1 0 0 1 0 1 1XOR Keystream0 1 0 1 1 0 1= Ciphertext1 1 0 0 1 1 0XOR Keystream0 1 0 1 1 0 1= Plaintext1 0 0 1 0 1 1

A hypothetical XOR patent

Suppose that I were the inventor of XOR encryption. I claim:1. A method of encrypting a digitized message M with a digitized keyword K to produce a digitized ciphertext C, where:

said message M has m binary digits,

said keyword K has k binary digits, and

there is a number n such that m = nk – p, where 0 ≤ p < k ≤ m,said method comprising the following steps:

(1) dividing said message M into n blocks of length k, except that the nth block is of length k – p;

(2) if p > 0, adding to the nth block of said digitized message M numbers representative of digitized letters comprising a nonsense word or words until p = 0 and m = nk;

(3) successively XORing each of said n blocks of said digitized message M with said digitized keyword K; and

(4) concatenating the results of the third step to produce a ciphertext C wherein C = M ⊕ K.

2. A method of decrypting a digitized ciphertext C that has been encrypted in accordance with claim 1, said method comprising the following steps:

(1) dividing said ciphertext C into n blocks;

(2) successively XORing each of said n blocks of said digitized cipherword C with said digitized keywored K; and

(3) concatenating the results of the second step to produce a digitized message M' wherein M' = C ⊕ K.

3. A method for automatically enabling a verification for a credit card transaction via an insecure communication medium, said method comprising the following steps:

(1) inputting credit card information via a credit card reader to a computer at a retailer location;

(2) encrypting said credit card information in accordance with claim 1 by means of said computer, thereby producing a ciphertext;

(3) transmitting said ciphertext via said insecure communication medium to a verification location associated with said credit card;

(4) at said verification location:

decrypting said ciphertext in accordance with claim 2, and

performing a verification procedure using said credit card information, thereby producing a go-nogo signal, and transmitting said go-nogo signal to said retailer location; and

(5) receiving said go-nogo signal at said retailer location and acting in accordance therewith.

The hypothetical specification of the hypothetical patent does not propose use of any unconventional hardware. The only references to hardware in the claims suggest conventional devices, such as a conventional computer and conventional credit card reader; the insecure communication medium is presumably the Internet. For a patent like this, see U.S. patent no. 5,826,245 and see

Card Verification Solutions, LLC v. Citigroup, Inc.(N.D. Ill. Oct. 28, 2014), found below.

Is the XORing “invention” patent elgible?

In theFlookcase, the Court said: “As the CCPA has explained, ‘if a claim is directed essentially to a method of calculating, using a mathematical formula, even if the solution is for a specific purpose, the claimed method is nonstatutory.’ ” In July 2014, just afterAlice, a Federal Circuit panel inDigitech Image Technologies, LLC v. Electronics for Imaging, Inc., held a patent ineligible, quotingFlookand adding, “Without additional limitations, a process that employs mathematical algorithms to manipulate existing information to generate additional information is not patent eligible.” Even beforeFlook, in theBensoncase, the Court held patent ineligible a claim to a mathematical calculation (conversion of BCD format numbers to binary format). TheFlook-Mayo-Aliceline of cases clearly hold that an implementation of an abstract idea, such as a mathematical calculation, must add something inventive to the idea in implementing it in order to be patentent eligible. How does this case law apply to the XORing encryption patent?

After readingAliceand the post-Alicecases, does the assurance in the oral argument that encryption processes will survive theAlicecase appear to be a sound prediction? Is there “something extra” in the XOR claims that transforms the number crunching into patent-eligible subject matter? Is there any inventive concept beyond the mathematics? Is there more than the use of a generic computer employed to perform ordinary arithmentic operations? Do the credit card reader or Internet aid me as the hypothetical inventor?

If you think XORing is too simple to prove anything, let us consider a more serious invention—RSA encryption. RSA encryption is the most widely used form of encryption for Internet transactions. Some students have objected, however, to the technical nature of the following discussion. If your eyes glaze over when number theory is ventilated, skim over the following or just go directly to the last few paragraphs.

Public key/private key encryption

This is the modern kind of encryption in which we are seriously interested. Public key/private key encryption is based on the multiplication of large prime numbers, which produces a product that is very hard to factor. This property of large primes provides the basis of an important encryption scheme—the RSA encryption scheme, which was discovered—orinvented(that is the legal issue)—in the mid-1970s. The acronym RSA is based on the initial letters of the surnames of the “inventors”—Ronivest, AdiRhamir, and LeonardSdleman. MIT received U.S. Patent No. 4,405,829, on September 20, 1983, for a “Cryptographic communications system and method” that used the RSA encryption algorithm. This note explores the question whether the RSA patent was valid under § 101. (I say “was” because the patent has expired.)A

Modular arithmetic

Much encryption (including RSA) is based on a branch of number theory calledmodular arithmetic. The normal 12-hour clock dial illustrates modular arithmentic with the modulus 12. In this system there is no 0; the numeration starts again after 12—its successor is 1. Thus, 4 hours after 12 is 4; 4 hours after 11 is 3. A number such as 4 has no multiples of 12 in it, so we can simply write it as 4. The number 16 is (1)(12) + 4, which is also equivalent to 4 in this system. (If you divide 16 by 12, you get 1 with a remainder or residue of 4.) Similarly, the number 28 is (2)(12) + 4, which is also equivalent to 4. The number 12 always acts as if 0 in this cyclical system.

The notation for this is 4 ≡ (4 mod 12); 16 ≡ (4 mod 12); 28 ≡ 4 mod 12, etc. In other words, 4 + 12n is congruent to or equivalent to 4; (4 + 12n) ≡ 4. They are all at 4 o'clock on the 12-hour dial shown above.

A few points about modular arithmentic are important to encryption. Euclid of Alexandria (circa 300 BC) discovered what is called the Fundamental Theorem of Arithmetic. Each number has a unique prime factorization. A prime number has only itself and 1 as factors. Any other number (a so-called composite number) can be decomposed or factored into a set of prime numbers. When multiplied together, this set of primes equals the relevant number. For example, 39 = 3 • 13, 68 = 17 • 2^{2}, 180 = 2^{2}• 3^{2}• 5. Each of those factorizations is unique. You cannot factor 68, for example, into 3 times some other prime (or composite) number, such as 13. The only possible complete factorization of 68 is 17 • 2^{2}.

Leonhard Euler introduced the totient or phi function, φ(n), in 1763. (This note will sometines use T or T(N), rather than φ(n), to refer to the totient in connection with RSA encryption.) The totitent of n is is a count of all positive integers k that do not share any common factor with the number n, or are “relatively prime” to the number. 1 ≤ k < n, where k and n have no common factor. Examples:

φ(8). There are four numbers—1, 3, 5, and 7—that share no factors with 8. (We count 1 but not 0 or 8.) The numbers 2 and 6 (i.e., 3•2), however, share the factor 2 with 8. Therefore, φ(8) = 4.

φ(9). There are six numbers—1, 2, 4, 5, 7 and 8—that share no factors with 9. However, 3 and 6 share the factor 3 with 9. Therefore, φ(9) = 6.

φ(1) = 1.

For any prime number p, φ(p) = p – 1.

For the product of any two primes p and q, φ(p•q) = φ(p)•φ(q) = (p–1)•(q–1).

This last item leads us to a simple way to find the totient of the product of two primes. Euler showed that φ(p•q) = φ(p)•φ(q). That is, the totient of the product of any two primes, p and q, is the product of their totients. Thus, φ(p•q) = φ(p)•φ(q) = (p–1)•(q–1). 77 is the product of the two primes 7 and 11. φ(77) = φ(7•11) = φ(7)•φ(11) = 6•10 = 60. Therefore, φ(77) = 60. We will put that away for later use with RSA encryption.

If we take two very large primes p and q, and multiply them, so p•q = N, we can easily determine φ(N). Since we know p and q, we simply multiply (p–1)•(q–1). But because N is large, others from whom p and q are kept secret cannot determine φ(N) without an inordinate amount of trial and error—many, many years of effort.

These aspects of number theory concerning primes interesect with some principles of modular arithmetic theory to provide the basis for RSA encryption. This scheme is based on a theorem that Euler proved in the 18th century, Euler’s Totient Theorem, which is to this effect:m

^{φ(n)}= 1 mod n = 1The following is is an example illustrating the operation of this theorem:

Let m = 5, n = 8. Then, substituting these numbers into the theorem, 5

^{φ(8)}= 1 mod 8.φ(8) = 4. Therefore, 5

^{φ(8)}= 5^{4}= 625.625 ≡ 1 mod 8 ≡ 1, because 625 = 8•78+

^{1}/_{8}.What this theorem comes down to for RSA purposes is, surrprisingly, that you can exponentiate a number and use modular arithmetic, and then do it again with another exponent, and if the two exponents have the right relationship to one another, and to n in the above equation, then the final result is the original number, but the intermediate result is indecipherable. If the exponentiated number represents a message, this provides a way to encrypt and decrypt the message.

The concept of the RSA encryption procedure

To begin with, encryption and decryption functions F and F^{-1}must be devised. To encrypt a message using the RSA algorithm or method, a would-be encryptor chooses two large primes, p and q—which are kept secret—and multiplies them to produce a number N. (All numbers in this discussion are positive integers.) The number N will be the modulus of the encryption system. Because p and q are both large and are primes, their product N, even though made public, will be practically impossible to factor, so that p and q will remain secret. (According to the specification of the RSA patent, a 200-bit number requires 10^{9}years to factor, using known computer-implemented methods.)

The next steps are to use p and q to devise the public key e (used for encryption) and the private key d (used for decryption) in the system. First, a number T is calculated. T = φ(N) = (p–1)•(q–1). The public key will be any number e such that: e < T, and e shares no factors with T. (T and e are “coprime.”)

The private key will be a number d such that, when it is multiplied by e (the public key), it is congruent to 1 mod T. In other words, the following congruence must hold:(d•e) ≡ (1 mod T).

When you divide the product d•e by T, the residue has to be 1. For some integer x, 1 + x•T = d•e; x may be 1, so that in that case d = (1+T)/e; if x is 2, d = (1+2T)/e, and so on. As will presently appear, x is selected by an empirical process. The relationship between e and d is such that two exponentiations in the RSA procedure will offset one another or effectively cancel eachother out, as explained below. The underlying reason is that 1 mod T ≡ 1, so that exponentiating by d•e has the net effect of multiplying by 1 . . . that is, no effect.

To encrypt plaintext P (a number) into ciphertext C (another number), we raise P to the e power, P^{e}, and then determine its residue mod N. That residue is ciphertext C. Thus,

(1)(P^{e}mod N) ≡ C.The ciphertext C can be decrypted to produce the plaintext P. The ciphertext C is raised to the d power, C

^{d}, and then its residue mod N is determined, which is supposed to be plaintext P. Thus, (C^{d}mod N) ≡ P. At least that is so in principle. Perhaps we should restate that as

(2)(C^{d}mod N) ≡ P′and ascertain whether P′ really = P. Is there a proper inverse relationship?

We can eliminate C from the decryption equation by substituting in the value of C from the encryption equation(1)into the decryption equation(2). Thus,

(3)[P^{e}mod N]^{d}mod N = P ′.Is P = P ′? For that to be true, what we need is, in effect, for the operation {[( )

^{e}mod N]^{d}mod N} to be equivalent to 1, which means that the exponentiations to e and d should offset one another. That will make P ′ = P.

We have the additional information that that (d•e) ≡ (1 mod T(N)), where T(N) is φ(N). Based on all of the foregoing information, it can be proved that RSA works. A proof can be found in a Wikipedia article on RSA.

A “suggestion” of a proof of RSA follows (but it is not a proper proof). Briefly:

(P ^{e}mod N)^{d}mod N = P^{e•d}mod N( x ^{y})^{z}= x^{yz}and consolidate mod Ns= P ^{(1 mod T)}mod NSubst. d•e ≡ 1 mod T into the exp. of P above = P ^{1}mod N = P mod NBut [1 mod T = 1]; subst. that into above eq. = P But x mod y = x, so [P mod N = P]. Subst. that into above eq. (P ^{e}mod N)^{d}mod N = PQ.E.D. Let us instead see if and how RSA works, using actual numbers.

The RSA encryption procedure illustrated using specific numbers

An illustrative example follows. Alice wants to send a message P to Bob. She must first convert the plaintext message P into numbers, which will be encrypted to ciphertext C. Somebody, not necessarily Alice or Bob, must design the encryption system of functions F and F^{-1}. For sake of simplicity, assume this is Alice.

Alice must choose two large (here, small to make it easier for the student to follow) primes, p and q. Let p = 61 and q = 53.

The modulus N = p•q = 61 • 53 = 3233.

φ(N) = (p – 1) • (q – 1) = 60 • 52 = 3120. This is the totient T.

The public key e will be 17; that prime number has no common factor with T, which is 3120.

The decryption key, d, must be a number such that, when multiplied by e, its product is congruent to 1 mod 3120. That is, d • e ≡ 1 mod T. Restated, a number d must be selected such that (17 × d) = 1 + some multiple x times the totient 3120. By tedious trial and error, we find that the combination x = 15 and d = 2753 works, because (17 • 2753) = 46,801 = 1 + 46,800; 46,800 = (3120 • 15). (One way to do this is to take the equation d•e = 1 + x•T, solve for d: d = (1/e)•(1 + x•T). Then substitute in values of x: 1, 2, 3, … until d is solved as an integer.)

Therefore, in summary, the important numbers are as follows: the product p•q is N = 3233; public key e = 17; and the private key d = 2753. Those are the three numbers Alice and Bob need to send messages. The numbers p and q will always be kept secret, known only to the system designer. T is used to calculate the keys, but only the system designer needs to know it. The public key e used for encrypting messages can be known by anyone, as can N. The private key d needs to be known only by the person using it for decryption purposes. The public key is known by anybody who will encrypt and send a message.

Now Alice is ready to send Bob a message P. Let’s say that plaintext P = 65. Alice must convert P into a ciphertext C that cannot be decrypted without private key d.

Now, Alice makes this calculation: P

^{e}mod N = C. 65^{17}mod 3233 = 2790. 2790 is the ciphertext C for plaintext 65.Alice can send

2790to Bob over an insecure line. Bob can then decrypt ciphertext2790using only N = 3233 and d = 2753, as follows: C^{d}mod N = P. 2790^{2753}mod 3233 = 65. This result equals the original plaintext number.That is how RSA encryption works. It worked in the example because the relation of d and e was such that the decryption operation acted as an inverse to the encryption operation.

The LCM solution

In the original RSA paper, the decryption method was that stated above. In the patent, however, a slightly different decryption process is given. The specification of the RSA patent explains the decryption aspect of the system in terms of what the patent calls a “l cm,” which is described in the specification as a “least common multiple” of two numbers, such as the numbers (p–1) and (q–1). Specifically, the patent describes the relation of e and d, the encrypting and decrypting keys, as: e•d ≡ 1 (mod(1 cm((p–1), (q–1)))), rather than as 1 mod φ.

The customary notation forleast common multipleis LCM or lcm (i.e., LCM in lower case letters, not the numeral 1 and cm). As will appear, it is almost certain that the “1 cm” of the RSA patent is a typographic error for LCM written in lower case letters. (Perhaps, somebody confused LCM in lower case letters with “one centimeter,” abbreviated and in figures.)

There exists a different possible solution for the decryption key d, based on the least common multiple (lcm). We previously used the equations—d•e = 1 mod T = 1 + xT, and found an x such that it provided an integer value for d. There is a more general formulation of m^{φ(n)}= 1 mod n = 1. It is m^{λ(n)}= 1 mod n = 1, where λ(n) is the smallest integer for which this equation is true. (λ(n) is known as the Carnichael function. The Carmichael function is also known as the reduced totient function or the least universal exponent function and is sometimes designatedψ(n).)

Now, T or φ(n) is not the smallest integer that is the least common multiple of (p–1) and (q–1). For example, φ(143) = φ(11•13) = 10•12 = 120. But 60, only half the totient φ, is a multiple of both 10 and 12. In fact, for any pair of primes p and q, p and q are each odd and (p–1) and (q–1) are each even, making the latter of the form 2a and 2b, so that their product is 4ab. 2ab is thus a multiple of both 2a and 2b. But φ = (2a+1)•(2b+1) = 4ab + 2a + 2b + 1 > 2ab. Thus, the Euler totient approach always requires use of a value at least twice the lcm (least common multiple) approach.

λ and lcm are functions with properties similar to φ:

- λ[lcm (a, b)] = lcm [λ(a), λ(b)]

- λ(p•q) = lcm [λ(p), λ(q)]

- lcm [λ(p), λ(q)] = lcm [(p–1), (q–1)]
Therefore, λ(p•q) = lcm [(p–1), q–1)]. Based on reasoning similar to that applying to φ(pq), d•e = 1 mod λ(p•q), so that d•e = 1 mod lcm (p–1, q–1). It makes no difference which system is used, but they are not necessarily compatible, so that encrypting in one system and then trying to decrypt in the other may not work.

There is still a third approach, but it is equivalent to the others. A function gcd, meaninggreatest common denominator, can be used in place of lcm. λ(n) = lcm(p–1, q–1) = (p–1)(q–1)/gcd(p–1, q–1).

The RSA patent

Here is a representative encryption method claim for the RSA patent:

23. A method for establishing cryptographic communications comprising:

encoding a digital message word signal M to a ciphertext word signal C, where

M corresponds to a number representative of a message and 0 ≤ M ≤ n-1,

where n is a composite number of the form n = p•q, where p and q are prime numbers, and

where C is a number representative of an encoded form of message word M, wherein said encoding step comprises transforming said message word signal M to said ciphertext word signal C, whereby C≡M^{e}(mod n), where e is a number relatively prime to (p-1)•(q-1).In this claim message word M corresponds to plaintext P in the preceding discussion; n corresponds to N. The decoding process (decryption) is claimed in claim 24, as follows:

24. The method according to claim 23 comprising the further step of:

decoding said ciphertext word signal C to said message word signal M, wherein said decoding step comprises the step of:

transforming said ciphertext word signal C, whereby M ≡ C

^{d}(mod n), where d is a multiplicative inverse of e (mod {1 cm [(p-1), (q-1)]}).A representative system claim follows:

1. A cryptographic communications system comprising:

A. a communications channel,

B. an encoding means coupled to said channel and adapted for transforming a transmit message word signal M to a ciphertext word signal C and for transmitting C on said channel,

where M corresponds to a number representative of a message and

0 ≤ M ≤ n–1

where n is a composite number of the form n = p•q, where p and q are prime numbers, and

where C corresponds to a number representative of an enciphered form of said message and corresponds to

C ≡ M

^{e}(mod n)where e is a number relatively prime to 1 cm(p–1, q–1), and

C. a decoding means coupled to said channel and adapted for receiving C from said channel and for transforming C to a receive message word signal M'

where M' corresponds to a number representative of a deciphered form of C and corresponds to

M' ≡ C

^{d}(mod n)where d is a multiplicative inverse of e (mod[1 cm{(p–1),(q–1)}]).

The specification does not propose use of any unconventional hardware. The only references to hardware suggest conventional devices. For example, the specification says. “The communications channel 10 may include, for example, a conventional broad-band cable with associated modulator and demodulator equipment at the various remote terminals to permit data transfer between teminals connected to the channel and the channel itself.” At another point, the specification says, “These blocks in the device 12 are conventionally arranged together with timing control circuitry to perform ‘exponentiation by repeated squaring and multiplication.’ ”

Is RSA patent elgible?

In theFlookcase, the Court said: “As the CCPA has explained, ‘if a claim is directed essentially to a method of calculating, using a mathematical formula, even if the solution is for a specific purpose, the claimed method is nonstatutory.’ ” In July 2014, just afterAlice, a Federal Circuit panel inDigitech Image Technologies, LLC v. Electronics for Imaging, Inc., held a patent ineligible, quotingFlookand adding, “Without additional limitations, a process that employs mathematical algorithms to manipulate existing information to generate additional information is not patent eligible.” Even beforeFlook, in theBensoncase, the Court held patent ineligible a claim to a mathematical calculation (conversion of BCD format numbers to binary format). TheFlook-Mayo-Aliceline of cases clearly hold that an implementation of an abstract idea, such as a mathematical calculation, must add something inventive to the idea in implementing it in order to be patentent eligible. How does this case law apply to the RSA patent?

After readingAliceand the post-Alicecases, does the assurance that encryption processes will survive theAlicecase appear to be a sound prediction? Is there “something extra” in the RSA claims that transforms the number crunching into patent-eligible subject matter? Is there any inventive concept beyond the mathematics? Is there more than the use of a generic computer employed to perform ordinary arithmentic operations?

In theCard Verificationcase mentioned earlier, the court declined to dismiss on the pleadings. It said:[A] review of the diagrams demonstrates incorporation of a computer, nonsecure network, and pseudorandom tag generating software. A plausible interpretation of the patent is that computing devices, software, keyboards, and credit card readers would be required to use the invention.

Although simply implementing an abstract idea on a computer is not a patentable application of the idea, seeAlice, a plausibly narrowing limitation is that of required pseudorandom tag generating software. The question whether a pseudorandom number and character generator can be devised that relies on an algorithm that can be performed by a human with nothing more than pen and paper poses a factual question inappropriate at the motion to dismiss stage. Without discovery on the issue, the Court is bound to make all reasonable inferences in favor of Card Verification. Here, an entirely plausible interpretation of the claims include a limitation requiring pseudorandom tag generating software that could not be done with pen and paper. Accordingly, Card Verification has plausibly alleged a method that does not comprise a “mental process.”Even assuming that a psuedorandom number could not be obtained by glancing at the second hand of one’s watch or by flipping coins, and that this is enough to avoid a mental-steps invalidation, does that matter? Is not being a set of mental steps a sufficient condition for eligibility or a necessary comndition? The

Card Verificationcourt also added that the claimed process plausibly requires a sufficiently concrete transformation so as to ground the abstract idea to a particular inventive implementation, because adding random number tags to credit card number data in the computer could be seen as “fundamentally altering the original confidential information.” Is using that pro-eligibility factor consistent with the analysis thatAliceprescribes? For example, theMayoCourt said that “we have neither said nor implied that the [machine-or-transformation] test trumps” the exclusions from patent eligibility of natural law [or, presumably, abstract ideas]. How does all of this apply to the patent eligibility of RSA and other encryption patents?

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