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`BEFORE THE PATENT TRIAL AND APPEAL BOARD
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`FACEBOOK, INC., INSTAGRAM, LLC and WHATSAPP INC.,
`Petitioners
`
`v.
`
`BLACKBERRY LIMITED
`Patent Owner
`
`______________
`
`U.S. Patent No. 7,372,961
`Issue Date: May 13, 2008
`
`Title: METHOD OF PUBLIC KEY GENERATION
`______________
`
`
`DECLARATION OF JONATHAN KATZ, PH.D.
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`Facebook's Exhibit No. 1002
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`Declaration of Jonathan Katz, Ph.D. in Support of
`Petition for Inter Partes Review of
`U.S. Patent No. 7,372,961
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`
`
`I.
`
`INTRODUCTION AND QUALIFICATIONS .............................................. 5
`A. Qualifications and Experience ............................................................. 5
`B. Materials Considered ............................................................................ 7
`PERSON HAVING ORDINARY SKILL IN THE ART ............................ 10
`II.
`III. LEGAL PRINCIPLES USED IN THE ANALYSIS ............................... 13
`B.
`Prior Art .............................................................................................. 13
`C.
`Claim Construction............................................................................. 14
`D.
`Legal Standards for Obviousness ....................................................... 16
`IV. RELEVANT TECHNOLOGY BACKGROUND ................................... 21
`B.
`Random Number Generation ............................................................. 22
`C.
`Cryptography and Its Reliance on Random Number Generators ...... 23
`D. Modular Reduction and the Modulo Bias Problem ........................... 29
`E.
`Rejecting Random Numbers Above a Desired Range, aka
`Rejection Sampling ............................................................................ 34
`V. OVERVIEW OF THE ’961 PATENT ...................................................... 46
`A.
`The Specification ................................................................................ 46
`B.
`The Challenged Claims ...................................................................... 51
`C.
`Interpretation of Claim Limitations in the ’961 Patent ...................... 53
`VI. APPLICATION OF THE PRIOR ART TO THE CHALLENGED
`CLAIMS OF THE ’961 PATENT ............................................................. 59
`B.
`Brief Description and Summary of the Prior Art ............................... 60
`1.
`Brief Summary of DSS (EX. 1004) ......................................... 60
`2.
`Brief Summary of Rose (EX. 1006) ........................................ 66
`3.
`Brief Summary of Menezes (EX. 1005) .................................. 70
`4.
`Brief Summary of Schneier (EX. 1008) .................................. 72
`5.
`Brief Summary of Rao (EX. 1018) and Floyd (EX. 1019) ...... 73
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`Declaration of Jonathan Katz, Ph.D. in Support of
`Petition for Inter Partes Review of
`U.S. Patent No. 7,372,961
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`4.
`5.
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`C. Ground 1: Comparison of the Prior Art to the Challenged
`Claims Based on DSS, Schneier, and Rose........................................ 75
`1.
`Independent Claim 1 ................................................................ 75
`2.
`Dependent Claim 2: The method of claim 1 wherein
`another seed value is generated by said random number
`generator if said output is rejected. ........................................ 116
`Dependent Claim 5: The method of claim 1 wherein said
`order q is a prime number represented by a bit string of
`predetermined length L. ......................................................... 116
`Independent Claim 15: ........................................................... 117
`Dependent Claim 16: The computer readable medium
`of claim 15 wherein another seed value is generated by
`said random number generator if said output is rejected. ...... 120
`Dependent Claim 19: The computer readable medium of
`claim 15 wherein said order q is a prime number
`represented by a bit string of predetermined length L. .......... 121
`D. Ground 2: Comparison of the Prior Art to the Challenged
`Claims Based on DSS, Schneier, and Menezes ............................... 121
`2.
`Independent Claim 1 .............................................................. 122
`3.
`Dependent Claim 2: The method of claim 1 wherein
`another seed value is generated by said random number
`generator if said output is rejected. ........................................ 135
`Dependent Claim 5: The method of claim 1 wherein said
`order q is a prime number represented by a bit string of
`predetermined length L. ......................................................... 136
`Independent Claim 15: ........................................................... 136
`5.
`Dependent Claims 16, 19 ....................................................... 136
`6.
`Grounds 3-4: Comparison of the Prior Art from Grounds 1-2, in
`Further View of Rao and Floyd........................................................ 137
`2.
`Independent Claim 23: ........................................................... 137
`3
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`3.
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`6.
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`4.
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`E.
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`Declaration of Jonathan Katz, Ph.D. in Support of
`Petition for Inter Partes Review of
`U.S. Patent No. 7,372,961
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`3.
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`Dependent Claim 24: The cryptographic unit of claim
`23 wherein another seed value is generated by said
`random number generator if said output is rejected. ............. 145
`Dependent Claim 27: The cryptographic unit of claim
`23 wherein said order q is a prime number represented by
`a bit string of predetermined length L. .................................. 146
`VII. CONCLUSION
`146
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`4.
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`Declaration of Jonathan Katz, Ph.D. in Support of
`Petition for Inter Partes Review of
`U.S. Patent No. 7,372,961
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`I, Jonathan Katz, PhD., declare as follows:
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`1.
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`I have personal knowledge of the facts stated in this declaration, and
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`could and would testify to these facts under oath if called upon to do so.
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`I.
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`INTRODUCTION AND QUALIFICATIONS
`A. Qualifications and Experience
`I received an undergraduate degree in mathematics from the
`2.
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`Massachusetts Institute of Technology in 1996. My coursework included several
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`classes in computer science. I entered the Ph.D. program in computer science at
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`Columbia University in 1998. My doctoral work focused on cryptography.
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`3.
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`In early 2000, I taught a course “Introduction to Programming in C” at
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`Columbia University, and in early 2001, I taught a graduate-level “Introduction to
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`Cryptography” course there. In 2002, I received my Ph.D. (with distinction) in
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`computer science; my doctoral thesis was entitled “Efficient Cryptographic
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`Protocols Preventing ‘Man-in-the-Middle’ Attacks.”
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`4.
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`From May 1999 to March 2000, while I was conducting my doctoral
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`research, I worked as a security consultant for Counterpane Systems. In connection
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`with my work for Counterpane, I discovered security flaws in the popular email
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`encryption software Pretty Good Privacy (PGP). This work was covered in the press
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`U.S. Patent No. 7,372,961
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`and led to two published papers and a refinement of existing standards for email
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`encryption. I also designed and implemented secure cryptographic protocols for
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`clients. In March 2000, I began work as a Research Scientist in the cryptography
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`group at the Mathematical Sciences Research Center of Telcordia Technologies.
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`5.
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`Since 2002, I have worked as a professor in the Department of
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`Computer Science at the University of Maryland. In that capacity I regularly teach
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`courses in cryptography and cybersecurity at both the undergraduate and graduate
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`level, conduct research in those fields, and supervise graduate-student research. I
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`was appointed Director of the Maryland Cybersecurity Center in 2013. In that
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`capacity I regularly interact with cybersecurity researchers at other academic
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`institutions around the world, as well as with industry professionals working in the
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`area of cybersecurity.
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`6.
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`In 2007, I co-authored a
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`textbook Introduction
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`to Modern
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`Cryptography, now in its second edition. This text has been used in courses on
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`cryptography worldwide. I also authored the book Digital Signatures in 2010. Since
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`2011, I have served as an editor of the Journal of Cryptology, the leading journal in
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`the field. In 2016 and 2017, I served as co-program chair of the annual Crypto
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`conference, the leading and longest-running academic conference focused on
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`Declaration of Jonathan Katz, Ph.D. in Support of
`Petition for Inter Partes Review of
`U.S. Patent No. 7,372,961
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`cryptography. I am currently serving as co-program chair of the ACM Conference
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`on Computer and Communications Security, one of the top conferences in the field
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`of cybersecurity. A list of other program committees and editorial boards I have
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`served on can be found as part of my curriculum vitae.
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`7.
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`8.
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`I have attached a more detailed list of my qualifications as Exhibit A.
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`I am being compensated for my time working on this matter at my
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`standard hourly rate plus expenses. I have no personal or financial stake or interest
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`in the outcome of the present proceeding, and the compensation is not dependent on
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`the outcome of this IPR and in no way affects the substance of my statements in this
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`Declaration.
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`B. Materials Considered
`The analysis that I provide in this Declaration is based on my education
`9.
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`and experience in cryptography, software, and mathematics, as well as the
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`documents I have considered, including U.S. Patent No. 7,372,961 (the “’961
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`Patent”) [EX1001], which states on its face that it issued from an application filed
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`on December 26, 2001, in turn claiming priority back to a foreign application filed
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`on December 27, 2000. For purposes of this Declaration, I have assumed December
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`Declaration of Jonathan Katz, Ph.D. in Support of
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`U.S. Patent No. 7,372,961
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`27, 2000 as the effective filing date for the ’961 patent. I have cited to the following
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`
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`documents in my analysis below:
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`Description of Document
`Ex. No.
`1001 U.S. Patent No. 7,372,961 B2 to Scott A. Vanstone et al. (filed
`December 26, 2001, issued May 13, 2008)
`
`1003
`1004
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`File Wrapper of U.S. Patent No. 7,372,961
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`Federal Information Processing (FIPS) Publication 186, Digital
`Signature Standard (DSS) (May 19, 1994)
`1005 Excerpts from Alfred J. Menezes et al., Handbook of Applied
`Cryptography (1997)
`
`1006 USENET article by Greg Rose to sci.crypt and sci.math, Re: “Card-
`shuffling” algorithms (March 10, 1993)
`1007 Excerpts from Donald E. Knuth, The Art of Computer Programming,
`Vol. 2 (3d ed. 1998)
`
`1008 Excerpts from Bruce Schneier, Applied Cryptography (2d ed. 1996)
`1009 USENET article by Steve Brecher, Re: How can I generate random
`numbers? (October 12, 1996)
`
`1010 USENET article by Steve Brecher, Re: Help: Random numbers?
`(November 13, 1996)
`1011 USENET article by Trevor L. Jackson, III, Re: Random Generator
`(April 7, 2000)
`
`1012 Declaration of Aviel Rubin, Ph.D Regarding Claim Construction of the
`’961 Patent, filed in Blackberry Limited v. Facebook, Inc. et al., No.
`2:18-cv-01844-GW-KS (C.D. Cal.), filed February 28, 2019
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`Declaration of Jonathan Katz, Ph.D. in Support of
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`U.S. Patent No. 7,372,961
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`Description of Document
`Ex. No.
`1013 U.S. Patent No. 5,732,138 to Landon Curt Noll et al. (filed January 29,
`1996, issued March 24, 1998)
`1014 Excerpts from Jenny A. Fristrup, USENET: Netnews for Everyone
`(1994)
` 1015 Federal Information Processing (FIPS) Publication 186-2, Digital
`Signature Standard (DSS) (January 27, 2000)
`(https://web.archive.org/web/20000815231203/http:/csrc.nist.gov/fips/f
`ips186-2.pdf)
`1016 Brian W. Kernighan & Dennis M. Ritchie, The C Programming
`Language (2d ed. 1988)
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`1017
`
`John von Neumann, Various Techniques Used in Connection with
`Random Digits, Summary by G.E. Forsythe, National Bureau of
`Standards Applied Math Series, 12 (1951), pp 36-38, reprinted in von
`Neumann's Collected Works, 5 (1963), Pergamon Press pp 768-770.
`1018 Excerpts from Thammavarapu R. N. Rao, Error Coding for Arithmetic
`Processors (1974)
`1019 Excerpts from Nancy A. Floyd, Essentials of Data Processing (1987)
`1020 Amendments to Claims and Remarks, October 30, 2007, filed in U.S.
`Appl. Ser. No. 10/025,924 (issuing as the ’961 patent)
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`1028 Excerpts from Merriam Webster’s Collegiate Dictionary (10th ed.
`1998)
`1029 Excerpts from The American Heritage Dictionary of the English
`Language (4th ed. 2000)
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`Declaration of Jonathan Katz, Ph.D. in Support of
`Petition for Inter Partes Review of
`U.S. Patent No. 7,372,961
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`II.
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`PERSON HAVING ORDINARY SKILL IN THE ART
`I understand that an assessment of the claims of the ’961 patent should
`10.
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`be undertaken from the perspective of what would have been known or understood
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`by a person of ordinary skill in the art as of the earliest claimed priority date of the
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`patent claims which, as noted, I have assumed is December 27, 2000.
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`11. Counsel has advised me that to determine the appropriate level of one
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`of ordinary skill in the art, I may consider the following factors: (a) the types of
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`problems encountered by those working in the field and prior art solutions thereto;
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`(b) the sophistication of the technology in question, and the rapidity with which
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`innovations occur in the field; (c) the educational level of active workers in the field;
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`and (d) the educational level of the inventor.
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`12. The relevant technical field for the ’961 patent is key generation and
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`cryptography. (’961, 1:5-6.) Based on the factors listed above, in my opinion a
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`person of ordinary skill in the art as of December 2000 would have had a bachelor’s
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`degree or the equivalent in electrical engineering or computer science (or equivalent
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`degree) and at least two years of experience with computer-based cryptographic
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`techniques. A person could also have qualified as a person of ordinary skill in the
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`art with (1) less technical experience and more formal education (such as a master’s
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`degree or Ph.D.) or (2) more technical experience and less formal education.
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`Additionally, the requisite technical experience and formal education could have
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`been acquired concurrently. As part of such person’s basic education and/or
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`experience, a person of ordinary skill in the art would have obtained a working
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`knowledge of computer programming and would have learned how to implement
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`cryptographic algorithms in software, how to test those algorithms, and how to adapt
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`them for practical applications (e.g., for digital communications).
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`13. My opinions regarding the level of ordinary skill in the art are based
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`on, among other things, my more than 20 years of experience in cryptography and
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`related fields, my understanding of the basic qualifications that would be relevant to
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`an engineer or scientist tasked with investigating methods and systems in the
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`relevant area, and my familiarity with the backgrounds of colleagues, co-workers,
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`and employees, both past and present, who work in that area.
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`14. Although my qualifications and experience exceed those of the
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`hypothetical person having ordinary skill in the art defined above, my analysis and
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`opinions regarding the ’961 patent have been based on the perspective of a person
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`of ordinary skill in the art in December 2000.
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`15. My opinions herein regarding the understanding of a person of ordinary
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`skill in the art and my other opinions set forth herein would remain the same if the
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`person of ordinary skill in the art were determined to have somewhat more or less
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`education and/or experience than I have identified above.
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`16.
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`I have reviewed the “Declaration of Aviel Rubin, Ph.D Regarding
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`Claim Construction of the ’961 Patent,” which contains a declaration by an expert
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`retained by the Patent Owner. (Rubin Decl., Ex. 1012.) Dr. Rubin opines in that
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`declaration that a person of ordinary skill in the art for purposes of the ’961 patent
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`“would have had a bachelor of science degree in Computer Science or the equivalent,
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`and approximately three years of work or research experience in the field of
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`cryptography or an equivalent subject matter; or an advanced degree (masters or
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`doctorate) in Computer Science or the equivalent, and less work or research
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`experience (dependent in part on the level of degrees achieved) in the field of
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`cryptography or an equivalent subject matter.” (Rubin Decl., Ex. 1012, ¶34.)
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`17.
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`In my opinion, Dr. Rubin’s formulation of a person of ordinary skill in
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`the art is not materially different from mine. His formulation adds one additional
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`year of work or research experience (“approximately three years”), a difference that
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`I do not consider material to my analysis. My opinions as expressed in this
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`Declaration of Jonathan Katz, Ph.D. in Support of
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`Declaration would therefore remain the same regardless of whether I applied my
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`own formulation or the one provided by Dr. Rubin.
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`III. LEGAL PRINCIPLES USED IN THE ANALYSIS
`I am not a patent attorney, nor have I independently researched the law
`18.
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`on patent validity. Attorneys for the Petitioner explained certain legal principles to
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`me that I have relied upon in forming my opinions set forth in this report.
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`B.
`19.
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`Prior Art
`I understand that the law provides categories of information that
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`constitute prior art that may be used to anticipate or render obvious patent claims.
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`To be prior art to a particular patent, a reference must have been made, known, used,
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`published, or patented, or be the subject of a patent application by another, before
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`the priority date of the patent. I am informed that admissions by the patent
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`application regarding the presence of features in the prior art, whether contained in
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`the patent specification or made during the prosecution history, can be relied upon
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`to show anticipation or obviousness of a claim. I also understand that a person of
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`ordinary skill in the art is presumed to have knowledge of the relevant prior art.
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`Declaration of Jonathan Katz, Ph.D. in Support of
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`20.
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`I understand that the Petitioner has assumed, for purposes of my
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`Declaration, that the challenged claims of the ’961 patent are entitled to a December
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`27, 2000 priority date.
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`C. Claim Construction
`I understand that under the applicable legal principles, claim terms are
`21.
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`generally given their ordinary and customary meaning, which is the meaning that
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`the term would have to a person of ordinary skill in the art at the time of the
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`invention, i.e., as of the effective filing date of the patent application. I further
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`understand that a person of ordinary skill in the art is deemed to read claim terms
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`not only in the context of the particular claim in which a claim term appears, but in
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`the context of the entire patent, including the specification.
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`22.
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`I am informed by counsel that the patent specification, under the
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`applicable legal principles, is often the single best guide to the meaning of a claim
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`term, and is thus highly relevant to the interpretation of claim terms. And I
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`understand for terms that do not have a customary meaning within the art, the
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`specification usually supplies the best context of understanding their meaning.
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`23.
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`I am further informed by counsel that other claims of the patent in
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`question, both asserted and unasserted, can be valuable sources of information as to
`
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`the meaning of a claim term. Because the claim terms are normally used consistently
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`throughout the patent, the usage of a term in one claim can often illuminate the
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`meaning of the same term in other claims. Differences among claims can also be a
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`useful guide in understanding the meaning of particular claim terms.
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`24.
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`I understand that the prosecution history can further inform the meaning
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`of the claim language by demonstrating how the inventors understood the invention
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`and whether the inventors limited the invention in the course of prosecution, making
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`the claim scope narrower than it otherwise would be. Extrinsic evidence, such as
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`my expert testimony, may also be consulted in construing the claim terms.
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`25.
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`I have been informed by counsel that, in Inter Partes Review (IPR)
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`proceedings, a claim of a patent shall be construed using the same claim construction
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`standard that would be used to construe the claim in a civil action filed in a U.S.
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`district court (which I understand is called the “Phillips” claim construction
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`standard), including construing the claim in accordance with the ordinary and
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`customary meaning of such claim as understood by one of ordinary skill in the art
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`and the prosecution history pertaining to the patent.
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`26.
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`I have been instructed by counsel to apply the “Phillips” claim
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`construction standard for purposes of interpreting the claims in this proceeding, to
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`Declaration of Jonathan Katz, Ph.D. in Support of
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`the extent they require an explicit construction. The description of the legal
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`
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`principles set forth above provides my understanding of the “Phillips” standard as
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`provided to me by counsel.
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`D. Legal Standards for Obviousness
`I have been provided the instructions reproduced in part below, taken
`27.
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`from the Federal Circuit Bar Association Model Instructions regarding obviousness.
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`I follow these instructions in my analysis, with the caveat that I have been informed
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`that the Patent Office will find a patent claim invalid in inter partes review if it
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`concludes that it is more likely than not that the claim is invalid (i.e., a preponderance
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`of the evidence standard), which is a lower burden of proof than the “clear and
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`convincing” standard that is applied in United States district court (and described in
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`the instructions below):
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`4.3c OBVIOUSNESS
`
`Even though an invention may not have been identically
`disclosed or described before it was made by an inventor,
`in order to be patentable, the invention must also not have
`been obvious to a person of ordinary skill in the field of
`technology of the patent at the time the invention was
`made.
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`Declaration of Jonathan Katz, Ph.D. in Support of
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`U.S. Patent No. 7,372,961
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`[Alleged infringer] may establish that a patent claim is
`invalid by showing, by clear and convincing evidence, that
`the claimed invention would have been obvious to persons
`having ordinary skill in the art at the time the invention
`was made in the field of [insert the field of the invention].
`
`In determining whether a claimed invention is obvious,
`you must consider the level of ordinary skill in the field
`[of the invention] that someone would have had at the time
`the [invention was made] or [patent was filed], the scope
`and content of the prior art, and any differences between
`the prior art and the claimed invention.
`
`Keep in mind that the existence of each and every element
`of the claimed invention in the prior art does not
`necessarily prove obviousness. Most, if not all, inventions
`rely on building blocks of prior art. In considering whether
`a claimed invention is obvious, you may but are not
`required to find obviousness if you find that at the time of
`the claimed invention [or the patent’s filing date] there was
`a reason that would have prompted a person having
`ordinary skill in the field of [the invention] to combine the
`known elements in a way the claimed invention does,
`taking into account such factors as (1) whether the claimed
`invention was merely the predictable result of using prior
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`art elements according to their known function(s); (2)
`whether the claimed invention provides an obvious
`solution to a known problem in the relevant field; (3)
`whether the prior art teaches or suggests the desirability of
`combining elements claimed in the invention; (4) whether
`the prior art teaches away from combining elements in the
`claimed invention; (5) whether it would have been obvious
`to try the combinations of elements, such as when there is
`a design need or market pressure to solve a problem and
`there are a finite number of identified, predictable
`solutions; and (6) whether the change resulted more from
`design incentives or other market forces. To find it
`rendered the invention obvious, you must find that the
`prior art provided a reasonable expectation of success.
`Obvious
`to
`try
`is not sufficient
`in unpredictable
`technologies.
`
`In determining whether the claimed invention was
`obvious, consider each claim separately. Do not use
`hindsight, i.e., consider only what was known at the time
`of the invention [or the patent’s filing date].
`
`In making these assessments, you should take into account
`any objective evidence (sometimes called “secondary
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`considerations”) that may shed light on the obviousness or
`not of the claimed invention, such as:
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`
`
`(a) Whether the invention was commercially successful as
`a result of the merits of the claimed invention (rather than
`the result of design needs or market-pressure advertising
`or similar activities);
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`(b) Whether the invention satisfied a long-felt need;
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`(c) Whether others had tried and failed to make the
`invention;
`
`(d) Whether others invented the invention at roughly the
`same time;
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`(e) Whether others copied the invention;
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`(f) Whether there were changes or related technologies or
`market needs contemporaneous with the invention;
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`(g) Whether the invention achieved unexpected results;
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`(h) Whether others in the field praised the invention;
`
`(i) Whether persons having ordinary skill in the art of the
`invention expressed surprise or disbelief regarding the
`invention;
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`(j) Whether others sought or obtained rights to the patent
`from the patent holder; and
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`(k) Whether the inventor proceeded contrary to accepted
`wisdom in the field.
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`Federal Circuit Bar Association Model Jury Instructions §4.3c (2014).
`
`28.
`
`I am also informed that the United States Patent Office supplies its
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`examining corps with a Manual of Patent Examining Procedure that provides
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`exemplary rationales that may support a conclusion of obviousness, including:
`
`(a) Combining prior art elements according to known
`methods to yield predictable results;
`
`(b) Simple substitution of one known element for another
`to obtain predictable results;
`
`(c) Use of known technique to improve similar devices
`(methods, or products) in the same way;
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`(d) Applying a known technique to a known device
`(method, or product) ready for improvement to yield
`predictable results;
`
`(e) “Obvious to try” – choosing from a finite number of
`identified, predictable solutions, with a reasonable
`expectation of success;
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`(f) Known work in one field of endeavor may prompt
`variations of it for use in either the same field or a different
`one based on design incentives or other market forces if
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`the variations are predictable to one of ordinary skill in the
`art; or
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`(g) Some teaching, suggestion, or motivation in the prior
`art that would have led one of ordinary skill to modify the
`prior art reference or to combine prior art reference
`teachings to arrive at the claimed invention.
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`MPEP §2143. I apply these principles in my analysis below.
`
`IV. RELEVANT TECHNOLOGY BACKGROUND
`29. The ’961 patent, entitled “Method of Public Key Generation,” purports
`
`to disclose and claim a method for generation of a key for use in a cryptographic
`
`function. (’961, claim 1.) The ’961 patent focuses primarily on a single aspect of
`
`cryptographic functions – the generation of an appropriate cryptographic key. The
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`patent thus falls generally in the fields of key generation and cryptography.
`
`30. Cryptographic keys are often generated using a random number
`
`generator. (‘961, 1:41-43) Random number generation has long been practiced in
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`computer science and cryptography. In this section, I will provide a brief
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`background discussion of technologies pertinent to the ’961 patent prior to
`
`December 2000.
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`B. Random Number Generation
`31. Random number generation refers to the generation of a sequence of
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`numbers that are or appear random, i.e., lack any pattern. Random number
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`generation was a highly developed field within mathematics and computer science
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`prior to the filing of the ’961 patent.
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`32. A simple example of a random number generator is a die with six sides
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`showing 0, 1, 2, 3, 4, and 5.1 Each roll of the die generates a
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`random number in the range from 0 to 5. If the die is unweighted,
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`then each side is equally likely, i.e., each roll is equally likely to
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`result in any of the numbers 0, 1, 2, 3, 4, or 5. When each value in a given set of
`
`possible values is equally likely to occur, the output of the random number generator
`
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` 1
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` In my discussion, I will be using a die with faces showing 0-5 instead of 1-6
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`because the lowest unsigned value representable in a computer is zero, not one.
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`Thus, unsigned numbers on computers range from 0 to their maximum value.
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`is said to have a “uniform distribution” in that set. When I refer to this six-sided die
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`
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`further below, I always assume that it is an unweighted die for wh