A Method for Obtaining Digital Signatures and Public-Key Cryptosystems
Communications of the ACM, 1978 · paper · Computer science
Presents an encryption method in which publicly revealing the encryption key does not reveal the decryption key. This allows messages to be sent without couriers and signed so that signatures can be verified but not forged. Security rests partly on the difficulty of factoring.
Checked: PDF from people.csail.mit.edu, fetched 4 Oct 2026. Page numbers are those of that PDF.
0 findings
None that passed a second check.
11 minor (typos and the like; most checked by the judge only)
Wording Section VII.B recommends "using 100-digit (decimal) prime numbers p and q, so that n has 200 digits" and later says p and q "should differ in length by a few digits"; the two are not reconciled, and a product of two 100-digit numbers can have 199 or 200 digits.
- p. 9 Section VII.B, first paragraph
We recommend using 100-digit (decimal) prime numbers p and q, so that n has 200 digits.
- p. 9 Section VII.B, paragraph beginning "To gain additional protection"
should differ in length by a few digits
Judge's note: Checked the page 9 image: both sentences are in the same subsection as quoted. Read literally they cannot both hold. Page 12 softens the length to "We recommend that n be about 200 digits long", which suggests the figures are meant as approximate, but nothing in the paper states that 100 digits is nominal. Loose wording, not an error in the method.
Second check: another model agreed
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Wording "However, all such d′ differ by the least common multiple of (p − 1) and (q − 1)": equivalent exponents differ by multiples of the lcm, not all by the lcm itself.
- p. 13 Section IX.C, third paragraph
all such d′ differ by the least common
Judge's note: Checked the page 13 image: the sentence is printed as quoted. Values d′ equivalent to d are those congruent to d modulo lcm(p−1, q−1), so only neighbouring values differ by exactly the lcm. The sentence can be read as referring to the spacing between consecutive values, and the argument it supports (such d′ are sparse) is unaffected. Imprecise wording only.
Second check: another model agreed
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Typo Title-page footnote reads "Author's Address: Laboratory for Computer Science, ... E-mail addresses: rivest@theory.lcs.mit.edu": singular "Author's" for a three-author paper, and plural "addresses" followed by one address.
- p. 1 Footnote, last line
Author’s Address: Laboratory for Computer Science, Massachusetts Institute of Technology, Cam-
- p. 1 Footnote, last line
E-mail addresses: rivest@theory.lcs.mit.edu
Judge's note: Checked the page 1 image. The footnote is printed as quoted, the byline lists three authors, and only one e-mail address follows the plural label. Cosmetic number mismatch in this re-typeset version; either form could have been carried over from an earlier layout, so observation rather than correction.
Second check: none (judge only)
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Typo Equations (1) and (2) are printed with an unbalanced parenthesis: "D(E(M) = M." and "E(D(M) = M."; they should read D(E(M)) = M and E(D(M)) = M.
- p. 2 Section II, property (a), equation (1)
D(E(M) = M.
- p. 2 Section II, property (d), eq. (2)
E(D(M) = M.
Judge's note: Checked the page 2 image: both displayed equations open two parentheses and close one. The same expressions are balanced on page 7 ("D(E(M))", "E(D(M))") and page 3 ("D_A(E_A(M)) = M"). Not an extraction artefact. The meaning is unambiguous, so cosmetic.
Second check: none (judge only)
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Typo Section II opens "In a "public key cryptosystem" each user" without the hyphen, against "public-key cryptosystem" in the title, the heading directly above and the rest of the paper.
- p. 2 Section II, first sentence
In a “public key cryptosystem” each user
- p. 2 Section I, second paragraph
an implementation of a “public-key cryptosystem,”
Judge's note: Checked the page 2 image: the phrase is mid-line and unhyphenated, while Section I on the same page and the Section II heading print "public-key" / "Public-Key". Hyphenation inconsistency only.
Second check: none (judge only)
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Typo Misspelling: "every message is the cipertext for some other message" should read "ciphertext".
- p. 3 Section II, first paragraph on the page
every message is the cipertext for some other
- p. 3 Section II, paragraph on permutations
cipertext for some other
- p. 3 Section II, last paragraph, sentence on permutations
every message is the cipertext for some other message
Judge's note: Checked the page 3 image: "cipertext" is printed, and the same sentence continues with the correctly spelled "every ciphertext is itself a permissible message". Not a ligature or extraction artefact.
Second check: none (judge only)
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Typo Missing article: "is the Euler totient function giving number of positive integers less than n" should read "giving the number of positive integers".
- p. 7 Section VI, sentence after equation (3)
is the Euler totient function giving number of positive integers less than n
Judge's note: Checked the page 7 image: the sentence after equation (3) is printed without "the", within a single line, so it is not a line-break or extraction loss.
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Typo Section VII.D defines the starting values with different relation symbols: "where x0 ≡ φ(n), x1 = d"; x0 is simply set equal to φ(n) (the example has "x0 = 2668").
- p. 10 Section VII.D, first paragraph
x0 ≡ φ(n), x1 = d
Judge's note: Checked the page 10 image: "x_0 ≡ φ(n), x_1 = d" is printed, with ≡ and no modulus for x0 and = for x1 in the same clause; the worked example in Section VIII uses "x0 = 2668". The ≡ could be read as "defined as", so this is a notational inconsistency without effect on the algorithm.
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Typo Subsection heading IX.C ends with a full stop, "Determining d Without Factoring n or Computing φ(n).", unlike every other heading.
- p. 12 Section IX.C heading
Determining d Without Factoring n or Computing φ(n).
Judge's note: Checked the page 12 image: the heading ends in a period. The neighbouring headings "Factoring n" (page 11), "Computing φ(n) Without Factoring n" (page 12) and "Computing D in Some Other Way" (page 13) have none. Purely cosmetic.
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Typo Reference 1 has a comma directly before the parenthesised date: "IEEE Trans. Inform. Theory IT-22, (Nov. 1976), 644-654."; either the comma is stray or an issue number has dropped out.
- p. 14 Reference 1
Theory IT-22, (Nov. 1976), 644-654.
Judge's note: Checked the page 14 and 15 images. Reference 1 prints "IT-22, (Nov. 1976)". No other entry has a comma immediately before the date: references 2, 4 and 9 go from volume to date with no comma, and reference 10 has "Science 195, 4283 (March 1977)" with a number after the comma. The punctuation inconsistency is visible in the file. The reviewer's claim that the missing issue number is 6 is outside knowledge and is not relied on here.
Second check: none (judge only)
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Typo Reference 12 gives no volume number: "SIAM J. Comptng. (March 1977), 84-85.", while every other journal entry prints a volume before the date.
- p. 15 Reference 12
SIAM J. Comptng. (March 1977), 84-85.
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that the volume is 6, which cannot be verified from the files. What the files do show: on the page 15 image reference 12 goes straight from the journal name to "(March 1977)", whereas references 1, 2, 4, 9 and 10 all give a volume ("IT-22", "Computer 10", "Cryptologia 1", "76", "Science 195"). The omission is real and internally evident; only the authors or the cited journal can supply the value. Not an extraction loss.
Second check: none (judge only)