Inverses mod 26 1 3 5 7 9 11 15 17 19 21 23 25 x22121 1 9 21153 19 7 23 11 5 17 25 Here is the complete multiplication table for Zinverses follows from the positions of the 1s in this table Con ID: 380269
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(The Integers mod 26) An element 1 (mod invertible 1 (mod implies 1 (mod 1 (mod Also, for any integer 1 (mod implies 1 (mod if and only if effective algorithm for computing the inverse when it exists. If is prime, then every nonzero element of Z is composite, there are fewer invertible elements. We define ) to be the number of elements of {1,2, ..., 1} that are relatively prime to , i.e., the number of invertible . If we can factor ers of distinct primes. Then )(1)(1). In the special case that ) simplifies to 1)( are: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25. Inverses mod 26 1 3 5 7 9 11 15 17 19 21 23 25 −1 1 9 21153 19 7 23 11 5 17 25 Here is the complete multiplication table for Zinverses follows from the positions of the 1s in this table. Conversely, if we can compute ), then we can factor , at least in the special case that is the product of two primes. This will turn out to be critical when we look at the RSA algorithm. 0 1 2 3 4 5 6 7 8 9101112131415161718 19 20 21222324250 0 0 0 0 0 0 0 0 0000000000 0 0 00000 0 1 2 3 4 5 6 7 8 9101112131415161718 19 20 2122232425 2 0 3 6 9 12 15 18 21 24 1471013161922252 5 8 1114172023 0 4 8 12 16 20 24 2 6 10141822048121620 24 2 610141822 5 0 6 12 18 24 4 10 16 22 281420061218244 10 16 22281420 0 7 14 21 2 9 16 23 4 11182561320181522 3 10 172451219 8 0 9 18 1 10 19 2 11 20 3122141322514236 15 24 71625817 0 10 20 4 14 24 8 18 2 12226160102041424 8 18 21222616 11 0 12 24 10 22 8 20 6 18 4162140122410228 20 6 18416214 0 13 0 13 0 13 0 13 0 130130130130130 13 0 13013013 14 0 15 4 19 8 23 12 1 16 5209241321762110 25 14 31872211 0 16 6 22 12 2 18 8 24 1442010016622122 18 8 241442010 17 0 18 10 2 20 12 4 22 14 6241680181022012 4 22 14624168 0 19 12 5 24 17 10 3 22 1581201362518114 23 16 9221147 20 0 21 16 11 6 1 22 17 12 7223181383241914 9 4 252015105 0 22 18 14 10 6 2 24 20 1612840221814106 2 24 20161284 23 0 24 22 20 18 16 14 12 10 864202422201816 14 12 108642 0 25 24 23 22 21 20 19 18 171615141312111098 7 6 54321 We can obtain all the invertible elements of Z as powers of some single invertible element. Powers of 7 (mod 26) 1 2 3 4 5 6 7 8 9 10 11 12 7 23 5 9 1125193 21 17 15 1 (We could have used 11, 19, or 15 in place of 7.) This property does not hold in Z is an odd prime and However, it is always true (for any invertible implies