**Topic :**Cryptography Mini-Tutorial

**Author :**William Moore

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one may simply try out all the possible values of e less than sqrt(1081) (the so-called Sieve of Eratosthenes). That is, one requires a value of e such that there is a whole-number d such that d * e = 1081. For really big primes, there are more efficient ways to obtain d,e. In the present example, e=23, d=47, and

d * e = 23 * 47 = 1081 = (1) mod 1080 = (1) mod ((p-1)*(q-1)).

That is, '1 mod 1080' denotes that the remainder of 1081 divided by 1080 is 1. Since d * e = 1 can be written equivalently as d = (e-1), we can assert that:

d = (23-1) mod(30*36) = 47.

Let the plaintext message be the number t=13. Then we may encrypt the plaintext message, t, according to the formula in Table 1 as:

c = (t3) mod n = (1323) mod 1147 = 520.

That is:

t1 = 13, and (t1) mod 1147 = 13.

t2 = 169, and (t2) mod 1147 = 169.

t3 = 2197, so that (t3) mod 1147 = 1050.

The calculation may be continued on a small calculator by noting that:

[t4 mod 1147] equals [t * [(t3) mod 1147] mod 1147]

[t5 mod 1147] equals [t * [(t4) mod 1147] mod 1147]

....

c = [1323 mod 1147] equals [t * [(t22) mod 1147] mod 1147]

c = (1323) mod 1147 = 520.

(In this manner, the whole-numbers in the intermediate calculations never exceed 11472 in size.)

When the receiver obtains the ciphertext, c=520, it may be decrypted by the formula (Table 1):

t = (cd) mod n = (52047) mod 1147 = 13.

Note that not even the sender can decrypt the initial message, t, after it has been encrypted into ciphertext, c.

What is the legal status of asymmetric cryptography?

At this time, every significant public-key encryption algorithm is patented, and several legal challenges to these patents to date have all been decided in favor of the patent holders. The U.S. patent for the RSA public-key encryption algorithm expires on September 20, 2000.

What U. S. legislation applies to confidentiality in medical records?

U. S. Code of Federal Regulations, 45 CFR Subtitle A (10-1-95 Edition), part 46.101 (b) (4).

U. S. Senate 1416, H.R. 2690. The Genetic Privacy and Non-Discrimination Act of 1995. November 15, 1995 (Senate); November 29, 1995 (H.R.).

U. S. Senate 1360. The Medical Records Confidentiality Act of 1995. October 24, 1995.

U. S. Senate 1898. The Genetic Confidentiality and Non-Discrimination Act of 1996. June 24, 1996.

U. S. Senate Bill 422. The Genetic Confidentiality and Non-Discrimination Act of 1997. March 11, 1997

What are the intermediate calculations for the RSA encryption example?

(131) mod 1147 = 13.

(132) mod 1147 = 169

(133) mod 1147 = 1050.

(134) mod 1147 = 1033.

(138) mod 1147 = 379.

(1316) mod 1147 = 266.

(1320) mod 1147 = 645.

(1323) mod 1147 = 520.

(1324) mod 1147 = 1025.

What are the intermediate calculations for the RSA decryption example?

(5201) mod 1147 = 520.

(5202) mod 1147 = 855.

(5204) mod 1147 = 386.

(5208) mod 1147 = 1033.

(52016) mod 1147 = 379.

(52032) mod 1147 = 266.

(52040) mod 1147 = 645.

(52044) mod 1147 = 71.

(52047) mod 1147 = 13.

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