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Download Cryptography and Data Security by Dorothy Elizabeth Robling Denning PDF

By Dorothy Elizabeth Robling Denning

Publication by means of Denning, Dorothy Elizabeth Robling

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We have presented the algorithm in the preceding form for clarity. The algorithm is easily extended to find solutions to general equations a x mod n = b when gcd(a, n) = 1. First, the solution x0 to " a x mod n = 1" is found. 21 Euclid's algorithm for computing the greatest common divisor. 22 Euclid's algorithm extended to compute inverses. =x+n end is t h e u n i q u e s o l u t i o n to " a x m o d n = b " in t h e r a n g e [1, n formulas: 1]. 7) If gcd(a, n) v~ 1, t h e e q u a t i o n " a x m o d n = b" will e i t h e r have no s o l u t i o n or will have m o r e t h a n one s o l u t i o n in t h e r a n g e [1, n - 1] as d e s c r i b e d by t h e following t h e o r e m .

To see why, consider a ciphertext C = EA(M), where E4 is a public enciphering transformation and M is a plaintext message in a set of n possible messages M1 . . , M,. Even if it is computationally infeasible to determine the private deciphering transformation DA, it may be possible to determine M by computing Ci = EA(Mi) for i = 1, 2, . . until C = Ci, whence M = M i. This type of attack would work, for example, if M is known to be an integer salary less than $100,000 because there would be at most 100,000 messages to try.

This checks, because 3 • 5 mod 7 = 1. m If 4~(n) is known, the inverse x of a (mod n) can be computed using Eq. 20. Alternatively, x can be computed using an extension of Euclid's algorithm for computing the greatest common divisor [Knut69]. With this approach, it is not necessary to know 4~(n). 21. 22. The algorithm computes gcd(a, n) by computing gi+l = gi-1 rood gi for i = 1, 2 , . . until g; = 0, where go = n, g, = a, and "'gi = UiFl "~" via" is the loop invariant. W h e n gi = O, gi-, = god(a, n).

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