By E. F. Brickell, J. H. Moore, M. R. Purtill (auth.), Andrew M. Odlyzko (eds.)

This publication is the lawsuits of CRYPTO 86, one in a sequence of annual meetings dedicated to cryptologic learn. they've got all been held on the college of California at Santa Barbara. the 1st convention during this sequence, CRYPTO eighty one, geared up by means of A. Gersho, didn't have a proper court cases. The complaints of the subsequent 4 meetings during this sequence were released as: Advances in Cryptology: court cases of Crypto eighty two, D. Chaum, R. L. Rivest, and A. T. Sherman, eds., Plenum, 1983. Advances in Cryptology: lawsuits of Crypto eighty three, D. Chaum, ed., Plenum, 1984. Advances in Cryptology: lawsuits of CRYPTO eighty four, G. R. Blakley and D. Chaum, eds., Lecture Notes in desktop technological know-how #196, Springer, 1985. Advances in Cryptology - CRYPTO '85 complaints, H. C. Williams, ed., Lecture Notes in machine technology #218, Springer, 1986. A parallel sequence of meetings is held each year in Europe. the 1st of those had its lawsuits released as Cryptography: court cases, Burg Feuerstein 1982, T. Beth, ed., Lecture Notes in desktop technological know-how #149, Springer, 1983.

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Cn . . Zn i = 1, ... , k j = 1, ... , n (t-error correcting algebraic code) Then, for j= 1, ... + mkGk{ + . . + mkGk$ + Z I + ~ 2 To solve k unknowm (rn l , m 2 , . . , mk ), k operations are required because k equations are sufficient to solve the equations if the code is maximal distance separable (MDS) code. Otherwise, at most k’ = n-d+l equations are required to solve for k-unknowns [Pless ,821. 39 Since t is smaller than n-k, it is possible that the cryptanalyst could select k equations containing no errom from n equations.

Lemma 1 : The number of P’s that transform ATE’S into non-ATE’S is at least (n - - I)! if 2 < t <_ 2 where n is the length of 2’ ATE and t is the length of adjacent errors. Outline of Proof: Let vector V be an ATE of length n. , bt}, from V where b = these positions as an ordered set, B = (1, t, 2t, 121. , bt, 2). We reorder This map- ping is illustrated in the figure below. v- = +-I B = V’ = ---- + +---- - - 1 2 t 2t 3t (1, t, 2t, . . , bt, I- - ---+____I bt n (ATE) b = 2) - - ---+____ B_--+ ______ - - - _-_I (non-ATE) We consider a permutation map of vector V to V’ with B embedded in V’.

Let M1 and M 2 are two plaintext differing in one p i t i o n only, that is, M, - M, = (00 . . 010 . . 0 ) ithposition for i = 1, . . ,k then, - + - 22) (Es. 1) is the ith row vector of G’. The Hamming weight of ( 2 , - 2,) is at most 2t. Since t is much smaller than n, the majority of the bits of the vector C, - C2 correspond directly with 9;’ . C, C, = where gi’ gj’ (21 We can let C1 - C2 represent one estimate of g j ’ several times a number of estimates of g;’ mates of gj’ . By repeating the step can be obtained.