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R = [r 5 ,r4,r 3,r 2 ,r 1 ,r 0 ] = [2030,2730,4290,6006,10010,15015]. Let x= [0,1,1,0,1,0]. I) = 17030. Conversely, givenS 17030, one may decipher by ( 13) : = 17030 = 0 mod 13 __. :z: 5 = 0. 17030 = 2 mod 11 __. :z: 4 = 1. 17030 = 6 mod 7 ... :z: 3 = 1. 17030 = Omod5 -+:z: 2 = 0. 17030 = 2 mod 3 ... :z: 1 = 1. 17030 = 0 mod 2 -+ :z: 0 = 0. Remark. In the preceding discussion, we described two types of reducibility yielding two types of easy knapsacks, viz. the superincreasing knapsacks and the divisiblility knapsacks.

The process is then restarted from any remaining input point (x 1 ,x 0 ). It is not difficult, from the above discussion, to design a total substitution network on the alphabet L = L: 1 x L: 0 : clearly, g 0 and g 1 should be replaced by Universal Logical moduLes (ULM's) able to realize arbitrary Boolean functions defined on E 1. Similarly. the network S should be able to realize an arbitrary substitution on L: 1 . This is illustrated in figure 9a which represents a total substitution network on L: = ~0.

This is for example the case of the DES. If Zi = DES~e(m;), then (zi. mi), ... , (z;, mi), ... g k. Another important remark should be made about one-way function used in practice: when we say for example that the exponential z = y~ is a one-way function of x, we mean that the best available algorithm for computing the logarithm x = log11 z over a finite field is significantly more complex than computing the exponential. This may be restated in terms of computational complexity: in the present state of the art, the computation of the logarithm has a much higher upper complexity bound than the computation of the exponential.

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