Two algorithms for the formal check of election results authenticity are described. The fact of frauds in a number of regions of Russia at 2016 legislative elections is proved by means of the last digits frequency analysis for integer electoral characteristics. Consideration of the dependence between turnout and fraction of void ballots confirms large scale of frauds and enables one to estimate the total amount of ballots added. Definitions of the reconstruction of adulterated results problem are clarified. New method for the solution of this problem in a weak (sociological) definition is proposed. It bases on the assumption of presence for electoral districts of linear dependence between the common turnout and the turnout for opposition. Deviations from such dependences are not only an evidence of electoral frauds but also enable one to calculate for it the amounts of both added and transmitted ballots. The results of two last legislative elections are reconstructed and the effect of administrative support is estimated with the help of the method proposed.