The present invention relates to a cosmetic composition comprising: i) one or more dicarbonyl compounds corresponding to formula (I) below, and/or derivatives thereof and/or hydrates thereof and/or salts thereof: in which formula (I), R is as defined in the description, in a specific amount; ii) one or more acids different from the compound(s) i) as defined previously; and iii) one or more specific alkalinizing agents. The subject of the invention is also a process for straightening keratin fibres using ingredients i), ii) and iii) with a step of straightening by means of a straightening iron at a temperature of at least 150° C., preferably ranging from 150 to 250° C.