Conversion of oxalic acid to malonic acid The reagents I have used here are $\ce{LiAlH_4}$ which reduces the carboxylic groups into alcohol.Used $\ce{HIO_4}$ which breaks up the molecule into methanal. Then  I applied Grignard reagent which consisted ethyl for the alkyl part.Then I tried to oxide the terminal carbon into $\ce{COOH}$  but could not find a suitable reagent.

What is a better approach and a method to solve this question.
The Arndt-Eistert procedure suggested above seems very applicable to this problem. To use it we need ethyl oxalyl chloride. This can be prepared in 2 steps from oxalic acid by first esterification to form diethyl oxalate  procedure here then reaction with PCl5 to form ethyl oxalyl chloride procedure here. This can undergo the Arndt-Eistert mechanism here with excess diazomethane then either irradiation or treatment with silver oxide to give monoethyl malonate. Simple base hydrolysis then gives the required.