Undefined $E_y(1,r_{i,j,1})$ notation in cryptography paper, suspect ElGamal-like
An ElGamal ciphertext encrypting a message $m$ (as you say) is a tuple $(g^k,m\cdot y^k)$ when working in a group $G$ generated by $g$ with public key $y$.

Some people make the "randomizer" $k$ explicit by writing $E_y(m,k)=(g^k,my^k)$ to denote that they can control and know this input and that this input is not freely chosen by the encryption algorithm (typically if one uses ElGamal encryption as a commitment and requires proofs of knowledge about the plaintext - I guess this is what the guys do). So, people often write $E_y(m)$ if they do not care what the value of $k$ is and write $E_y(m,k)$ if they want to know what the value of $k$ is. Then, $E_y(1,k)$ is just an encryption of the identity in $G$ giving ciphertext $(g^k,y^k)$.