The　security　of　modern　block　ciphers　substantially　relies　on　the　cryptographic　properties　of　its　S-boxes　(vectorial　cryptographic　functions),　which　are　always　the　only　source　of　nonlinearity.　It　is　optimal　to　choose　differentially　4-uniform　permutations　as　S-boxes　of　block　ciphers　in　real　applications.　The　inverse　function　is　the　most　famous　differentially　4-uniform　permutation　with　many　desirable　cryptographic　properties.　The　vectorial　functions　of　affine　equivalent　to　the　inverse　function　over　F28　are　frequently　selected　as　the　S-boxes　of　many　important　block　ciphers,　such　as　AES,　Camellia,　CLEFIA　and　SMS4.　Now　the　research　on　the　S-boxes　of　affine　equivalent　to　the　inverse　function　focuses　the　counting　method　of　the　minimum　number　of　active　S-boxes　for　several　consecutive　rounds　of　block　ciphers.　Unlike　the　previous　research　works,　this　paper　investigates　the　counting　problem　of　affine　equivalent　to　the　inverse　function　over　Fpn.　If　the　exact　number　of　affine　equivalent　to　the　inverse　function　is　calculated,　the　designer　of　cryptographic　algorithm　knows　that　how　many　the　S-boxes　of　affine　equivalent　to　the　inverse　function　should　be　selected　in　real　applications.　The　inverse　function　over　finite　field　F2n　is　generalized　to　the　inverse　function　over　finite　field　Fpn,　where　p≥2　is　a　prime　number.　This　is　a　generalization　of　the　inverse　function.　Firstly,　the　product　＂*＂　of　(T1,R1)　and　(T2,R2)　is　defined　as　(T2,R2)*(T1,R1)(•)/(•)=(T2∘T1,R1∘R2),　where　(T1,R1),(T2,R2)∈Aff　n　-1(Fq)×Aff　n　-1(Fq),　Aff　n　-1(Fq)　is　the　n×n　invertible　affine　transformation　group　over　finite　field　Fq,　q=pm,　p≥2　is　a　prime　number,　m≥1　is　a　positive　integer,　and　＂∘＂　denotes　the　product　of　the　mapping.　This　paper　proves　that　Aff　n　-1(Fq)×Aff　n　-1(Fq)　is　a　group　and　the　pairs　of　invertible　affine　transformations　(V,W)∈Aff　n　-1(Fq)×Aff　n　-1(Fq)　satisfied　by　F=V∘F∘W　form　a　subgroup　of　the　group　Aff　n　-1(Fq)×Aff　n　-1(Fq)　with　respect　to　the　operation　＂*＂.　Secondly,　when　p≥3　and　n≥2,　or　p=2　and　n≥4,　for　the　inverse　function　F(x)=x-1=x　pn-2∈Fpn[x],　we　utilize　the　above　results　and　some　properties　of　finite　fields　to　prove　that　there　exists　the　pairs　of　invertible　affine　transformations　(ν,μ)∈Aff　n　-1(Fp)×Aff　n　-1(Fp)　such　that　F=ν∘F∘μ,　where　the　linearized　polynomials　of　invertible　affine　transformations　μ　and　ν　must　be　μ(x)=S　t　x　pt　and　ν(x)=S　t　pn-tx　pn-t,　0≠S　t∈Fpn,　t=0,1,...,n-1.　Then　the　pairs　number　of　invertible　affine　transformations　(ν,μ)　is　n(pn-1).　The　group　Aff　n　-1(Fp)×Aff　n　-1(Fp)　can　be　partitioned　into　equivalence　classes　by　using　the　pairs　of　invertible　affine　transformations　(ν,μ)　form　the　subgroup.　The　number　of　coset　representatives　of　the　group　Aff　n　-1(Fp)×Aff　n　-1(Fp)　relative　to　the　subgroup　is　equal　to　the　number　of　affine　equivalent　to　the　inverse　function.　In　this　case,　the　number　of　affine　equivalent　to　the　inverse　function　is　([pn(n+1)/2∏i=1　n(pi-1)]2)/(n(pn-1)).　Thirdly,　when　p=2　and　n=3,　for　the　inverse　function　F(x)=x-1=x　23-2∈F23[x],　the　pairs　number　of　invertible　affine　transformations　(ν,μ)∈Aff　3　-1(F2)×Aff　3　-1(F2)　satisfied　by　F=ν∘F∘μ　is　168,　which　is　calculated　by　the　computer.　The　group　Aff　3　-1(F2)×Aff　3　-1(F2)　can　be　partitioned　into　equivalence　classes　using　the　subgroup　that　is　formed　by　the　pairs　of　invertible　affine　transformations　(ν,μ).　The　number　of　coset　representatives　of　the　group　Aff　3　-1(F2)×Aff　3　-1(F2)　relative　to　the　subgroup　is　equal　to　the　number　of　affine　equivalent　to　the　inverse　function.　In　this　case,　the　number　of　affine　equivalent　to　the　inverse　function　is　10752.　Our　results　show　that　there　exists　269×255×[∏i=1　7(2i-1)]2cryptographic　functions　of　affine　equivalent　to　the　inverse　function　over　finite　field　F28　to　be　used　in　the　S-boxes　of　block　ciphers　in　real　applications.　©　2019,　Science　Press.　All　right　reserved.