Example: Modulo 11: Since Z p is cyclic, there is a generator. Let g be a generator of Z p.
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1 Qudrtic Residues Defiitio: The umbers 0, 1,,, ( mod, re clled udrtic residues modulo Numbers which re ot udrtic residues modulo re clled udrtic o-residues modulo Exmle: Modulo 11: Itroductio to Number Theory i i mod There re six udrtic residues modulo 11: 0, 1, 3, 4, 5, d 9 There re five udrtic o-residues modulo 11:, 6, 7,, 10 c Eli Bihm - My 3, Itroductio to Number Theory (1 c Eli Bihm - My 3, Itroductio to Number Theory (1 Qudrtic Residues (cot Lemm: Let be rime Exctly hlf of the umbers i Z re udrtic residues With 0, exctly +1 umbers i Z re udrtic residues There re t most +1 udrtic residues, sice 0 1 ( 1 (mod ( (mod i ( i (mod i Thus, ll the elemets i Z s t most +1 udrtic residues There re t lest +1 udrtic residues, otherwise, for some i j / it holds tht i ( i j ( j, i cotrst to Lgrge theorem tht sttes tht the eutio x i 0 hs t most two solutios (mod Qudrtic Residues (cot Sice Z is cyclic, there is geertor Let g be geertor of Z 1 g is udrtic o-residue modulo, sice otherwise there is some b such tht b g (mod Clerly, b 1 (mod, d thus g b 1 (mod However, the order of g is 1 Cotrdictio QED g, g 4,, g ( mod re udrtic residues, d re distict, therefore, there re t lest udrtic residues 3 g, g 3, g 5,, g ( mod re udrtic o-residues, sice if y of them is udrtic residue, g is lso udrtic residue c Eli Bihm - My 3, Itroductio to Number Theory (1 c Eli Bihm - My 3, Itroductio to Number Theory (1
2 Euler s Criterio Theorem: Let be rime, d let Z The, is udrtic residue modulo iff 1 (mod ( If is udrtic residue, there is some b such tht b Thus, (b b 1 (mod (mod Euler s Criterio (cot ( If is udrtic o-residue: For y r there is uiue s such tht rs (mod, ie, s r, d there is o r r such tht s r Sice is udrtic o-residue, r s (mod Thus, the umbers 1,, 3,, 1 re divided ito distict irs (r 1, s 1, (r, s,, (r, s, such tht r i s i, d we get r 1 s 1 r s r s 1 ( 1 (mod by Wilso s theorem QED c Eli Bihm - My 3, Itroductio to Number Theory (1 c Eli Bihm - My 3, Itroductio to Number Theory (1 Qudrtic Residues Modulo Let d be lrge rimes d let (s i RSA Theorem: Let m Z If m is udrtic residue modulo, the m hs exctly four sure roots modulo i Z Assume α m (mod The gcd(m, 1 gcd(α, 1 gcd(α, 1 α Z d sice the m α (mod m α (mod m α (mod m hs two sure roots modulo (α mod d α mod d two sure roots modulo (α mod d α mod Qudrtic Residues Modulo (cot Look t the systems of eutios x ±α (mod x ±α (mod which rereset four systems (oe of ech ossible choice of ± Ech system hs uiue solutio modulo which stisfies d thus stisfies x m (mod x m (mod x m (mod All the four solutios re roots of m modulo These re ll the roots Otherwise there must be more th two roots either modulo or modulo QED c Eli Bihm - My 3, Itroductio to Number Theory (1 c Eli Bihm - My 3, Itroductio to Number Theory (1
3 Qudrtic Residues Modulo (cot Coclusio: Exctly urter of the umbers i Z re udrtic residues modulo Legedre s Symbol Defiitio: Let be rime such tht Legedre s symbol of over is +1, if is udrtic residue modulo ;, if is udrtic o-residue modulo By Euler: (mod c Eli Bihm - My 3, Itroductio to Number Theory (1 c Eli Bihm - My 3, Itroductio to Number Theory (1 Legedre s Symbol (cot Proerties of Legedre s symbol: 1 (mod ( ( 1 ( c 1 c ( 3 ( 1, if 4k + 1;, if 4k + 3 ( (mod ( 4k+1 ( 4k+3 ( k 1, if 4k + 1; ( k+1, if 4k ( ( (give without roof 5 ( ( ( b b Legedre s Symbol (cot Let g be geertor modulo The, i, g i (mod d j, b g j (mod is udrtic residue iff i is eve, b is udrtic residue iff j is eve, d b is udrtic residue iff i + j is eve Thus, by Euler: b ( i+j ( i ( j b (mod c Eli Bihm - My 3, Itroductio to Number Theory (1 c Eli Bihm - My 3, Itroductio to Number Theory (1
4 Legedre s Symbol (cot 6 The recirocity lw: if re both odd rimes the Jcobi s Symbol Jcobi s symbol is geerliztio of Legedre s symbol to comosite umbers (give without roof ( Defiitio: Let be odd, d let 1,,, k be the rime fctors of (ot ecessrily distict such tht 1 k Let be corime to Jcobi s symbol of over is 1 k I rticulr, for c Eli Bihm - My 3, Itroductio to Number Theory (1 c Eli Bihm - My 3, Itroductio to Number Theory (1 Remrks: Jcobi s Symbol (cot 1 Z is udrtic residue modulo iff the Legedre s symbols over ll the rime fctors re 1 Whe Jcobi s symbol is 1, is ot ecessrily udrtic residue 3 Whe Jcobi s symbol is -1, is ecessrily udrtic o-residue Proerties of Jcobi s symbol: Jcobi s Symbol (cot Let m d be itegers, d let d b be corime to m d Assume tht is odd d tht the fctoriztio of is 1 k 1 b (mod ( ( b ( 1 1 (1 is udrtic residue modulo y 3 ( ( 1 k (( (( (( k oeig retheses: ( i 1 i S S {1,,,k} c Eli Bihm - My 3, Itroductio to Number Theory (1 c Eli Bihm - My 3, Itroductio to Number Theory (1
5 Jcobi s Symbol (cot S {1,,,k} S i S ( i 1 + i {1,,,k} ( i [( 1 1( 1 ( k 1 + ] + ( ( ( k where ll the terms with S (i the brckets re multiles of four, d ll the i 1 re eve Thus, d 1 ( ( 1 k + + ( k 1 (mod, ( ( 1/ ( ( / ( ( k/ ( ( 1/+( /++( k / ( (/ c Eli Bihm - My 3, Itroductio to Number Theory (1 Jcobi s Symbol (cot 4 ( ( We sw tht ( (, thus: 1 It remis to show tht ( 1 k k k (mod 1 (1 + ( 1 1(1 + ( ( ( 1 + ( 1 1( 1 But ( 1 1 d ( 1, thus 64 ( 1 1( 1 Therefore, ( ( 1 (mod 16 c Eli Bihm - My 3, Itroductio to Number Theory (1 Ad, Jcobi s Symbol (cot 1 3 (1 + ( 1 1(1 + ( 1(1 + ( 3 1 (mod ( ( 1 + ( 3 1 (mod 16 etc, thus, 1 + ( ( ( k 1 (mod 16 Jcobi s Symbol (cot 5 The first multilictio roerty: ( ( ( m m (if is corime to m it is corime to m d to ; the rest is derived directly from the defiitio 6 The secod multilictio roerty: ( ( ( b b (if b is corime to, the both d b re corime to ; the rest is derived sice this roerty holds for Legedre s symbol k 1 (mod c Eli Bihm - My 3, Itroductio to Number Theory (1 c Eli Bihm - My 3, Itroductio to Number Theory (1
6 Jcobi s Symbol (cot 7 The recirocity lw: if m, re corime d odd the m ( m First ssume tht m is rime, thus, 1 m k By the recirocity lw of Legedre s symbol we kow tht Thus, i ( ( i i ( 1 ++ k 1 }{{ k } c Eli Bihm - My 3, Itroductio to Number Theory (1 ( We sw i roerty 3 tht, thus, 1 Jcobi s Symbol (cot ( 1 1 Now for y odd m: QED m ( 1 ( ( m + + ( k 1 l l m ( ( 1 ++ l (mod, c Eli Bihm - My 3, Itroductio to Number Theory (1 Alictio of Jcobi s Symbol: Jcobi s Symbol (cot Usig the roerties of Jcobi s symbol, it is esy to clculte Legedre s symbols i olyomil time Exmle: ( 37 7 (( ((+1 3 (( is rime, therefore ( c lso be comuted by: (mod Comlexity: Jcobi s Symbol (cot The oly reuired rithmetic oertios re modulr reductios d divisio by owers of two Clerly, divisio (rule 6 reduces the umertor by fctor of two A modulr reductio (usig rule 7 d the rule 1, reduces the umber by t lest two: s if > b the b + r b + r > r + r, thus r < /, ie, mod b < / Therefore, t most O(log modulr reductios/divisios re erformed, ech of which tkes O((log time This shows tht the comlexity is O((log 3, which is olyomil i log A more recise lysis of this lgorithm shows tht the comlexity c be reduced to O((log c Eli Bihm - My 3, Itroductio to Number Theory (1 c Eli Bihm - My 3, Itroductio to Number Theory (1
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