JP4866389B2 - 閉ガロア体組合せ - Google Patents
閉ガロア体組合せ Download PDFInfo
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- JP4866389B2 JP4866389B2 JP2008139004A JP2008139004A JP4866389B2 JP 4866389 B2 JP4866389 B2 JP 4866389B2 JP 2008139004 A JP2008139004 A JP 2008139004A JP 2008139004 A JP2008139004 A JP 2008139004A JP 4866389 B2 JP4866389 B2 JP 4866389B2
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- number sequence
- galois field
- sequence
- galois
- field
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F7/00—Methods or arrangements for processing data by operating upon the order or content of the data handled
- G06F7/60—Methods or arrangements for performing computations using a digital non-denominational number representation, i.e. number representation without radix; Computing devices using combinations of denominational and non-denominational quantity representations, e.g. using difunction pulse trains, STEELE computers, phase computers
- G06F7/72—Methods or arrangements for performing computations using a digital non-denominational number representation, i.e. number representation without radix; Computing devices using combinations of denominational and non-denominational quantity representations, e.g. using difunction pulse trains, STEELE computers, phase computers using residue arithmetic
- G06F7/724—Finite field arithmetic
- G06F7/725—Finite field arithmetic over elliptic curves
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- Mathematical Analysis (AREA)
- Mathematical Optimization (AREA)
- Pure & Applied Mathematics (AREA)
- Computational Mathematics (AREA)
- Theoretical Computer Science (AREA)
- Computing Systems (AREA)
- Mathematical Physics (AREA)
- General Engineering & Computer Science (AREA)
- Error Detection And Correction (AREA)
- Detection And Correction Of Errors (AREA)
Description
Y1=X1・p1 (1)
Y2=X2・p1 (2)
ここで、Y1およびY2は、乗算演算202、204を介して得られる積である。
Z1=Y1+C1=(X1・p1)+C1 (3)
Z2=Y2+C2=(X2・p1)+C2 (4)
ここで、Z1およびZ2は、加算演算の結果である。Y1およびY2は、乗算演算202、204を介して得られる積である。C1およびC2は、有限体またはガロア体GF[p]から選択される非ゼロ定数である。X1およびX2は、RNS剰余値、p進記数法表現、または混合基数記数法表現として表されたデータ入力である。pは、ガロア体GF[p]の有限体サイズを定義する素数の定数である。
RV1=Z1・Z2(mod pk+1)=[(X1・p1)+C1]・[(X2・p1)+C2](mod pk+1) (5)
ここで、RV1は、ガロア体乗算210の結果である。pk+1は、拡大されたガロア体GF[pk+1]のサイズによって定義される値を有する係数である。ガロア体GF[pk+1]からの2つの要素が乗算される時に、その積は、そのガロア体GF[pk+1]に含まれるもう1つの要素であることを理解されたい。効果的に、乗算演算が結果Z1、Z2を使用して実行されている間に、有限の構造サイズが維持される。たとえば、pk+1が、16の値を有するように選択され、積Z1・Z2が、45の値をもたらす場合に、算術演算45 modulo 16の結果は、13と等しい。値13は、ガロア拡大体GF[16]の有限に多数の可逆要素{1,3,…,15}の一要素である。
RV2=−C1・C2(mod p) (6)
ここで、RV2は、非ゼロ定数C1、C2を表す値である。pは、ガロア体GF[p1]の有限体サイズを定義する素数値を有する係数である。pが1と等しく(p=1)、C1かけるC2が1と等しい(C1=C1=C1・C2=1)場合には、逆数が両方のデータ入力X1、X2について必ず存在するので、このガロア拡大体での乗算212を実行する必要はない。
Q=RV1−RV2 (7)
CO=Q・p−1 (8)
ここで、Qは、RV1からRV2を引くことから得られる結果である。p−1は、素数pの逆数である。
ここで、pは、ガロア体GF[pm]の有限体サイズを定義する素数定数である。X1およびX2は、p進記数法表現または混合基数記数法表現の値として表されたデータ入力である。C1およびC2は、ガロア体GF[p]から選択された非ゼロ値である。pk+1は、ガロア体GF[pk+1]の有限体サイズを定義する数値である。2と等しいp(p=2)について、拡大ガロア体GF[2k+1]でのすべての積が、奇数であることに留意されたい。したがって、数式(8)で定義される演算は、modulo 2k+1かけるRV1のk+1ビット結果をk個の最上位ビット(MSB)に切り詰めることによって、2進算術で単純に実行することができる。より一般的には、最下位ビット(LSB)が、RV1からRV2を引いた結果から除去される。
従来の暗号系(システム)が、通常は、ディジタルハードウェアで実施される非対称暗号化処理技法または非対称暗号化解除処理技法を必要とすることを理解されたい。成句「非対称処理」は、本明細書で使用される時に、関連する動作を実行するのに必要な時間、処理構造、または計算リソースにおける大きい差を指す。この非対称暗号化処理技法または非対称暗号化解除処理技法は、多くの暗号の機能性および応用可能性を制限する可能性がある。たとえば、RSAアルゴリズムは、データを暗号化するための公開鍵および秘密鍵の選択を伴う。この選択は、通常、所期のユーザによるメッセージの暗号化解除を、メッセンジャによる暗号化ステップよりはるかに簡単にする。ディジタルハードウェアは、通常、比較的多数のディジタル論理ゲートを含む。したがって、従来の暗号系は、比較的高い電力消費および比較的長いデータ処理時間を有する。対称処理を可能にする暗号化法は、ハードウェアの設計再利用ならびに機能および応用可能性におけるより高い柔軟性を可能にする。
502 メッセージ生成器
504、510 乱数生成器
506 CGF暗号化システム
508 CGF暗号化解除システム
Claims (9)
- 通信システム内で少なくとも2つの入力数列を組み合わせる方法であって、
演算手段により、ガロア体GF[p k ]のメンバとして表される複数の第1デジットを含む第1数列および前記ガロア体GF[p k ]のメンバとして表される複数の第2デジットを含む第2数列を受け取ることと、
前記演算手段により、前記複数の第1デジットを用いた第1アフィン変換と前記複数の第2デジットを用いた第2アフィン変換とを利用した写像を実行することと、
前記演算手段により、前記第1アフィン変換の第1の結果と前記第2アフィン変換の第2の結果とを組み合わせることによって出力数列を生成することと、を含み、
前記第1の結果と前記第2の結果は、ガロア拡大体GF[p k+1 ]でのガロア体乗算演算を利用して組み合わされ、pは、素数であり、kは、整数値であり、pk+1は、前記ガロア拡大体GF[pk+1]の有限体サイズを定義する、
方法。 - p k は、前記ガロア体GF[pk]の有限体サイズを定義する素数になるように選択される、請求項1に記載の方法。
- 前記演算手段により、前記受け取るステップの前に、前記第1数列および前記第2数列をp進記数法表現で表すことをさらに含む、請求項1に記載の方法。
- 前記演算手段により、前記受け取るステップの前に、前記第1数列および前記第2数列を混合基数記数法表現で表すことをさらに含む、請求項1に記載の方法。
- 前記第1数列および前記第2数列は、加重記数法で表された少なくとも2つのデジットからなるように選択される、請求項1に記載の方法。
- 前記生成するステップが、前記第1数列および前記第2数列のそれぞれからのデジットの別個の部分集合を使用する複数のガロア体乗算演算を選択的に実行することによって、前記第1数列および前記第2数列を組み合わせることをさらに含む、請求項1に記載の方法。
- 前記第1数列は、ペイロードデータまたは乱数列になるように選択される、請求項1に記載の方法。
- 前記第2数列は、擬似乱数列または擬似無秩序乱数列になるように選択される、請求項1に記載の方法。
- ガロア体GF[p k ]のメンバとして表される複数の第1デジットを含む第1数列および前記ガロア体GF[p k ]のメンバとして表される複数の第2デジットを含む第2数列を受け取り、
前記複数の第1デジットを用いた第1アフィン変換と前記複数の第2デジットを用いた第2アフィン変換とを利用した写像を実行し、
前記第1アフィン変換の第1の結果と前記第2アフィン変換の第2の結果とを組み合わせることによって出力数列を生成する、
ように構成された少なくともひとつの演算手段を備え、
前記第1の結果と前記第2の結果は、ガロア拡大体GF[p k+1 ]でのガロア体乗算演算を利用して組み合わされ、pは、素数であり、kは、整数値であり、pk+1は、前記ガロア拡大体GF[pk+1]の有限体サイズを定義する、
閉ガロア体暗号(CGFC)システム。
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US11/756,086 US7995757B2 (en) | 2007-05-31 | 2007-05-31 | Closed galois field combination |
US11/756,086 | 2007-05-31 |
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JP2008299330A JP2008299330A (ja) | 2008-12-11 |
JP4866389B2 true JP4866389B2 (ja) | 2012-02-01 |
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US (1) | US7995757B2 (ja) |
EP (1) | EP2003546B1 (ja) |
JP (1) | JP4866389B2 (ja) |
CA (1) | CA2632857C (ja) |
DE (1) | DE602008006296D1 (ja) |
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EP2003546A2 (en) | 2008-12-17 |
EP2003546A3 (en) | 2010-04-07 |
US7995757B2 (en) | 2011-08-09 |
CA2632857A1 (en) | 2008-11-30 |
US20090044080A1 (en) | 2009-02-12 |
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JP2008299330A (ja) | 2008-12-11 |
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